id
int64 10
788
| tag
stringclasses 6
values | content
stringlengths 0
4.2k
| solution
stringclasses 101
values | answer
stringclasses 101
values |
|---|---|---|---|---|
314 | THERMODYNAMICS | Entropy is an important physical quantity for studying the properties of thermodynamic processes of ideal gases. The entropy formula for an ideal gas can be written as:
$$
S = \nu C_{V} \ln T + \nu R \ln V + S_{0}
$$
where \( C_{V} \) is the molar heat capacity at constant volume for the ideal gas, and \( \nu \) is the number of moles of the gas. \( S_{0} \) is related to the choice of the zero point of entropy and is a constant that does not change with the state variables \( T \) and \( V \).
Assume the atmospheric environment has a sufficiently large volume \( \textstyle V_{0} \to \infty \) (but fixed), thus having a constant pressure \( p_{0} \) and temperature \( T_{0} \). There is an insulated container with a volume of \( V \) in the atmosphere. The container has a small hole that can be controlled to open or close. Inside the container is an ideal gas with an adiabatic index \( \gamma = C_{p}/C_{V} \), with pressure \( p = \alpha p_{0} \) less than the external pressure but with temperature consistent with the external temperature.
When the small hole is opened, the external gas enters the container. This process is slow enough for the gas inside the container to reach equilibrium, but occurs rapidly enough in terms of heat transfer between the inside and outside of the container (thereby not allowing enough time for heat transfer). Determine the total entