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611 | MECHANICS | In a viscous fluid, a slender rod experiences two types of resistance. In this problem, we refer to the resistance acting in the normal direction to the rod as resistance, and the resistance acting along the rod as friction. The resistance per unit length is approximately $-\mu v_n$, and the friction per unit length is $-\mu v_t /2$, where $v_n$ and $v_t$ are the velocity components perpendicular and parallel to the rod axis, respectively, and $\mu$ is a constant related to the fluid viscosity and the rod's diameter. Consider a slender rod of length $l$. The instantaneous center of the rod is point P (which may not necessarily be on the rod), and the radial and tangential distances from point P to the center of the rod are $a$ and $b$, respectively. The rod rotates in the plane around the instantaneous center P with an angular velocity $\omega$, and point P is instantaneously at rest. The resistance and friction will exert forces and moments on the rod. The model described above can be used to calculate the resistance and its moment acting on the rod. Derive the moment of fluid resistance (including normal resistance and tangential friction) about the instantaneous center P. Express the result using the symbols defined in this problem.
| ||
785 | MECHANICS | A thin cylinder with a radius of \( R \) and a height of \( H \) is situated on the horizontal surface, with its axis parallel to the vertical direction. Inside, there is a solid small ball with a radius of \( r \) and a mass of \( m \) that is in **pure rolling** motion while remaining in contact with the cylinder wall. At \( t = 0 \), the center of the ball is at a height of \( H/2 \) from the horizontal ground. The velocity of the ball in the vertical direction is upward with a magnitude of \( v_0 \), and the vertical acceleration is 0 (note that the **angular velocity perpendicular to the cylinder wall direction is non-zero**, but it is not required to be given here). The horizontal velocity is \( v_1 \). During the subsequent motion, the ball **does not collide with the ground**, **does not leave the cylinder**, and the cylinder itself **does not slip or topple over**. The gravitational acceleration is given as \( g \). If the thin cylinder is fixed on the ground, find the **minimum value** of the coefficient of friction between the ball and the cylinder. | ||
733 | MODERN | When a photon is scattered by an electron, if the initial electron has sufficient kinetic energy such that energy is transferred from the electron to the photon during the scattering process, this scattering is called inverse Compton scattering. Inverse Compton scattering occurs when a low-energy photon and a high-energy electron collide head-on. The rest mass of an electron is known to be $m_{e}$, and the speed of light in vacuum is $c$. If an electron with energy $E_{e}$ and a photon with energy $E_{y}$ collide head-on, If the incident photon energy is $2.00\mathrm{eV}$ and the electron energy is $1.00\times10^{9}~\mathrm{eV}$, find the energy of the photon after scattering. It is known that $m_{e}=0.511\times10^{6}~\mathrm{eV}/c^{2}$. When necessary in calculations, approximations such as $\sqrt{1-x} \approx 1 - \frac{1}{2}x$ can be used, and relativistic effects should be considered. | ||
298 | MODERN | People discovered that Mercury's orbit is constantly precessing, rather than following the fixed orbit predicted by Newton's laws. This problem was discovered in 1859, and for 50 years people were unable to provide a satisfactory solution. At the beginning of the 20th century, Einstein's special theory of relativity emerged, offering new insights for explaining Mercury's precession. M represents the mass of the sun (considered stationary), $\mathrm{m}$ represents the rest mass of Mercury, E represents the total energy, L represents Mercury's angular momentum, G is the gravitational constant, and c is the speed of light. Under special relativity, inertial mass and gravitational mass are not equivalent; gravitational mass equals rest mass.
Please derive the precession angle $\Delta\theta_{t1}$ for one period based on special relativity. | ||
558 | ELECTRICITY | In 1917, Stewa and Tolman discovered that for a closed coil wound around a cylinder, when the cylinder rotates about its central axis with a constant angular acceleration, a current will flow through the coil.
There is now a coil with many turns, each turn having a radius of \( r \), and each turn is wound with a thin metal wire having a resistance \( R \). The coil is uniformly wound around a very long hollow glass cylinder, and the inside of the cylinder is a vacuum. Each turn of the coil is fixed on the cylinder with adhesive, with \( n \) turns per unit length, and the plane of the coil is perpendicular to the central axis of the cylinder. Starting from a certain moment, the cylinder and coil rotate about the central axis of the cylinder with a constant angular acceleration \( \beta \). Given the electron mass \( m \) and elementary charge \( e \). We assume that only electrons can move freely, while positive ions cannot.
Find the magnetic induction \( B \) at the central axis of the cylinder after a sufficiently long time. | ||
146 | MECHANICS | Three perfectly elastic balls are initially arranged in a straight line at rest. Striking the first ball with mass $m_1$ gives it a velocity of $v_{1}$ along the line, causing it to collide with the second ball of mass $m_2$, which in turn collides with the third ball of mass $m_3$. To maximize the velocity of the third ball after the collisions, given $m_{2} \neq m_{3}$, what should the mass of the second ball be? | ||
283 | OPTICS | When a strong laser beam shines on a semi-transparent plate, due to uneven heating of the material, the transmitted light can self-focus to a point behind the plate. This effect is known as the thermal lens effect. In materials where the refractive index increases with temperature, this effect is characterized by a positive thermo-optic coefficient $\gamma = \mathrm{d}n/\mathrm{d}T$.
A semi-transparent disk, with radius $a$, thickness $b$, and light absorption rate $A$, is composed of material with thermal conductivity $k$ and thermo-optic coefficient $\gamma$. The outer edge of the disk is in thermal contact with a circular metal frame (not shown in the figure), which is maintained at a constant temperature $T_{h}$. A parallel laser beam with radius $\sigma$ and power $P_{L}$ vertically irradiates the center of the disk. The intensity distribution of the beam across its cross-section is uniform.
Under steady-state conditions, neglecting thermal convection and thermal radiation, the laser beam will focus at a point. Find the distance $f$ from this point to the disk. | ||
690 | MODERN | In the rotational spectroscopy analysis of diatomic molecules, we usually consider the two constituent atoms of the molecule as point masses, assuming their masses are respectively ${m}_{1}, m_{2}$, and that the connection between the two atoms is rigid, assumed to be at a distance $r$ apart. The quantization condition for angular momentum is given by: $$ \mathrm{L}^{2}=l(l+1)\hbar^{2} $$ where $L$ is the angular momentum, and $l$ is any positive integer. Knowing that transitions occur only between two adjacent energy levels, try to find the frequency of the photon absorbed during the transition from the $l$-th level to a higher energy level in the rotational spectrum of this molecule. | ||
509 | MECHANICS | A cylindrical rod with a radius of $R$ is fixed to a vertical wall (its base is fixed to the wall, and the axis of the cylinder is perpendicular to the vertical wall). The length of the cylinder is $L$ ($R \ll L$). The cylinder has a mass $m$, Young's modulus $E$, and the effect of gravity is neglected. A torque $M$ is applied at the right end of the cylinder, causing the right end to experience a very small displacement $d$ ($d \ll L$) downward.
After removing $M$, assume that during vibration, the shape of the cylinder satisfies $y=\frac{\theta x^2}{2L}$, where $x$ represents the distance from a point on the axis of the undeformed cylinder to the wall, $y$ represents the vertical displacement of the point at position $x$ from its equilibrium position, and $\theta$ is the central angle corresponding to the approximate circular arc formed by the deformation of the cylinder. Determine the period of small vibrations of the cylinder. | ||
435 | MECHANICS | When studying the motion of viscous fluids, when the flow velocity is not large, the motion of viscous fluids can be divided into a layer-by-layer motion. There is relative sliding between these layers, with viscous resistance acting upon each other. This type of motion is called laminar flow. Due to the viscous resistance between the layers, the distribution of flow velocity and the magnitude of the flow rate will be affected. As shown in the figure, there is a horizontally placed cylindrical tube with uniform diameter. The radius of the cylindrical tube is $r$, and the length of the tube is $L$. The tube undergoes uniformly accelerated linear motion to the right, with an acceleration of $a$. In the tube, there is an incompressible viscous fluid undergoing laminar flow. The fluid density is $\rho$. The left end of the fluid block is subjected to a pressure $P_{1}$, and the right end to a pressure $P_{2}$ ($P_{1}>P_{2}$). The velocity is greatest at the center of the tube, and as the distance from the central axis increases, the flow velocity gradually decreases. At the point adjacent to the tube wall, the fluid velocity is zero due to adherence to the wall surface.
Supplementary Knowledge:
Newton's law of viscosity states that the magnitude of the viscous force between two adjacent layers within a fluid is proportional to the contact area $A$ between the two layers and is also proportional to the velocity gradient at the contact surface of the two layers $\frac{\mathrm{d}\nu}{\mathrm{d}r}$, namely
$$
f=\eta A{\frac{\mathrm{d}\nu}{\mathrm{d}r}}
$$
Using the horizontal cylindrical tube as the reference frame, calculate the flow velocity $\nu(y)$ at a distance $y$ from the axis for the incompressible viscous fluid. | ||
309 | MECHANICS | Precession refers to the phenomenon where the axis of rotation of a spinning rigid body rotates around a certain center due to the influence of external forces. A round coin rolls purely on a horizontal surface, and under certain conditions, the homogeneous coin will move with an inclined state in a looping motion, maintaining an angle $\theta$ between its plane and the horizontal plane, and precess uniformly around a vertical axis. The center of the coin is O.Assume the coin precesses uniformly around a vertical axis passing through point $S$ (with $S$ at the same height as point $O$), the gravitational acceleration is $g$, the coin's mass is $m$, and the radius is $r$. The radius of the coin's looping motion is $R$. Answer the following question.
Find the expression for the precession angular velocity of the coin, $\omega_{pr}$. | ||
474 | MECHANICS | Please solve the following physics problem. Use \boxed{} to enclose the final answer: From the ceiling, a uniform rod is hinged vertically downward. The rod has a length of $l$ and a mass of $m$. A second identical rod is hinged to the end of the first rod. The entire system is subject to vertical downward gravity with an acceleration of $g$. The double-rod system is evidently stable, so in the case of small oscillations, it can perform simple harmonic motion near the equilibrium position with a certain natural frequency. In the eigenmode, the angles $\theta_{1}$ and $\theta_{2}$ between the rods and the vertical direction have a common angular frequency. Find the larger angular frequency of such a system. | ||
236 | MECHANICS | As one of the most common means of transportation, bicycles are renowned for their excellent stability and portability and have long been widely popular among the public. Even today, with transportation methods and technology rapidly changing, bicycles are still considered the most labor-saving and superior non-motorized means of transportation and are highly regarded for their outstanding environmental protection benefits. Early research on bicycles focused on gear coupling, pedal acceleration, and similar processes, but the study of bicycle control stability has been challenging. Why does a bicycle with two wheels remain stable while moving and not topple over? For over a hundred years, this question has intrigued many famous mechanicians, physicists, and mathematicians, resulting in more than a hundred renowned papers published in various languages such as English, German, French, Russian, and Italian. In 1897, the French Academy of Sciences even offered a reward for this research. Today, the study of the control and riding stability of bicycles with different structures and parameters continues to be a hot topic in the engineering field.
(In this problem, the rider does not hold the handlebar and imparts no force on it)
A ridden bicycle can ensure its own stability and not fall over. Based on our own experience, the faster the speed, the better the stability. However, is it possible to attach some devices to the bicycle to make it stable and upright even when stationary? Below, we provide a schematic of a bicycle structure with an attached momentum wheel, where a brushless motor is fixed to the bicycle body, connected to a momentum wheel, and can be driven by the motor to achieve any desired angular velocity.
Next, we simplify this system. Since we are now studying the upright stability of the bicycle in a non-moving state, we can consider the front and rear wheels, body, and motor as a rigid body, abstracted as a homogeneous rigid rod that can only rotate in the plane around the contact point with the ground. The rod has a length $L$, mass $M$, and the momentum wheel connected to the rod's end can be driven by the motor to achieve any angular velocity. The momentum wheel is viewed as a disk with mass $m$ and radius $r$. (Ignore lateral slip at point $O$)
If the bicycle body obtains a clockwise rotation angle $\theta$ , the angular velocity that the previously stationary momentum wheel subsequently needs to acquire is
In fact, when the bicycle tilt angle changes, the angular velocity and angular acceleration of the momentum wheel should also change correspondingly. We assume that sensors installed on the bicycle body can record the tilt angle $\theta$ and feedback to the motor, allowing the momentum wheel to acquire an angular acceleration of size $\dot{\omega}=K\theta$. To bring the bicycle back to a balanced position, solve for the dependency of $\omega$ on time $t$. ( $\theta<<1$, ignore any resistance, at $t=0$, $\theta=\theta_{0},\dot{\theta}=0$, and the angular velocity of the momentum wheel is 0) | ||
557 | ELECTRICITY | Two conductor spheres with equal radius \( R \) come into contact to form an isolated conductor. The distance between the centers of the two spheres is exactly \( 2R \). Find their capacitance \( C \). | ||
722 | MODERN | The two identical long straight wires 1 and 2, which are placed vertically and fixed parallel, are spaced apart by a distance of a ($0 << a$, much smaller than the length of the wires). Both wires carry a steady current of the same direction and magnitude, with the current flowing upwards. The positive ions in the wires are stationary, and the charge of the positive ions per unit length of the wire is $\lambda$; the conduction electrons forming the current move uniformly downwards along the wire at a speed of $\boldsymbol{\mathrm{\Delta}}v_{0}$, with the charge of the conduction electrons per unit length of the wire being $-\lambda$. It is known that an infinitely long uniformly charged straight wire with a charge per unit length of $\eta$ produces an electric field of magnitude $E=k_{e}\frac{2\eta}{r}$ at a distance $r$ from it, where $k_{e}$ is a constant; and when an infinitely long straight wire carries a steady current $I$, the magnetic field strength generated at a distance $r$ from the wire is $B=k_{m}\frac{2I}{r}$, where $k_{m}$ is a constant. Use the length contraction formula from special relativity to find the ratio of the constants $k_{e}$ and $k_{m}$. Hint: Ignore gravity; the charge of positive ions and electrons is independent of the choice of inertial reference frame; the speed of light in a vacuum is $c$. | ||
569 | ELECTRICITY | On an infinitely large superconducting plate with a friction coefficient of $\mu$ placed horizontally on the ground, there is a uniformly charged insulating spherical shell with a total charge of $Q$, radius $R$, and mass $M$. Fixed smooth panels are positioned on both sides of the spherical shell, ensuring that its center of mass does not move in the horizontal direction. It is known that during subsequent motion, the sphere will not leave the superconducting plate. Initially, the sphere is given an angular velocity of $\omega_{0}$.
Determine the relationship between the angular displacement and angular velocity of the sphere following this setup. Express the angular displacement $\theta$ in terms of the instantaneous angular velocity $\omega$, initial angular velocity $\omega_{0}$, and physical quantities such as $M$, $Q$, $R$, $\mu$, and $g$.
(Hint: Consider the electric and magnetic images created by the superconducting plate.) | ||
779 | MECHANICS | In this problem, we are given a zero-natural-length elastic rope with a constant elastic coefficient \( k \), a natural length of 0, and uniformly distributed mass (it can be considered that the natural length is almost 0). The total mass is \( m \), and it is suspended in Earth's gravitational field. We are to determine its shape under the following conditions.
Shape of a Zero-Natural-Length Elastic Rope in a Gravitational Field:
If the rope is close to the Earth's surface and its stretched length is much smaller than the Earth's size, the gravitational acceleration is \( g \), and the suspension points are at the same height with a spacing of \( D \), determine the shape of the rope.
(For this question, it is recommended to set up a Cartesian coordinate system where the rope passes through the points (0, 0) and (D, 0).)
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418 | ELECTRICITY | An infinitely long cylindrical conductor shell with a radius of $R$ carries a positive charge of $+\lambda$ per unit length. It is placed in front of an infinitely large grounded conducting plate, with the distance from the axis to the conductor plate being $d$. Given the vacuum permittivity constant as $\varepsilon_{0}$, we establish a rectangular coordinate system with the foot of the perpendicular $O$ as the origin. The direction opposite to the normal vector of the infinite plane points inward to the plane, defined as the z-direction. The direction parallel to the cross-section of the cylinder on the infinite plane is the y-direction, and the direction perpendicular to the cross-section is the x-direction. In the $xOy$ plane, find the distribution of the charge density $\sigma(y)$ on the conducting plate. | ||
247 | MECHANICS | A uniform rectangular thin plate oscillates around one of its edges, where this edge makes an angle $\theta$ with the vertical direction, and the length of the other side is $b$. Given the gravitational acceleration $g$, find the period of small oscillations near the equilibrium position. | ||
299 | MECHANICS | Two infinitely long smooth rails intersect at point $O$, and there are two blocks with mass $m$ each on the rails. When passing through point $O$, the blocks do not collide with the other rail and pass directly through. One block on a rail is initially at a distance $b$ from point $O$ and remains stationary. The block on the other rail is initially far enough from point $O$ and slides towards $O$ with an initial velocity magnitude $v_{0}=\lambda\sqrt{G m/b}$. Consider the gravitational force between the two blocks, but ignore the gravitational force from the rails and gravity. Question: What is the final velocity magnitude of the block that initially moves with velocity $v_{0}$ after a sufficiently long time? | ||
539 | MECHANICS | **Transverse Waves on a String**
In this problem, we describe waves using the negative phase convention. In this convention, the most general form of a traveling wave is expressed as
$$
\psi(x,t)=A e^{i(\pm k x-\omega t)}
$$
where $\omega$ is the angular frequency, $k$ is the magnitude of the wave vector, the $"+"$ sign represents a wave traveling to the right, and the $"-"$ sign represents a wave traveling to the left. $A$ is the complex amplitude (its magnitude corresponds to the amplitude, while its phase angle contains the information about the phase). Next, we will consider transverse waves on a string, whose wave equation is
$$
\frac{\partial^{2}\psi}{\partial x^{2}}-\frac{1}{u^{2}}\frac{\partial^{2}\psi}{\partial t^{2}}=0
$$
where $u$ is the wave velocity. When $k$ and $\omega$ in the traveling wave expression satisfy $\omega/k=u$, it is a solution to the wave equation above. Due to the linearity of the wave equation, the superposition of solutions to the traveling wave is also a solution to this equation.
Both ends A and B of a string are fixed to supports. We establish a coordinate axis with the midpoint (point O) as the origin and the positive direction of the $x$-axis pointing to the right. Thus, the coordinates of endpoints A and B are $±l/2$, where $l$ is the length of the string. At a point with a coordinate of $a$ on the string, a driver C with an angular frequency of $\omega$ is installed, causing all mass elements on the string to oscillate with an angular frequency of $\omega$.
Let the amplitude reflection coefficient at the string's connection point with the supports be $\boldsymbol{r}$, which is defined as the ratio of the complex amplitude of the reflected wave to that of the incident wave at the reflection point. Assume that the incident wave between C and B is
$$
A e^{i(k x-\omega t)}
$$
We can introduce an inhomogeneous term into the wave equation to model the effect of the driver $C$:
$$
\frac{\partial^{2}\psi}{\partial x^{2}}-\frac{1}{u^{2}}\frac{\partial^{2}\psi}{\partial t^{2}}=F\delta(x-a)e^{-i\omega t}
$$
Here, $F$ characterizes the amplitude of the driving force. The function $\delta(x-a)$ is a delta function with the following properties: it exhibits a very narrow, tall pulse centered near ${\boldsymbol{x}}={\boldsymbol{a}}$, with the area under the pulse curve equal to 1. Using this property, we can write the relationship satisfied by the wave functions $\psi_{1}$ and $\psi_{2}$ to the left and right of C. This relationship is also known as the continuity condition.
Finally, express $A$ in terms of $F$, $k$, $l$, and $a$. | ||
652 | MECHANICS | A bottle with a height of $H$ is filled with water. One side of the bottle is uniformly distributed with small holes, with a number of holes per unit length being $n$. The area of each hole is $s$ (but its dimension is much smaller than $H$). Each hole sprays water outward, and it is assumed that the direction of water sprayed from the holes is along the horizontal direction. The decrease in water surface height is not considered during the spraying process. Find the amount of water hitting the ground per unit time and per unit length at a distance $x$ from the bottle.
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436 | ADVANCED | There is a mechanism that simulates a helicopter rotor, but it has only one blade. One end of the blade is hinged at a fixed rod $O$ and can rotate around the fixed point $O$ in the plane $β$ formed by the rod and the blade's center of mass. The center of mass is denoted as point $C$. The mass of this rigid body is $m$, and the distance $OC$ is $r$. A three-dimensional Cartesian coordinate system is established at the center of mass $C$. The $OC$ direction is designated as the $y$-axis, the axis perpendicular to the $y$-axis and within plane $β$ is the $z$-axis, and the axial direction is denoted as the $x$-axis. The principal moments of inertia of the rigid body around the three axes are $J_x, J_y, J_z$. Assuming the fixed axis rotates with a constant angular velocity $\Omega$, determine the angle between $OC$ and the vertical rod at stable equilibrium (by default, take the acute angle). | ||
160 | MECHANICS | A smooth horizontal plane rotates uniformly with angular velocity $\omega$ around a fixed axis passing through point $O$, and the axis is perpendicular to this plane. There is a particle $P$ with mass $m$ on the plane which is attracted by point $A$. The distance between $A$ and $O$ is $c$, and the gravitational force is proportional to the distance from the particle to point $A$, with the proportional constant being $4\omega^{2}m$. At $t=0$, the particle is launched from $\left({\frac{8}{3}}c,0\right)$ with velocity ${\frac{4}{3}}c\omega$ perpendicular to the $OA$ axis. Determine the trajectory equation $r(\theta)$ of the particle relative to the fixed coordinate system. | ||
732 | THERMODYNAMICS | Consider a thermodynamic cycle composed of a monoatomic ideal gas, which undergoes the following four processes on the \(p\)–\(V\) diagram:
- **Process A→B:** Isobaric expansion at pressure \(rp\) (where \(r>1\)), with temperature rising from \(T_C\) to \(T_H\);
- **Process B→C:** Isothermal expansion at temperature \(T_H\);
- **Process C→D:** Isobaric compression at pressure \(p\);
- **Process D→A:** Isothermal compression at temperature \(T_C\). It is known that when the gas absorbs heat, it is always in contact with the heat reservoir at temperature \(T_H\), and when it releases heat, it is always in contact with the heat reservoir at temperature \(T_C\). In the above thermodynamic cycle, it is known that the efficiency of an ideal Carnot engine working between a high-temperature reservoir \(T_H\) and a low-temperature reservoir \(T_C\) is \[ e_C=1-\frac{T_C}{T_H}. \] Please write the expression for the ratio \(\frac{e}{e_C}\), where \(e\) is the efficiency of the cycle engine, using \(T_C, T_H, r\) to express your answer. | ||
486 | OPTICS | A perfect plano-convex thin lens is placed in a vacuum, with a refractive index of \(n\). A broad parallel light beam is incident perpendicularly on the flat side of the lens, and the transmitted light rays are strictly focused at point \(F\), where the focal length is \(f\). A parallel light beam with a cross-sectional diameter of \(D\) is incident perpendicularly on one side of the lens, with its axis aligned with the principal axis of the lens. Assume the incident light intensity is \(I\), and neglect the reflection and absorption of light by the lens. Calculate the force exerted by the light field on the lens. | ||
366 | ELECTRICITY | In a uniform magnetic field $\vec{B}=B_0 \hat{z}$, an ideal conductive spherical shell with a radius of $a$ rotates around the z-axis at an angular velocity $\omega$. Find the electromotive force between its north pole and equator. | ||
276 | ELECTRICITY | The strict formulation of Green's reciprocity theorem is as follows: When the charges on $n$ conductors are $Q_{1}, Q_{2}, \ldots, Q_{n}$, the potentials on each conductor are $\varphi_{1}, \varphi_{2}, \ldots, \varphi_{n}$; if the charges on the conductors are reset to $q_{1}, q_{2}, \ldots, q_{n}$, the corresponding potentials change to $\varphi_{1}^{\prime}, \varphi_{2}^{\prime}, \ldots, \varphi_{n}^{\prime}$, then the following equation holds:
$$
\sum_{i=1}^{n} Q_{i} \varphi_{i}^{\prime} = \sum_{i=1}^{n} q_{i} \varphi_{i}
$$
For an ellipsoidal conductor rotating around the minor axis, take the minor axis as the polar axis. At a distance from the center $r, r > a$, at an azimuthal angle $\theta$, a charge with an amount of $q$ is placed, and the conductor is grounded. Try to solve for the induced charge $q_{a}^{\prime}$ on the conductor in the stable state (to simplify the answer, introduce $\begin{array}{r} {\tan \alpha_{0} = \frac{2r c \cos \theta}{r^{2} - c^{2}}} \end{array}$ to replace $r^{2} - c^{2}$). | ||
296 | MECHANICS | A smooth steel wire is bent into a semicircle with a radius of $R$ and placed vertically. Two particles are placed on the wire, with weights $P$ and $Q$ respectively. The two particles are connected by a lightweight, inextensible rope of length $2L$. Find the angle $\theta$ that the rope makes with the horizontal plane at equilibrium. | ||
531 | THERMODYNAMICS | # 3. Asymmetric Adiabatic Process
Inside a horizontal cylinder, there is an enclosed piston. The piston is connected to a handle, allowing us to control the volume of the container. Initially, saturated water vapor at temperature $T_{0}$ is contained within the cylinder, with no liquid water present.
Assume that the water vapor can be regarded as an ideal gas made of polyatomic molecules. At temperature $\scriptstyle{T_{0}}$, the heat of vaporization of water is $L$, and for the purposes of this problem, it is assumed that $L$ is independent of temperature. The universal gas constant is $R$. The molar mass of water is $\mu$.
It is well known that when there is a slight change in temperature around $\scriptstyle{T_{0}}$, the relative change in saturated vapor pressure and the relative change in temperature satisfy the following relationship:
$$
\varepsilon_{p}=\frac{\Delta p}{p_{0}}=\alpha\varepsilon_{T}=\alpha\frac{\Delta T}{T_{0}}
$$
Next, assume the container and piston are adiabatic. When the volume of the container slowly increases so that its relative change reaches $\beta$, find the change in temperature $\Delta T_{2}$. | ||
286 | MECHANICS | Two concentric homogeneous steel wire loops are held by a soap film in the middle. The mass, radius, and surface tension coefficient of the soap film are $m_{1}$, $m_{2}$, $r_{1}$, $r_{2}$, and $\sigma$. There is no gravity in space. The problem is:
If the ambient temperature changes, one physical fact is that the surface tension coefficient of the soap film will not remain constant. Assume that the two loops are initially given very small velocities along the axial direction, $v_1$ and $v_2$. During the subsequent motion of the system, the surface tension coefficient of the soap film very gradually increases from $\sigma$ to $2\sigma$ and then remains unchanged. Calculate the maximum kinetic energy of the system during the final stable motion. Retain terms up to the second order of $(v_2 - v_1)$. | ||
408 | MECHANICS | Sometimes, a golf ball approaches the edge of the hole, rolls along the wall of the hole while moving downward, and then bounces out of the hole. We examine this absurd motion.
Consider a sufficiently rough ball rolling without slipping on the inner wall of an infinitely deep cylindrical hole. The mass of the ball is $m$, its moment of inertia is $I$, its radius is $a$, and the radius of the hole is $b > a$. The point of contact $P$ between the ball and the wall has an angular coordinate $\theta$ in cylindrical coordinates. The unit normal vector at point $P$ on the wall is denoted by $\hat{n}$, and gravity $g$ is directed along the $-\hat{z}$ axis, which is also the axis of the hole. The friction between the ball and the wall is sufficient to ensure rolling without slipping, meaning the velocity at the point of contact is $0$, but it is not sufficient to eliminate the rotation perpendicular to the wall, i.e., $w_{n} = \vec{\omega} \cdot \hat{n}$ does not vanish.
Assuming the initial precession angular velocity of the ball is $\omega_{0}$, i.e., $\dot{\theta}(0) = \omega_{0}$, determine the period $T$ of the ball's simple harmonic motion in the vertical direction. | ||
382 | MECHANICS | Uranus is a spherically symmetric planet with uniform mass distribution, density $\rho_{1}$, and radius $R$. In the cosmic space between distances $R$ to $2R$ from the planet's center, cosmic dust with a uniform density of $\rho_{2}$ is distributed.
Before leaving Uranus, a probe adjusts its orbit and performs uniform circular motion at a distance of $\frac{3}{2}R$ from the center. Ignoring all drag caused by collisions with the dust, if the probe experiences a small radial impulse, determine the precession angular velocity of its orbit. | ||
715 | ELECTRICITY | In the $xy$ plane, there is a fixed charged ring with radius $R$, on which the charge $Q > 0$ is uniformly distributed. At the center of the ring, a point charge with mass $m$ and charge $q > 0$ is placed. The point charge is constrained to move along a smooth tube that passes through the center of the ring and forms an angle $\theta$ with the $z$-axis. Now, move the point charge along the tube to a distance $A$ away from the center ($A \ll R$). Neglect the effects of gravity, and let the electrostatic constant be $k$. Given that $\theta > \arctan{\sqrt{2}}$, find the oscillation period of the point charge (expressed in terms of $Q$, $q$, $m$, $k$, $R$, and $\theta$). | ||
692 | ELECTRICITY | There is an ideal electret dielectric with intrinsic polarization strength $P_0$, which can also be polarized by an internal electric field. The equation of state is $\vec P=\varepsilon _0\chi\vec E+\vec P_0$. Place this dielectric sphere in a vacuum, with an external uniform electric field $E_0$ (which can be oriented differently from $P_0$). Find the distribution of bound surface charge. Use $\cos\theta = \cos<\vec r, \vec E_0>, \cos \phi = \cos<\vec r, \vec P_0>$ and the symbols provided in the problem to express the answer. | ||
322 | ELECTRICITY | This problem does not consider relativistic effects.
An infinitely long leaky dielectric cylinder, with a radius of $R$ and an absolute permittivity of $\varepsilon$, initially stationary and overall uncharged. The carriers have a very small mass $m \to 0$, a charge of $q$, and a mobility of $\mu$ (i.e., the drift velocity of the carriers generated under a uniform electric field $E$ is $v=\mu E$). The number denstiy of the carriers is $n$, Within the cylinder, a cylindrical coordinate system $(z,r,\theta)$ is established, with the origin at the center of the circular top surface of the cylinder, the z-direction pointing axially outwards from the cylinder, and r pointing radially outwards. Throughout this process, the volume charge density $\rho$, and the current densities $j_{r}, j_{\theta}$ within the cylinder will vary with time $t$ and are functions of the spatial coordinates $r, \theta$. Due to translational symmetry in the z-axis direction, the current $j_{z}$ in the z-direction is always zero, and all non-zero physical quantities are independent of the coordinate $z$.
First, keep the cylinder stationary and slowly apply a uniform magnetic field $B$ along the cylinder's axis. Next, maintain the uniform magnetic field unchanged and suddenly apply a resistive torque to bring the cylinder to a complete stop, and marking this moment as $t=0$. Note that during this process, while the charges do not have time to transfer, the carriers (still due to the mass $m\rightarrow0$) will change their drift velocity under the new electromagnetic forces, thereby generating a new current distribution $j_{r}(t=0)$, $j_{\theta}(t=0)$. Find the current vector $\vec{j}$ within the cylinder at $t=0$. | ||
191 | MECHANICS | In 1975, in the graduate admissions exam of the Department of Physics at the University of Wisconsin, USA, there was a seemingly simple hard ball collision problem: Two solid spheres made of the same material, the radius of the lower one is $2a$, and the radius of the upper one is $a$. They fall from a height of $h$ above the ground (measured from the center of the larger sphere). Assuming the centers of both spheres are always on a vertical line and all collisions are elastic, what is the maximum height that the center of the upper sphere can reach? (Hint: assume the larger sphere first collides with the ground and bounces up before colliding with the smaller sphere.)
We do not know what the answer from the examiner was. However, when this question was included in a certain book, its solution assumed that after the larger sphere falls and hits the ground, it rebounds and collides with the smaller sphere that is still in a downward trajectory. The maximum height that the smaller sphere can reach upon rebounding and rising is the maximum height achievable by the smaller sphere. Request to solve for the maximum height $h_{1}$ of the center of the smaller sphere during its first rebound and ascent. | ||
749 | ELECTRICITY | In space, there are four infinitely large charged planes located at $x = a$, $x = -a$, $y = a$, and $y = -a$. The intersection of each plane with the $x$-axis or the $y$-axis serves as the center point of the plane. The charge distribution on all four planes is identical: at a distance $ \rho$ from the plane's center point, the surface charge density is $ \sigma = \frac{ \sigma_{0}}{(1+ \alpha^{2} \rho^{2})^{ \frac{3}{2}}}$, where $ \alpha > 0$ and $ \sigma_{0} > 0$. At the origin of coordinates, a particle with charge $q$ (where $q < 0$) and mass $m$ is placed. If the particle is constrained to move along the $z$-axis, determine the frequency of small oscillations $ \omega$ of the particle near the origin. | ||
290 | THERMODYNAMICS | The Boltzmann constant is $k$.
Imagine a general potential energy existing in space, which consists of positive power laws $r$ for three directions:
$$
E_{p}=\alpha(|x|^{r}+|y|^{r}+|z|^{r})
$$
A classical ideal gas with temperature $T$ exists in this potential energy. Find the average potential energy $\langle E_{p}\rangle$ of each molecule. | ||
187 | MECHANICS | Under the gravitational influence of the moon and sun, the phenomenon of the sea's periodic rise and fall twice a day is called tides. To simplify the model, we will ignore Earth's rotation, and this question only discusses the tidal phenomenon in the Earth-Moon system. Let the mass of the moon be $m$, the gravitational acceleration on the Earth's surface be $g$, the Earth's radius be $R$, the distance from the Earth-Moon barycenter be $r_{\mathrm{m}}$, and the gravitational constant be $G$. Assume that the Earth can be modeled as a rigid sphere with uniformly distributed mass, covered by a layer of seawater. The height of this seawater layer is negligible compared to the mass of Earth and the seawater, in comparison to the mass of the rigid Earth. The seawater covering the Earth's surface will become ellipsoidal. We establish a coordinate system $O x y z$ with the Earth's center of mass as the origin, where the $z$-axis is along the Earth-Moon line. Let $\theta$ be the angle with the $z$-axis. Provide an expression for the maximum amplitude of the tidal rise and fall.
| ||
315 | ELECTRICITY | In a vacuum, two parallel conductor plates with an area of $S$ are separated by a distance of $l_{0}$ and face each other to form a capacitor. The two plates are connected by a spring with an unstretched length of $l_{0}$, a cross-sectional area of $A$, and $N$ turns. The spring has a stiffness coefficient $k$. The spring is conductive and can also be treated as an inductive component. Neglect edge effects $\langle\sqrt{S}\gg l_{0}\gg\sqrt{A}\rangle$ and radiation effects, and assume that the stress and strain in the spring are always uniformly distributed.
If the two plates are initially charged at $t=0$ with $\pm Q_{0}$, and the current through the spring is zero, then if an external force $F(t)$ (positive outward) is applied to keep the plates stationary in their initial position, the charge on the plates will vary periodically over time.
Now, the external force is very slowly reduced to zero. During this process, the plate separation can be considered fixed over short periods of time but will change over a long period from the initial length $l_{0}$ to a final length $l$. Find the value of $l$. | ||
624 | THERMODYNAMICS | Given that the ambient temperature is $T$, the atmospheric pressure is $P_0$, and the initial pressure inside a tire is $P_i$ (which is greater than the atmospheric pressure), with the temperature remaining at the ambient temperature. The maximum pressure in the tire right after inflation is $P_{max}$, with the temperature at that moment being $T_{max}$ (both $P_{max}$ and $T_{max}$ are unknown). After cooling slowly back to the ambient temperature, the measured pressure is $P_f$. Assumptions: 1. Air is an ideal gas with a heat capacity ratio $\gamma$. 2. The volume of the tire remains constant. 3. The inflation process is adiabatic. A small hand pump is used to inflate the tire. The pump's volume is much smaller than that of the tire, and the initial pressure of the gas inside is the atmospheric pressure. Due to the small volume of the pump, many strokes are required for inflation. We assume the inflation process is as follows: the air in the pump is first adiabatically compressed, increasing the pressure from the atmospheric pressure $P_0$ to the current pressure in the tire, and then the air is introduced into the tire at constant pressure. Replace discrete summation with integration to derive the expression for the tire pressure $P_{max}$ right after inflation using the parameters given in this problem: $P_0, P_i, P_f, T, \gamma$. | ||
190 | MECHANICS | The Blue Star people have always fantasized about one day being able to abandon the inefficient transportation method of rockets and directly establish a "sky ladder" from the ground to the heavens. Some call it a "space elevator," while others refer to it as an "orbital lift." However, they do not realize that due to the limitations imposed by physical laws, building a space elevator on the Blue Star is unrealistic. In contrast, several years later, on Mars, which is farther from the sun, space elevators can "stand tall."
As early as the Book of Genesis in the Bible, humans hoped to jointly construct a towering tower that reaches from the ground to the sky—the Tower of Babel—to spread their fame. The person who first proposed the concept of a space elevator should be Russia's "father of rockets"—Tsiolkovsky. His envisioned space elevator extends from the ground to the geostationary orbit, tens of thousands of kilometers high. As the elevator rises, the gravity inside gradually decreases. When cargo reaches the endpoint with the elevator, its speed is enough to maintain synchronous orbital movement around the Earth. Therefore, the station at the geostationary orbit is in a completely weightless state.
In 1979, Arthur C. Clarke's science fiction novel, "The Fountains of Paradise," first brought this advanced concept of the space elevator into the public's view. Since then, space elevators have widely existed in various science fiction works. Whether it's the space elevator made with advanced nanomaterials in "The Three-Body Problem" or the three orbital lifts during the three-way division among Union, AEU, and Human Reform League in "Mobile Suit Gundam 00," their basic concepts and principles are the same.
Nowadays, we can occasionally hear some related news reports, whether it's the Japanese company "Obayashi Corporation" planning to complete the first space elevator by 2050, or a Canadian company proposing a new space elevator plan. Regardless of whether these are just commercial companies engaging in hype, such news does indeed cause more people to wonder—are we really close to the era of abandoning rockets and embracing space elevators?
Is this really the case? How far away are we from the technology required to build space elevators? We will conduct a detailed analysis in this question (the space elevators mentioned below are all built at the equator, and any bending is not considered).
In fact, whether a material is destroyed or not depends not only on the magnitude of the force but also on the area over which the force acts. This is easy to understand; under the same amount of tension, thinner pipes are more easily pulled apart. Therefore, we need to use "stress (force divided by the area of action)" to measure whether a space elevator will be damaged. If the cross-sectional area of the space elevator is uniform, then obviously the place where the force is greatest, in other words, the geostationary orbit area, is the most prone to breaking.
Do we have any way to make the stress inside the entire space elevator equal? Of course we do! We know that the tension increases with height, so the cross-sectional area of an equal strength (internally equal tensile stress) space elevator should also increase with height. Therefore, the optimized space elevator design changes from a uniformly thick "catenary" to a funnel-shaped "catenary" with a thick top and thin bottom.
In this question, we use the following model: A mass at the end of the space elevator is considered infinitely large to ensure that only tensile forces act inside the elevator, so there is no force acting at the contact point with the ground. The elevator is thick at the top and thin at the bottom, with each cross-section having an equal tensile stress of σ. The material density is ρ, and the material elongation is not considered. The cross-sectional area at the ground level is \(A_{0}\). Solve the following problem:
Find the expression for the cross-sectional area \(A_{x}\) at a distance \(X\) from the center of the Earth, expressed in terms of \(g\), \(R\), \(\omega_{0}\) (Earth's rotational angular velocity), \(\sigma\), \(\rho\), and \(A_{0}\). | ||
597 | ELECTRICITY | According to the principles of electrochemistry, the spontaneity of redox reactions in aqueous solutions can be determined using the standard electrode potential $E_0$. The standard electrode potential $E_0$ is related to the change in Gibbs free energy of the reaction. Under conditions of constant temperature and pressure, the decrease in the Gibbs free energy of the system is equal to the maximum work $W_{max}$ done by the system. During the course of a battery reaction, the decrease in Gibbs free energy is equal to the maximum electrical work done by the battery. At standard conditions, $-\Delta G_0 = W_{max}$, where $\Delta G_0$ represents the change in Gibbs free energy of the chemical reaction under standard conditions. The reaction equation for a silver-zinc battery is $\mathrm{Ag_2O+Zn=2Ag+ZnO}$, with a Gibbs free energy change set as $\Delta G_0 = -309.7 \text{ kJ/mol}$. Since both reactants and products are solids, near standard conditions, the variation of $\Delta G$ with temperature and pressure can be ignored. There is a type of silver-zinc button cell where the silver positive electrode and the zinc negative electrode are separated by a membrane material soaked with a KOH solution. The concentration of the KOH solution is $c=14 \text{ mol/L}$, the thickness is $h=0.5 \text{ mm}$, and the resistivity is $\rho=2.37 \times 10^{-4} \, \Omega \cdot \text{m}$. The cross-sectional area of the battery is $S=0.465 \text{ cm}^2$. The battery electrodes can be considered as a pair of parallel plate electrodes with a small gap $h$, neglecting edge effects. Consider only the resistance of the solution, ignoring the resistance of the materials of the battery's positive and negative electrodes. It is known that Avogadro's constant is $N_A=6.022 \times 10^{23} \text{ mol}^{-1}$, and the elementary charge is $e=1.602 \times 10^{-19} \text{ C}$. Short-circuit the battery, and determine the drift velocity $v$ of the charge carriers ($OH^{-}$ ions) in the electrolyte, expressed in terms of the physical quantities given in the problem, without requiring a numerical solution. | ||
114 | ELECTRICITY | Place an electric dipole $p$ at the origin and establish a polar coordinate system $(r,\theta)$ with its direction as the polar axis. A negatively charged particle with charge $-q$ and mass $m$ starts from the point in polar coordinates $r=R,$ $\theta=\pi/2$ with initial velocities ${V}_{r}$ and $V_{\theta}$. The particle then moves to a known point with polar coordinates $r_0,\theta_0$. Find the angular momentum ${L}$ of the particle at this point. The permittivity of free space is $\varepsilon_0$. | ||
503 | MODERN | Suppose the total mass of a galaxy is the sum of visible mass and dark matter mass, with only gravitational interaction between dark matter and visible matter. Consider a young galaxy, whose mass is primarily composed of visible interstellar gas and invisible dark matter (ignoring the mass of stars). The interstellar gas is composed of identical particles with mass \(m_p\), and the number density \(n(r)\) of these particles depends on the distance \(r\) from the galaxy center. The temperature \(T(r)\) of the interstellar gas is also a function of \(r\). We can assume that the interstellar gas is always in a state of hydrostatic equilibrium, with the pressure gradient force balanced by the gravitational force in the galaxy. It is known that all mass distribution in the galaxy is spherically symmetric. Given the gravitational constant \(G\), the pressure gradient \(\frac{dp}{dr}\) of the interstellar gas balances the gravitational force (pressure is generated only by interstellar gas, ignoring the pressure contribution from dark matter). Assume the interstellar gas is an ideal gas, and for simplification, consider the temperature of the interstellar gas to be uniform everywhere at \(T_0\), and the number density of interstellar gas particles satisfies \(n(r)=\frac{\alpha}{r(\beta + r)^2}\), where \(\alpha,\beta\) are known constants. Find the mass density \(\rho_d(r)\) of dark matter at a distance \(r\) from the galaxy center. | ||
748 | ELECTRICITY | In an LRC circuit, the change in current in the inductor coil leads to a change in the magnetic field, thereby affecting the voltage and current in the circuit. Inserting a non-magnetic metal rod or a ferromagnetic rod will change the magnetic field distribution of the inductor coil, subsequently influencing the circuit's response. For a non-magnetic rod, due to electromagnetic induction, the changing magnetic field will produce an induced current within the metal rod, accompanied by Joule heating, thus generating an equivalent non-contact resistance $R$. If edge effects are ignored, the magnetic field inside the coil is uniform, and the number of turns per unit length of the coil is $n$. Assume the conductivity of the metal rod is $\sigma$, the angular frequency of the power source (sine signal) is $\omega$, and the lengths of both the coil and the metal rod are $l_{0}$, with their axes coincident, the radius of the metal rod is $r_{1}$, and the radius of the coil is $r_{0}$. For the non-magnetic metal rod, derive the expression for the non-contact resistance (assuming $\mu_{0}\omega\sigma r_{0}^{2}\ll1$, take the lowest order approximation). | ||
563 | MECHANICS | When a child was bathing, they accidentally slipped in the tub. Next, we will analyze the possible motion of the child in the tub.We model the tub as a smooth semicircular groove with a radius of $R$. We also model the child as a rigid hemispherical body with a mass of $m$, a radius of $R$, and a uniformly distributed mass . During the motion, the hemispherical child coincides with the center of the bathtub sphere and does not leave the bathtub.
Now we assume that the hemisphere does not leave the bathtub during its motion. The child rotates about a fixed point through the sphere's center. The child's back remains at an angle $ \alpha $ (i.e., the angle between the normal vector of the semicircular cross-section and the vertical direction is $ \alpha $). During the motion of the hemisphere, the magnitude of the angular velocity remains constant, but its direction changes continuously. Determine the minimum angular velocity magnitude that allows the child's back to always remain at an angle $\alpha$. | ||
424 | MECHANICS | There is water vapor with density $ \rho_0 $ suspended in the air, and the density of water is $ \rho $. When raindrops fall, they are approximated as spherical and grow larger by absorbing water vapor (water vapor becomes water upon absorption). Please prove that the raindrops eventually tend toward uniformly accelerated linear motion and find their acceleration. The acceleration due to gravity is $ g $.
Based on the above model, we should first consider the resistance of dry air (excluding forces from interactions with water vapor):
$ f = k A v^2 $
where $ k $ is to be determined, $ A $ is the maximum cross-sectional area in the direction of the raindrop's motion, and $ v $ is the raindrop's speed. Additionally, the falling raindrops are not spherical, and assume $ A $ and the raindrop's mass $ m $ satisfy:
$ A = a m^\alpha $
where $ a $ and $ \alpha $ are known constants.
Under this model, find the raindrop's acceleration in the final state.
Note: Consider the effect produced by water vapor and treat $ k $ directly as a known parameter. | ||
394 | MECHANICS | A smooth bowl with a radius of $R$ is fixed, and the plane at the mouth of the bowl is horizontal. A smooth, homogeneous, thin rod $AB$ with length $L = \frac{4\sqrt{3}R}{3}$ Translation:
End B is located outside the bowl, while end A presses against a point inside the bowl. The rod achieves static equilibrium in a plane passing through the center of the sphere $O$. Points $D$ and $D^{\prime}$ on the rod are nearly coincident with the point of contact at the rim of the bowl, but $D$ is slightly lower-left, and $D^{\prime}$ is slightly upper-right. Let the angle between the rod and the horizontal plane be $\theta$.
The rod is suddenly cut at point $D$. Note that after being cut, point $D$ will gently rest on the inner surface of the bowl. Find the angular acceleration $\beta = {\ddot{\theta}}$ of the rod at this instant. | ||
158 | ADVANCED | ## **Theory of Surface States of Topological Insulators**
The surface states of topological insulators have unique electromagnetic wave propagation characteristics, involving **Berry phase**, **Chern number**, and **nonlinear effects**. These concepts can be understood through step-by-step derivation and analysis.
Consider the surface states of a three-dimensional topological insulator, whose low-energy effective Hamiltonian matrix is:
$$
H = v_F (\sigma_x k_y - \sigma_y k_x) + m \sigma_z
$$
Where $v_F$ is the Fermi velocity, $\sigma_x, \sigma_y, \sigma_z$ are the Pauli matrices, $k_x$ and $k_y$ are the wave vectors in momentum space, and $m$ is the mass term.
Let's define $\vert \psi(\mathbf{k})\rangle$ as the normalized eigenvector. Since there are two eigenvectors, for convenience we take the one with the negative eigenvalue (although the final results are clearly the same).
Berry connection is defined as:
$$
\vert A\rangle =\begin{pmatrix}
\langle \psi(\mathbf{k}) | \frac{\partial }{\partial k_x} | \psi(\mathbf{k})\rangle\\
\langle \psi(\mathbf{k}) | \frac{\partial }{\partial k_y} | \psi(\mathbf{k})\rangle
\end{pmatrix}
$$
Definition of Chern number**: Chern number C is the integral of the Berry curvature $\Omega(\mathbf{k})$ over the Brillouin zone, where the integration region can be considered as the entire plane:
$$
C = \frac{1}{2\pi} \int_{\text{BZ}} \Omega(\mathbf{k}) \, d^2 k
$$
Where Berry curvature $\Omega(\mathbf{k})$ is defined as:
$$
\Omega(\mathbf{k}) = \nabla_{\mathbf{k}} \times \langle \psi(\mathbf{k}) | i \nabla_{\mathbf{k}} | \psi(\mathbf{k}) \rangle
$$
Calculate the Chern number when $m>0$. | ||
519 | ELECTRICITY | Thunderstorm clouds are the source of chaotic and intense atmospheric electric fields. Under the combined influences of turbulent disturbances, precipitation, horizontal winds, and more complex convergence and uplift factors, their distribution becomes highly irregular. However, as an approximate model, we can consider only the vertical influences. This is because for the electric field of thunderstorm clouds, what truly concerns everyday life is essentially just the "lightning that reaches the ground." This portion of lightning can be understood by the vertical electric field surpassing the atmospheric breakdown limit, and the accumulation of the vertical electric field is of paramount importance.
Within thunderstorm clouds, negative ions are more likely to dissolve into raindrops and fall, resulting in a general pattern of positive charges above and negative charges below. As precipitation occurs, charge separation will continue to expand until the electric field becomes sufficiently large for lightning to occur.
Let us consider two cloud layers of equal thickness, with their thickness being much smaller than their vertical separation. Precipitation exists between them. Each raindrop has a radius of $r$, density $\rho$, carries negative charge, and the charge of each raindrop is proportional to its radius: $q = br$. The number density of raindrops is $n$. The friction force acting on a falling raindrop is proportional to the first power of its velocity: $f = -kv$, where $k$ is a constant. The resistance experienced by ions moving through the air can be expressed using the modified Stokes' formula: $F = -Ku$, where $K$ is a constant, and $u$ is the magnitude of the ion's velocity. Suppose all positive ions carry a single positive charge $e$, and compared to the electric field force, the gravitational force acting on the ions can be ignored. The gravitational acceleration is $g$, and the vacuum permittivity is $\epsilon_0$.
If lightning has not occurred, and the electric field between the cloud layers is considered uniform, with the field strength being zero at time $t = 0$, find the function of the electric field $E(t)$ between the cloud layers as a function of time. | ||
275 | ELECTRICITY | As one of the four fundamental interactions, electromagnetic interactions play an important role in determining the internal properties of matter in everyday life. In particular, the electromagnetic interactions become more complex when studying the interactions between molecules. Below, we will simply calculate a few types of intermolecular interactions.
Between two molecules, their charges polarize one another, causing a shift in the centers of positive and negative charges. Suppose the electron distribution of the molecules is approximately uniformly distributed within a sphere with radius $a$, and the electron cloud does not deform under the influence of an external electric field. The atomic polarizability $\alpha$ is defined as the ratio of the electric dipole moment induced by polarization in the molecule to the external polarizing electric field.
**Problem:** Calculate the case in which a nonpolar molecule $A$ is polarized by a molecule $B$ that possesses a permanent electric dipole moment $p_{B}$. Molecule $B$ is located at the origin of the coordinate system, with the dipole moment direction pointing in the positive direction of the polar axis. The polar coordinates of the nonpolar molecule $A$ are $(r, \theta)$. Calculate the force between the two molecules. | ||
501 | THERMODYNAMICS | The atmosphere is divided from low to high into the troposphere, the stratosphere... The temperature in the troposphere and the stratosphere varies with height according to the following formulas, respectively: $T_t(h) = T_0(1-\alpha h)$, $T_s(h) = T_t(h_t)[1+\beta (h-h_t)]$, where $h$ is the height above the ground, $h_t$ is the distance from the ground to the top of the troposphere (known), and $\alpha, \beta, T_0$ are known constants. To simplify the problem, we assume that the air behaves as an ideal gas and that no convection occurs, with the gas distributed in a stable and equilibrium state. Derive the relationship for the variation of air density in the stratosphere with height, $\rho_s(h)$. It is known that the atmospheric pressure at the ground is $p_0$, the molar mass of air is $\mu$, the ideal gas constant is $R$, and the gravitational acceleration is a constant $g$. | ||
663 | THERMODYNAMICS | The equation of state for a certain gas can be expressed within a certain range as: \( p V = n R(T + a T^{2}) \). Here, \( a \) is a constant, \( R \) is the universal gas constant, and \( n \) is the number of moles. The molar mass of this gas is \(\mu\). The relationship between internal energy and state is known to be: \( U = \int_{V_{0}}^{V}\left(T{\left(\frac{\partial p}{\partial T}\right)}_{V}-p\right)\mathrm{d}V+f(T) \). In this problem, \( f(T) \) is zero. Assume that the surface of a certain planet consists entirely of this gas, the surface gravitational acceleration of the planet is \( g \), and the temperature changes with altitude according to the relation \( T = T_{0}(1+\alpha h) \), where \(\alpha\) is a constant and \( h \) is the altitude. The gas density at an altitude of zero is \(\rho_{0}\). Find the expression for air density as a function of altitude, the expression must not contain the variable \( T \). | ||
667 | ELECTRICITY | According to the theory of special relativity, we can derive that if the electric field strength in the $S^{\prime}$ frame is $\vec{E}^{\prime}$ and the magnetic induction strength is ${\vec{B}}^{\prime}$, then in the $S$ frame (where the $S^{\prime}$ frame moves along the positive $x$-axis of the $S$ frame with a velocity $ u$), the electric field strength is $\vec{E}$ and the magnetic induction strength is $\vec{B}$, and there are the following formulas: $$ \left\{\begin{array}{l l}{E_{x}=E_{x}^{'}}\ {E_{y}=\gamma\left(E_{y}^{'}+ u B_{z}^{'}\right),}\ {E_{z}=\gamma\left(E_{z}^{'}- u B_{y}^{'}\right)}\ {B_{z}=\gamma\left(B_{z}^{'}+\frac{ u}{c^{2}}E_{y}^{'}\right)}\end{array}\right.\left\{\begin{array}{l l}{\displaystyle B_{x}=B_{x}^{'}}\ {\displaystyle B_{y}=\gamma\left(B_{y}^{'}-\frac{ u}{c^{2}}E_{z}^{'}\right)}\ {\displaystyle B_{z}=\gamma\left(B_{z}^{'}+\frac{ u}{c^{2}}E_{y}^{'}\right)}\end{array}\right. $$ Now consider a point charge moving along the $x$-axis direction with a velocity $ u=\beta c$. Find the electric field strength $\vec{E}$ at point $\boldsymbol{A}$ located at a vector position $\vec{r}$ from the charge, as observed in the $S$ frame.The result is indicated by $\beta$, not $u$ (the angle between $\vec{r}$ and $\vec{ u}$ is denoted by $\theta$). | ||
183 | ADVANCED | A cylindrical insulating container of mass 𝑀 is stationary in a vacuum, with one end sealed. Initially, an insulating piston of mass 𝑚 and negligible width divides the container into two equal parts. The sealed part contains 𝑛 moles of a monoatomic ideal gas with a temperature 𝑇 and molar mass $𝑀_0$. Assume the container is smooth. It is assumed that during the expansion process, the state of the gas can be approximated under thermal equilibrium conditions. At the moment the piston leaves the container, the gas and the container will move at a velocity 𝑣, while the piston moves at a velocity 𝑢. After all the gas has left the container, the container's final velocity further changes from 𝑣 to 𝑣 + 𝑣′. Use the kinetic theory of gases to estimate 𝑣′. Assume the container's final velocity is much smaller than the thermal velocity of the molecules. The gas constant is 𝑅. There is no heat exchange among the gas, container, and piston. Temperature changes of the gas after leaving the container can be considered negligible. Earth's gravitational force can be ignored. | ||
700 | ELECTRICITY | Coaxial thin-walled cylinders with radii of $R$ and $2R$ extend infinitely in the direction perpendicular to the plane of the paper, maintaining a potential difference of $U$ between the inner and outer cylinders (the inner cylinder is at a higher potential than the outer cylinder). The region between the inner and outer cylinders is vacuum and has a constant uniform magnetic field $B$ directed into the plane of the paper. Inside the region of the inner cylinder, there is a variable uniform magnetic field $B_{v} = kt$ directed into the plane of the paper, where $k$ is a positive constant, and $t$ is time. Considering a charged particle with mass $m$ and charge $+q$ moving in the region between the inner and outer cylinders, ignore the effect of gravity and the influence of the charged particle and its induced charge on the potential of the inner and outer cylinders. At $t=0$, the charged particle is released from rest from the outer surface of the inner cylinder. It is known that the charged particle first reaches the inner surface of the outer cylinder after rotating $180^{\circ}$ around the axis of the cylinder, with its velocity direction exactly tangent to the cylinder at that time. Determine the time $t$ when the particle first reaches the inner surface of the outer cylinder after being released (just provide the expression, it is not necessary to discuss the existence of a positive real solution). | ||
485 | MECHANICS | Consider the problem of the trajectory under the influence of an inverse-square force. It is well known that an electron, under the Coulomb force exerted by a near-static atomic nucleus, follows a classical elliptical trajectory:
$$
r = \frac{p}{1 + e \cos \theta}
$$
where $r$ is the radial distance and $\theta$ is the angle of the radial vector with respect to the $x$-axis. We aim to calculate the "average position" of the electron over one period $T$, defined as:
$$
\overrightarrow{O P} = \frac{1}{T} \int_{0}^{T} \overrightarrow{r} \, \mathrm{d}t
$$
Please compute the length $l$ of $OP$ (express the answer using the quantities provided in the problem). | ||
376 | MODERN | The mass of the star Proxima Centauri is \( M \), its surface temperature is \( T_s \), and its radius is \( R \). Planet B orbits Proxima Centauri in a circular orbit, with its orbital period given as \( T \). Assuming both the star and the planet are black bodies and perfect heat conductors, estimate the steady-state temperature of the planet. Known constants include the gravitational constant \( G \) and the Stefan-Boltzmann constant \( \kappa \). | ||
364 | OPTICS | Using illumination light with wavelength $\lambda$, perform Fraunhofer diffraction. The diffraction screen is composed of two overlapping square apertures, each with a side length of \(a\). Both square apertures have sides that are either horizontal or vertical. One square is located at the upper left position, and the other is at the lower right position. The overlapping part is exactly a square with a side length of \(\frac{a}{2}\). Only the square apertures and their overlapping section are transparent on the plane. A plane wave is incident perpendicular to the screen. Determine the light intensity distribution $\widetilde{I}(\theta_{1},\theta_{2})$ of the Fraunhofer diffraction field on the observation screen. Express the result using $\alpha=\frac{\pi a\mathrm{sin}\theta_{1}}{\lambda}$ and $\beta=\frac{\pi a\mathrm{sin}\theta_{2}}{\lambda}$. Ensure that the intensity is $I_0$ when $\theta_{1}\to 0, \theta_{2}\to 0$. | ||
521 | MECHANICS | Assume there exists a substance that always satisfies $p=\frac{1}{2}k\rho^2$, where $p$ is the pressure, $\rho$ is the density of the substance, and $k$ is a constant. In the universe, there is a spherically symmetric celestial body composed of this substance, and the body remains stable (every mass element experiences a net force of zero). Find the radius $R$ of this celestial body in its stable state. Assume that the law of universal gravitation always holds, with the gravitational constant being $G$. Hint: Derive the differential equation that $\rho(r)$ satisfies, where $r$ is the distance from a point to the center of the sphere, and use the substitution $u=r\rho$ to solve. | ||
291 | MECHANICS | Internal combustion engine and crankshaft connecting rod mechanism. The connecting rod $AB$ has a length of $l$; one endpoint $A$ rotates on a circle with radius $r$, with $O$ as the center of the circle. The piston is constrained to move along the line $OB$. The gas pressure in the cylinder is $P$, the atmospheric pressure is $p$, and the piston is circular with a radius of $R$. When the angle between $OA$ and $OB$ is $\psi$, with what force does the connecting rod rotate the crank? | ||
596 | ELECTRICITY | In recent years, some have proposed using plasma lenses to focus high-energy charged particles. A plasma lens can be considered as a long cylindrical good conductor with a radius of \( R \), carrying a uniform axial current \( I \) (which can be treated as infinitely long), with vacuum outside the cylinder. The origin is set at the middle point of the cylinder on the axis, and the axial direction is the \( z \)-axis (the direction of current is the +z direction), with the radial coordinate denoted as \( r \). At a certain moment, a relativistic particle with charge \( +q \) and momentum magnitude \( p_0 \) (the magnitude to be determined) is emitted from a particle source at \( z=0, r=r_0 (r_0>R) \), in the direction along the +z axis. It is assumed that the plasma is sufficiently sparse, so the particle only experiences the magnetic field generated by the plasma lens, and even if it enters within the cylindrical plasma lens, collisions between the particle and the plasma can be completely ignored. Since the plasma is considered a good conductor, the electric field within it has negligible effect on the charged particle. It is assumed that the magnetic permeability of the plasma equals the vacuum permeability \( \mu_0 \). During the focusing process, particles with smaller momentum may reverse direction within the plasma lens (i.e., have a momentum z-component opposite to the initial direction), a situation known as over-focusing of the plasma lens. To allow the charged particle to transmit in the axial direction without reversing, find the particle's minimum initial momentum \( p_0 \). Note: For convenience, you may need to assume the particle's rest mass or speed, but surprisingly, the final result depends only on momentum. This actually allows us to conveniently use readily available particles to simulate less common situations—for example, using a large and stable number of high-energy protons produced by an accelerator to simulate the extremely rare muons in cosmic rays, as long as their momentum is equal. | ||
723 | ELECTRICITY | Consider a regular polygon with 2017 sides, where a point charge \(q\) is placed at each of the 2016 vertices, and a point charge \(Q\) is placed at the center of the polygon. The distance from the center of the polygon to each vertex is \(a\). Utilizing symmetry and the principle of superposition of electric fields, determine the magnitude of the net electrostatic force on the point charge \(Q\). The answer should be expressed as a clear formula. | ||
698 | ADVANCED | To solve the surface wave equation for a viscous fluid under the approximation of small movements, we take the bottom boundary to be horizontal at $z=0$, with liquid depth $h$, kinematic viscosity $ u$, density $\rho$, and surface tension coefficient $\tau$. There exists only a uniform downward gravitational field $-g \vec e_z$. The surface equation corresponding to the displacement at $z=h$ is $\eta(\vec{x},t)$, where $\vec{x}$ represents the coordinates on the surface. Given the Laplace transform $\bar{\eta}_{k}(p)$ of the $k$ wavevector component $\eta_{k}(t)$ of the surface displacement $\eta(\vec{x},t)$. Initial conditions $\eta_{k}(0)$ and $\eta'_{k}(0)$ are provided. Hint: Solve the linearized Navier-Stokes equation $$ \partial_t \vec v = -\frac{1}{\rho} abla p - g \vec e_z + u abla^2 \vec v $$ Decompose the velocity field as $$ \vec v(x,z,t) = abla\times abla\times \vec e_z S(x,z,t) + abla\times \vec e_z Z(x,z,t) $$ You need to take the curl of the N-S equation twice to obtain a partial differential equation for $S$, requiring the no-stress boundary condition at the bottom boundary (this aids in smoothly transitioning to the ideal fluid case) $$ S(x,z,t)=\partial_z^2 S(z,x,t)=0|_{z=0} $$ | ||
138 | ELECTRICITY | Maxwell's electromagnetic theory tells the world that the electromagnetic field has gauge symmetry. Under the gauge transformation $$ \vec{E} \equiv- abla \varPhi- \frac{ \partial \vec{A}}{ \partial t}, \quad \vec{B} \equiv abla \times \vec{A}. $$ $$ \vec{A^{ \prime}}= \vec{A}- abla \varPsi, \quad \varPhi^{'}= \varPhi+ \frac{ \partial \varPsi}{ \partial t} $$ the electromagnetic field $ \vec{E}$ and $ \vec{B}$ are unchanged. In the Lorentz gauge $$ abla \cdot{ \vec{A}}+{ \frac{1}{c^{2}}}{ \frac{ \partial \phi}{ \partial t}}=0 $$ the electromagnetic potentials satisfy the source-included d'Alembert equation $$ abla^{2} \varPhi- \frac{1}{c^{2}} \frac{ \partial^{2} \varPhi}{ \partial t^{2}}=- \frac{ \rho}{ \epsilon_{0}}, \quad abla^{2} \vec{A}- \frac{1}{c^{2}} \frac{ \partial^{2} \vec{A}}{ \partial t^{2}}=- \mu_{0} \vec{J}. $$ However, if the gauge symmetry is broken, the equation satisfied by the electromagnetic potential takes the following form $$ abla^{2} \varPhi- \frac{1}{c^{2}} \frac{ \partial^{2} \varPhi}{ \partial t^{2}}- \mu^{2} \varPhi=- \frac{ \rho}{ \epsilon_{0}}, \quad abla^{2} \vec{A}- \frac{1}{c^{2}} \frac{ \partial^{2} \vec{A}}{ \partial t^{2}}- \mu^{2} \vec{A}=- \mu_{0} \vec{J}. $$ Or, using four-vectors, it can be written in the following form $$ \left( \boldsymbol{ \Pi}+ \mu^{2} \right) \boldsymbol{A}^{ u}= \mu_{0} \boldsymbol{J}^{ u}, $$ where $ \begin{array}{r}{ \exists \equiv{ \frac{1}{c^{2}}}{ \frac{ \partial^{2}}{ \partial t^{2}}}- abla^{2}} \end{array}$ In quantum mechanics, an equation in this form is called the Klein-Gordon equation. Study Coulomb's law in the case of $ \mu eq0$. The electric potential $ \varPhi$ satisfies $$ \left( abla^{2}- \mu^{2} \right) \varPhi \left( \vec{r} \right)=- \frac{q}{ \epsilon_{0}} \delta^{3} \left( \vec{r} \right). $$ It is known that the general solution of the inhomogeneous Helmholtz equation in the form $$ \left( abla^{2}+k^{2} \right)G \left({ \vec{r}} \right)=-4 \pi \delta^{3} \left({ \vec{r}} \right) $$ is $$ G \left( \vec{r} \right)=A \frac{ \mathrm{e}^{ \mathrm{i}k r}}{r}+B \frac{ \mathrm{e}^{- \mathrm{i}k r}}{r}+g, $$ where $A+B=1$, $g$ is a solution to the homogeneous Helmholtz equation. Thus, the modified formula for the electrostatic field of a point charge in Coulomb's law is $$ \varPhi \left( \vec{r} \right)= \frac{q}{4 \pi \epsilon_{0}} \frac{ \mathrm{e}^{- \mu r}}{r}. $$ Maxwell used the following experiment to measure deviations in the forces between charges relative to Coulomb's law: Place two concentric thin spherical conducting shells with radii $a$ and $b(a>b)$, and connect them with a thin wire. Charge the outer shell to a potential $U$, remove the power source, then remove the wire connecting the two shells, and finally ground the outer shell. At this point, the potential measured at the inner shell is no greater than $u$. Estimate the upper limit of the photon's static mass from this, retaining the leading order contribution. | ||
619 | THERMODYNAMICS | Heat engines and heat pumps utilize thermodynamic cycles of substances to achieve opposite functions: the former absorbs heat from a high-temperature source, converts part of the heat into work for external output, and releases the remaining heat to a low-temperature sink; the latter takes in external work to absorb heat from a low-temperature source and discharges it to a high-temperature sink, together with the heat converted from the external work. According to the second law of thermodynamics, whether for a heat engine or a heat pump, if the working substance during its cycle only interacts with two heat reservoirs at temperatures $T_{1}$ and $T_{2}$, the absorbed heat $Q_{1}$ and $Q_{2}$ satisfies the inequality $$ \frac{Q_{1}}{T_{1}}+ \frac{Q_{2}}{T_{2}} \leq0 $$ Here, the heat can be positive or negative, indicating absorbing heat from or releasing heat to a heat source, respectively. A certain heating device originally uses heat released by a boiler at temperature $T_{0}$ to directly heat a room, maintaining the room temperature at a constant $T_{1}$, which is higher than the outdoor temperature $T_{2}$. To improve energy efficiency, it is proposed to use the aforementioned machines based on existing energy to improve the heating scheme. Compared to direct heating, what is the theoretical limit of the reduction rate $ \eta$ in boiler energy consumption? | ||
550 | OPTICS | Solve the following physics problem, and frame the final answer in \boxed{}: General relativity is the astounding theory Einstein provided to theoretical physics in the early 20th century. It is profound for its unique concepts of spacetime and matter, elegant for its geometric mathematical structure, and distinguished for its successful predictions of physical phenomena, garnering the attention of later theoretical physicists. Mr. Yang Zhenning once said that the Yang-Mills theory (the main theory governing our understanding of the other three fundamental interactions) is actually similar in mathematical structure to general relativity.
As a crucial part of verifying general relativity, in 1919, Sir Arthur Eddington embarked on an expedition to Brazil and the Gulf of Guinea, leading an experiment observing the deflection of starlight during a solar eclipse, which directly announced the validity of general relativity to the world. Let's use a non-relativistic equivalent method to derive the classical results of this experiment. It can be proven that if a light ray originating from infinity behaves like a particle in general, incident with an impact parameter $b$ and its propagation direction being deflected under the gravitational field of a centrally fixed celestial body $M$, then its trajectory will be identical to that in a refractive index field $n(r)$. This refractive index field is:
$$
n^{2}=n_{0}^{2}\left(1+{\frac{2G M b^{2}}{c^{2}r^{3}}}\right)
$$
where $n_{0}$ is the refractive index when $r\rightarrow\infty$. Consider the following problem:
Let us study the influence of different $b$ on the shape of the orbit. Even if the radius of the central body $M$ can be neglected, it can be proven that when $b$ is smaller than a certain value, the radius of the light ray will decrease infinitely during its propagation and will not be able to escape the central body $M$. This phenomenon is called the capture of photons by the celestial body. Please strictly calculate, for a parallel beam of light originating from infinity, within what area range the light will be captured by the central body (capture cross-section)? | ||
442 | MODERN | A uniformly positively charged infinite straight conductor has a linear charge density $\lambda$. A particle with a static mass $m$ and negative charge $-q$ can, under the appropriate initial velocity, move in a uniformly spiraling motion on a cylindrical surface with radius $R$ due to the electrostatic force between it and the straight conductor. Based on this, consider solving for the angular frequency $\omega$ of small radial oscillations when the initial velocity of the negative point charge changes to the composition of a very small radial velocity $v_r \ll v_z$ in the radial direction. The result must be expressed using $$ k = \frac{\lambda q}{2\pi\varepsilon_0 m c^2} $$ and $v_z$. Both $k$ and $v_z$ are considered known. | ||
588 | ELECTRICITY | In a three-dimensional space without gravity, let us consider a thin metal spherical shell moving in a uniform magnetic field. The thickness of the metal spherical shell is \(h\), the radius of the metal spherical shell is \(a\), the density is \(\rho_m\), and the conductivity is \(\sigma\), where \(a \gg h\). The angular velocity \(\omega\) of the metal spherical shell is along the positive \(x\)-axis. There is a uniform magnetic field in the space along the positive \(z\)-axis, with magnitude \(B\). We assume that the current distribution is established instantaneously, and there is no transient process (i.e., when the angular velocity changes, the current at each moment is stable, satisfying \( abla\cdot \vec J=0\)). If the initial angular velocity of the spherical shell is \(\omega_0\), determine the relationship of the angular velocity with time \(t\). We assume that the spherical shell is not subject to any forces and torques other than electromagnetic forces, and we ignore the magnetic field generated by the shell's own current. Hint: Since the shell is very thin, the current only has a tangential component, so the problem can be transformed into a two-dimensional problem on the sphere's surface. You can use \(\vec J=\sigma(\vec E+\vec v\times B)\) to solve for the tangential electric field, back-calculate the current, and then complete the mechanical part of the calculation. | ||
768 | ELECTRICITY | A semicylindrical dielectric structure of length $l$ with inner and outer radii $a$ and $b$ respectively is composed of two different lossy dielectrics. Their relative permittivities and conductivities are $ \pmb{ \varepsilon}_{1}$ and $ \pmb{ \sigma}_{1}$ (in the region $0 \leq \varphi < \theta_{0}$), and $ \pmb{ \varepsilon}_{2}$ and $ \pmb{ \sigma}_{2}$ (in the region $ \theta_0 < \varphi \leq \pi$). The bottom sides of the semicylinder ($ \theta = 0$, $ \theta = \pi$) are coated with metallic films, between which a DC voltage $V_0$ is applied, reaching a steady state. The vacuum permittivity is given as $ \varepsilon_0$, and the relative permittivities of the two media are large, while their conductivities are small. Edge effects are neglected. Find the total charge at the interface $ \varphi = \theta_0$. The answer should not contain integral expressions. | ||
473 | MECHANICS | Please solve the following physics problem and put the final answer in \boxed{}: Consider a statics version of Buffon's needle problem: A steel needle of length $l^{\prime}=x l$ is randomly placed on infinitely thin horizontal iron racks that are evenly spaced at distance $l$, where $x\geq2$. Find the probability $p$ that the steel needle can rest stably on the racks without falling off. | ||
448 | MODERN | In the cosmic space, there is a spaceship that is approximately spherical in shape, initially at rest, with a radius of $R$ and a mass of $M$. The spaceship is propelled by radiation. Assume that the spaceship's speed is very high, so we must consider relativistic effects.
After a period of travel, the spaceship enters a region of uniform, stationary space dust, where the mass of dust per unit volume in the cosmic space reference frame is $p$. Upon contact with the spaceship, the dust adheres to the surface of the spaceship. For simplicity, let the spaceship's velocity at this moment be $v_{0}$ and its rest mass be $M_{1}$. If after entering the dust region, the spaceship shuts off its engines and moves without propulsion, assume that the mass of the dust attached to the spaceship is very small relative to the spaceship's own mass. Just upon entering the dust region, set the clock on the spaceship to $t^{\prime}=0$. Determine the relationship between the spaceship's speed $v$ and the time $t'$ on the spaceship. | ||
165 | MECHANICS | The plane of the uniform right-angle side $AOB$ is vertical, with side $AO$ having a length of $l_1$ and side $BO$ having a length of $l_2$. It can swing left and right around a horizontal axis passing through point $O$ and perpendicular to the plane of $AOB$, and at the same time, it can rotate around a vertical axis passing through point $O$. Assume that during the uniform rotation around the vertical axis, the angle between side $OA$ and the vertical line stabilizes at angle $\alpha$, and the gravitational acceleration is $g$. Find the angular velocity of the rotation. | ||
566 | ADVANCED | Alice and Bob are playing a game, roughly described as follows. Charlie is the host of the game. In each round, Charlie gives each of Alice and Bob a bit, \(x, y\), which means that each of them gets a number, 0 or 1. After receiving the bits, Alice and Bob each reply to Charlie with a bit, \(a, b\). To win the game, it is required that \(x\,\text{and}\, y = a\,\text{xor}\,b\). Alice and Bob can devise a strategy before the game, but they cannot communicate during the game.
We consider the following strategy:
A pair of entangled quantum bits can be described as
\[\ket{\phi}=\frac{1}{\sqrt{2}}\left(\ket{00}+\ket{11}\right)\]
Here, \(\ket{mn}\) refers to the state of the system composed of two quantum bits, meaning the quantum state of the system where the quantum bit in Alice's hand is in state \(\ket{m}\) and the quantum bit in Bob's hand is in state \(\ket{n}\). The game process is that Alice first performs a measurement based on the \(x\) she received, and then Bob performs a measurement based on the \(y\) he received. They determine their reply bits \(a, b\) based on the measurement results.
Let's define the following states:
\[\begin{align*}
\ket +&=\frac{1}{\sqrt2}\ket0+\frac{1}{\sqrt{2}}\ket 1&
\ket -&=\frac{1}{\sqrt2}\ket0-\frac{1}{\sqrt{2}}\ket 1\\
\ket{a_0}&=\cos\theta\ket0+\sin\theta\ket1&\ket{a_1}&=-\sin\theta\ket0+\cos\theta\ket1\\
\ket{b_0}&=\cos\theta\ket 0-\sin\theta\ket1&\ket{b_1}&=\sin\theta\ket 0+\cos\theta\ket 1
\end{align*}\]
where \(\theta=\pi/8\).
The specific measurement strategy is as follows:
**Alice's Measurement:**
* When Alice's random number \(x\) is 0, she uses the measurement basis \(\{\ket{1}, \ket{0}\}\).
* If she measures \(\ket{1}\), the value she sends is 1.
* If she measures \(\ket{0}\), the value she sends is 0.
* When Alice's random number \(x\) is 1, she uses the measurement basis \(\{\ket{+}, \ket{-}\}\).
* If she measures \(\ket{+}\), the value she sends is 0.
* If she measures \(\ket{-}\), the value she sends is 1.
**Bob's Measurement:**
* When Bob's random number \(y\) is 1, he uses the measurement basis \(\{\ket{b_1}, \ket{b_0}\}\).
* If he measures \(\ket{b_1}\), the value he sends is 1.
* If he measures \(\ket{b_0}\), the value he sends is 0.
* When Bob's random number \(y\) is 0, he uses the measurement basis \(\{\ket{a_0}, \ket{a_1}\}\).
* If he measures \(\ket{a_0}\), the value he sends is 0.
* If he measures \(\ket{a_1}\), the value he sends is 1.
Calculate the average probability of Alice and Bob's winning game with this quantum strategy. | ||
362 | MECHANICS | A cylinder with radius $r$ rolls purely on a horizontal surface. A rod $AB$ leans against it, with one point of the rod resting on the cylinder and the other end $A$ on the ground. There is no relative sliding between the rod and the cylinder, and the end $A$ of the rod does not leave the ground. Given that the velocity $v_0$ of the cylinder's center $O$ is a constant, and the angle between the rod and the ground is $\varphi$, find the angular acceleration $\varepsilon$ of the rod $AB$.
| ||
560 | MECHANICS | There is a railway curve with a curvature radius of $r$, and the distance between the two rails is $L$. There is a height difference between the inner and outer tracks. When the train passes through this curve at its rated speed $v_{0}$, the tracks do not experience lateral thrust. When the train passes through the curve at a speed $v(v>v_{0})$, to prevent the train from overturning, how high can its center of gravity be above its chassis? | ||
499 | MECHANICS | A rigid metal wire in the shape of a parabola is fixed in a vertical plane, with the parabola's equation given by \( y = ax^2 \) (the \( y \)-axis is oriented vertically upwards, and \( a \) is a constant to be determined). A uniform rigid thin rod of length \( 2l \) has small holes at each of its ends, \( A \) and \( B \), which just fit onto the metal wire. The contact between the holes and the wire is very smooth, with negligible friction.
If an impulse is given to the rod causing it to start moving, after a sufficient amount of time, it comes to rest at an equilibrium position, where the angle between the rod and the horizontal direction is \( \theta = 30^\circ \). The magnitude of the gravitational acceleration is known to be \( g \).
The rod remains stationary in this equilibrium position. Now, a small mouse starts to climb from the bottom end of the rod upwards. During its ascent, the rod remains stationary. Assume the mouse can be regarded as a point mass, and it does not contact the metal wire at the ends of the rod. Find the displacement of the mouse along the rod at time \( t \) (consider the time when the mouse starts climbing the rod as time zero) expressed as \( s(t) \). | ||
481 | MECHANICS | Two small steel balls, each with mass $m$, are connected by a string of length $2L$ and placed on a smooth horizontal surface. A constant pulling force $F$ is applied at the midpoint $O$ of the string. The direction of this force is horizontal and perpendicular to the initial direction of the connection. The string is very flexible and non-stretchable, and its mass is negligible. After several collisions, the two steel balls eventually continue to move while remaining in contact. Find the total energy lost due to collisions. | ||
579 | THERMODYNAMICS | Percolation is a geometric phase transition with profound significance in the study of localization and related topics. We use a model here to understand it. On a 2D lattice, lattice points can be occupied by black particles or be empty (white particles). The probability of placing a black particle is $p$, and the probability of being empty is $q=1-p$, where $N$ is the total number of particles.
It is clear that the probability of large clusters $\mathrm{\Delta}s$ appearing, where $n_{s}(p)$ at low values of $p$ is significant, decreases rapidly with $s$. However, when $p$ is large, one observes the emergence of macroscale $s$, which is the localization phenomenon. The transition occurring with $p$ is referred to as percolation. To study this problem, consider the influence between lattice points and transform it. We provide some approximate treatments: firstly, the number density of black particles is redefined as a dynamically changing $n$. We treat it as a function containing time $t$ and lattice coordinates $(x,y)$, and continuoize $(x,y)$ into a two-dimensional real number set instead of integer lattice coordinates. In our simple continuum model, the flux of number density between lattice points is exactly proportional to the gradient of the number density (consistent with the two-dimensional diffusion problem). However, in the actual discrete model, when localization occurs, a lattice point with certain probability is in a connected large cluster, with an internal number density always being 1; or the lattice point is located in an empty space or a small cluster with other probabilities, where the number density is less than 1. This links the continuous model with the discrete model: if the continuous model's $n(x,y)$ ultimately forms a stable distribution $n(r)$ rotationally symmetric around the origin and time-independent, where $r=\sqrt{x^{2}+y^{2}}$, in this model, take a circle of radius $r_{0}$ such that the total number of particles inside is $N$, representing the characteristic particle number of a large cluster, and the edge particle number density $n(r_{0})$ should be related to the critical probability $p_{c}$ at which localization begins. We directly assume $n(r_{0})\cdot\pi r_{0}^{2}=p_{c}N$. Now, let $N=1$, and use $p_{c},r_{0}$ to represent $n(r)$. | ||
476 | THERMODYNAMICS | The metal wire, with a length of $2l$ and a cross-sectional area of $A$, is connected to two steel terminal posts, $C$ and $D$, within the circuit. The metal wire is surrounded by materials with excellent thermal insulation. It is known that the resistivity of the metal wire is $\rho$ and its thermal conductivity is $\kappa$. A constant current $I$ is applied. Assume the temperatures of the two steel terminal posts, $C$ and $D$, are always maintained at room temperature $T_{0}$. After the entire system reaches a steady state, answer the following questions:
Description of the figure: The metal wire is a long cylindrical rod, surrounded by materials with excellent thermal insulation, and is connected at both ends to the steel terminal posts $C$ and $D$.
Take the midpoint of the metal wire as the origin, with the positive $x$-axis extending along the wire to the right. Derive the temperature distribution function along the metal wire.
**Hint**:
1. Heat conduction in the metal wire follows the linear heat conduction law: the heat flux through a cross-section of the wire in unit time is $\dot{Q}=-\kappa \frac{\mathrm{d}T}{\mathrm{d}x}A.$
2. You may need the integration formula: $\int \ln x \, dx = x(\ln x - 1) + C$ | ||
199 | THERMODYNAMICS | The thermal decoupling process $DM_1+DM_2\longleftrightarrow SM_1+SM_2$ has a dark matter annihilation cross-section $\sigma$, and the relative velocity of dark matter particles is $v$. Given that dark matter is cold, the cross-section can be expanded in a wave-like series as follows:
$$
\sigma v=a_s+a_p v^2+a_d v^4+\cdots
$$
It is known that the leading order is the $P$-wave, and we are given $\left\langle \sigma v\right\rangle=\frac{b_0 T}{m}$. Express $a_p$ in terms of $b_0$.
Hint: Under thermal equilibrium, the particle number density distribution function is $f_{eq}=\frac{1}{e^{E/T}\pm 1}\approx e^{-\frac{E}{T}}$, and the number density is $n_{eq}=\int e^{-\frac{E}{T}}\frac{d^3 \vec{p}}{(2\pi)^3}$. In this problem, dark matter particles are non-relativistic. | ||
373 | THERMODYNAMICS | At time $t=0$, an adiabatic, thin and lightweight piston, which is thin and lightweight, divides an adiabatic cylinder with a cross-sectional area $S$ into two equal parts, each with a volume of $V_{0}$. On the left side of the piston, there is an ideal gas with an adiabatic index of $\gamma=3/2$, and an initial pressure of $p_0$.
On the walls of the container, there is a certain viscous material characterized by a negligible heat capacity and volume. Its behavior is as follows: when the piston moves with a speed of magnitude $v$ in a certain direction, the material exerts a resistive force on the piston given by $F=-k v$. Moreover, the heat generated due to friction exhibits two possible behaviors:
- For a forward-directed viscous material, the heat is conducted along the original direction of motion to the adjacent gas on that side (if there is no gas, it is conducted to the external environment).
- For a backward-directed viscous material, the heat is conducted opposite to the original direction, transferring to the gas on the opposite side (if there is no gas, it is conducted to the external environment).
Consider the following scenario (retain results in analytical form without converting to numerical values):
The right side of the container is vacuum, and the material used is forward-directed viscous material. Calculate the time $t$ required for the piston to move to the far right. | ||
198 | ADVANCED | There is now a semi-classical Fermi gas system. It is approximated that the interaction between particles can be simplified to a hard sphere potential, with a characteristic radius $\lambda_{th}=\frac{h}{\sqrt{2\pi mkT}}$, and the system's free energy function can be written as:
$$
F=NkT\ln\frac{\lambda_{th}^3}{\frac{V}{N}-a\lambda_{th}^3}
$$
Given the definition of chemical potential $\mu=\frac{\partial G}{\partial N}$, the system's grand potential can be written as $\phi=F-\mu N$, and this system's grand potential has the following exact solution. Determine the constant $a$ in the free energy function. (You may use the approximation $\mu\ll kT$)
$$
\phi=-\frac{kTV(2\pi m)^{3/2}}{h^3}\cdot \frac{2}{\sqrt{\pi}}\int_{0}^{\infty}\ln\left[1+e^{-\frac{\epsilon-\mu}{kT}}\right]\epsilon^{\frac{1}{2}}d \epsilon
$$ | ||
761 | OPTICS | In a vacuum, there is a beam of circularly polarized light propagating along the $+z$ direction, represented by $\begin{array}{r}{\vec{E}=E_{0}\cos(\omega t-k z)\hat{x}+E_{0}\sin(\omega t-k z)\hat{y}}\end{array}$. From a quantum perspective, circularly polarized light consists of photons that possess non-zero spin angular momentum. The energy density of the electromagnetic field in a vacuum is known to be $\begin{array}{r}{w=\frac{1}{2}\varepsilon_{0}\vec{E}^{2}+\frac{1}{2\mu_{0}}\vec{B}^{2}}\end{array}$, and the spin angular momentum density is given by $\vec{s}=\varepsilon_{0}\vec{E}\times\vec{A}$, where $\vec{A}$ is the magnetic vector potential. Considering only the time-oscillating term of the magnetic vector potential, for a monochromatic plane electromagnetic wave, we have $\vec E=-\frac{\partial \vec A}{\partial t}$.\n\nNow, let this beam of light be incident perpendicularly on a wave plate with thickness $d$ and cross-sectional area $S$. The refractive indices of the wave plate for linearly polarized light in the $x$ and $y$ directions are $n_{o}$ and $n_{e}$, respectively. In this case, the angular momentum of the photons is no longer an eigenstate in a fixed direction but is in a superposition state of two opposite spin directions. Calculate the magnitude of the torque required to stabilize this wave plate. In this problem, we do not consider the absorption and reflection of light, i.e., only the phase change after passing through the wave plate needs to be considered. The speed of light in a vacuum is $c$. | ||
391 | THERMODYNAMICS | An adiabatic container is connected to a vacuum through a small hole. Initially, the container holds an ideal gas with a molar amount of $n_{0}$. The gas leaks very slowly, and the gas in the container is always in equilibrium. It is known that the adiabatic equation for an ideal gas is $p V^{\gamma} = \text{constant}$, where $\gamma = \frac{C_{v} + R}{C_{v}}$.
Assuming the container holds a single type of ideal gas with a molar heat capacity at constant volume of $C_{\scriptscriptstyle V} = \frac{5}{2}R$, find the molar amount $n_{1/2}$ remaining in the container when the pressure in the container is halved. | ||
738 | MECHANICS | A water bottle with a height of $H$ is filled with water. One side of the bottle is uniformly distributed with several small holes, with the number of small holes per unit length being $n$. Each small hole has an area of $s$ (the dimensions of the small hole are much smaller than $H$). Each small hole sprays water outward, and it is assumed that the direction of water sprayed from the small holes is along the horizontal direction. During the spraying process, the decrease in water level is not considered. Find the amount of water, per unit time and per unit length, that hits the ground at a distance $x$ from the side of the water bottle that is spraying water. (Do not consider the process of water splashing back up after hitting the ground, just provide the expression for the area that can receive the water.
| ||
705 | ELECTRICITY | Placed on the \(xOy\) plane is a fixed square wire loop carrying a constant current \(I\), with a side length of \(2a\). At its center, there is a small magnetic sphere, with a radius \(r (r \ll a)\), mass \(m\), and relative permeability \(\mu_r\). We assume the magnetization \(\vec{M}\) of the sphere is nearly unchanged. (The magnetization \(\vec{M}\) is not known.) The sphere's movement is constrained along the \(z\) axis, and it is subjected to a disturbance moving along the \(z\) axis. Find the period of small oscillations \(T\). | ||
402 | MECHANICS | In medical physics experiments, the capillary method is commonly used to measure the viscosity of low-viscosity liquids. The Ostwald viscometer, due to its ease of construction and simple operation, along with relatively high measurement accuracy, has become a common instrument for determining liquid viscosity using the capillary method. It is particularly suitable for studying liquids with small viscosity coefficients, such as water, gasoline, alcohol, plasma, or serum. It is widely used in clinical and pharmaceutical industries. The Ostwald viscometer consists of a $U$-shaped glass tube, with one side having a larger diameter and having a large glass bulb $A$, and the other side having a smaller diameter and having a small glass bulb $B$. The lower end of the small glass bulb $B$ has a cylindrical capillary tube with a length of $L$ and an inner radius of $R$. There are markings $m$ and $n$ (painted in red) at the upper and lower ends of the small glass bulb $B$. The height difference between $A$ and $B$ is $\Delta h$. During the experiment, the volume of liquid flowing through the capillary is the volume inside the small glass bulb $B$, which is the volume as the liquid level drops from $m$ to $n$. The average liquid level point is $^b$, and the liquid level point $a$ inside the large glass bulb $A$ is in the middle of the large glass bulb and remains basically unchanged during the experiment.
Neglecting capillary effects, the flow of the liquid can be considered equivalent to viscous fluid performing laminar flow in a horizontal fine circular tube with constant cross-section (cross-sectional radius is $R$) and a length of $L$, with the velocity at the contact position between the fluid and the tube wall being zero. Given that the Newtonian viscosity coefficient is $\eta$ and the capillary tube is placed vertically, derive the expression for the flow rate $q_{V}$ of the liquid in the capillary. | ||
595 | ELECTRICITY | There is a planar resistor network. The circuit is composed of 2014 sectors, with each side of the sector being a resistor with resistance $r$. Obviously, there are 2014 radii, with one end of the resistors on these radii connected at the center to form the center point; the other end is connected using arc resistors. Try to determine the resistance between two adjacent points on the sector arc in the circuit. (Hint: You can consider 2014 as a very large number and approximate it as an infinite network.) | ||
46 | MECHANICS | Considering the relativistic case of a two-dimensional elastic collision, a particle $m_{1}$ collides with $m_{2}$, with an initial velocity of $v_{0}$. Given that $m_{1} > m_{2}$, discuss the maximum angle between the exit angle of particle $m_{1}$ after the collision and its initial incident angle. | ||
203 | ELECTRICITY | In the $x y$ plane, there are electric dipoles $P_{2}$ and $P_{1}$ located at $(a,0)$ and $(-a,0)$, respectively. Their magnitudes are both $P$, and they form angles $\varphi_{2}(t)$ and $\varphi_{1}(t)$ with the $x$-axis. The permittivity of vacuum is given as $\varepsilon_{0}$.
Consider a special case where $\varphi_{1}(t) = \omega t$ and $\varphi_{2}(t) = \pi - \omega t$. A point charge with charge $-q$ and mass $m$ is located at the origin and experiences a small perturbation in the $x$ direction. It is known to undergo harmonic oscillation. Determine the angular frequency, under the condition that $\omega$ is much larger than the angular frequency of the perturbation experienced by the point charge but much smaller than the frequency range where radiation needs to be considered. | ||
397 | MECHANICS | A uniform thin rod is placed on the track of a rotating hyperbolic surface. The rod has a known length $L$ and mass $M$. The equation of the hyperbolic curve in the plane is:
$$
{\frac{y^{2}}{a^{2}}}-{\frac{x^{2}}{b^{2}}}=1
$$
The gravitational acceleration $g$ is known.
Given that the rod is in a non-horizontal stable equilibrium position, determine the angular frequency of small oscillations of the rod within the plane defined by the rod and the origin. | ||
480 | OPTICS | In the natural light Young's double-slit interference experiment, the front slit is called $S_0$, and the back double slits are called $S_1$ and $S_2$. Initially, no polarizers are placed at any positions. Now, polarizers $P_1$ and $P_2$ are placed behind slits $S_1$ and $S_2$, respectively, so that the transmission direction of $P_1$ forms an angle $\theta$ with $P_2$. Find the contrast of the interference fringes $\gamma$, expressed in terms of $\theta$. | ||
671 | ELECTRICITY | There is an isolated metal sphere with a charge of 0, having a radius of $r$. At point $A$, located at a distance $a$ from the sphere's center $O$, there is an electric dipole. The magnitude of the dipole moment is $p$, and the angle between the direction of the dipole and $\vec{OA}$ is $\theta$. Find the electric potential $\varphi(x)$ produced by the conductor sphere at a distance $x$ from the center $O$, in the direction along the line OA from the center to the dipole. | ||
606 | MODERN | In the upper half-plane of the $x$-axis, there is a magnetic field with strength $B$ directed into the page. A charged particle with rest mass $m_{0}$ and charge $-q (q > 0)$ is launched from the origin at high speed in the positive $y$-axis direction. A resistance with constant magnitude $q c B$ is applied opposite to the direction of velocity. It is known that the charged particle comes to a stop after turning an angle $\theta_{0}(\theta_{0} > \frac{\pi}{2})$ in the direction of velocity, and at point $A$ on the trajectory, the direction of the particle's velocity is parallel to the $x$-axis ($\theta_{A}=\frac{\pi}{2}$). Now, a smooth track is laid along the trajectory of the charged particle (taking $\theta_{0}=\pi$). Another charged particle, also with rest mass $m_{0}$ and charge $-q$, is launched from the origin along the inside of the track with the same initial momentum. At this time, a resistance proportional to velocity is applied, given by ${\vec{f}}_{0}=-q B{\vec{v}}$. The question is to determine the pressure $N_{A}$ exerted by the charged ion on the track at point $A$. | ||
162 | ADVANCED | 选用单位使得 $\hbar = c = 1$。考虑一个量子非相对论粒子，具有质量 $m = 1$（在某些单位中），电荷 $e = 1$，且没有自旋。粒子被约束在一个平面 $\mathbb{R}^2$ 上运动（具有欧几里得度量 $ds^2 = dx^2 + dy^2$），存在一个*与平面垂直的均匀磁场*。系统的哈密顿量是存在磁势 $(A_x, A_y)$ 时的标准形式，即
$$
H = \frac{1}{2} \left( (p_x - A_x)^2 + (p_y - A_y)^2 \right)。
$$
磁场的强度*未知*，但通过直接实验发现在相同强度的磁场作用下，对相同质量和电荷的粒子，该粒子在平坦的 2-环面 $T^2 = \mathbb{R}^2 / \mathbb{Z}^2$ 上运动，会产生一个具有 100 个基态的量子系统。
为确定性，我们固定规范条件
$$
A_y(x, y) = 0 \quad \text{和} \quad A_x(x, 0) = 0。
$$
某实验人员在时间 $t = 0$ 准备了这个系统的三个副本，处于状态下，具有波函数（在上述规范中）
$$
\psi(x, y; 0) = \frac{1}{\sqrt{\pi}} \exp \left[ ikx - \frac{1}{2}(x^2 + y^2) \right] \in L^2(\mathbb{R}^2)，
$$
然后检查系统在不同时间 $t$ 的演化。要求学生帮助预测他会发现什么。
写出在时间 $t = 1$ 的时间演化波函数 $\psi(x, y; 1)$； | ||
295 | MECHANICS | A particle with mass $m$ is released from rest at an unknown height $h$ and slides down a smooth track, colliding with a particle of mass $M$ located on a smooth horizontal surface. The coefficient of restitution for the collision is $e$. After the collision, $M$ will enter a vertical circular track with radius $R$ and a coefficient of friction $\mu$. This rough vertical circular track is tangent to the original smooth horizontal surface. The particle $M$ moves along the inner side of the track and detaches at the appropriate position to pass through the center of the circular track. The gravitational acceleration is $g$. Find the release height $h$ for mass $m$. | ||
368 | OPTICS | A thermometer is cylindrical (which can be considered as infinitely long), with an outer radius of $R = 20.00\mathrm{mm}$ and an inner radius of $r = 15.00\mathrm{mm}$. The region between the inner and outer walls consists of a transparent medium with refractive index $n$ to be measured. Inside the inner radius is opaque mercury that is capable of completely reflecting light. **The result should be expressed with four significant figures**.
The refractive index of the medium is measured as follows. A distant object is placed far away, and the line of sight $P$ is fixed in the direction of that object without adjusting the viewing direction anymore. The thermometer is inserted between the object and the line of sight, and the perpendicular distance $h$ from the center of the thermometer to the line of sight is gradually reduced. When this distance is $19.00\mathrm{mm}$, the human eye will first see the image of the object again along the line of sight. Based on this, the refractive index $n$ can be calculated. Furthermore, as $h$ continues to decrease, other possible images of the object can still be observed. Calculate the displacement $\Delta h$ of the thermometer along the $h$ direction when the next image of the object is seen. |
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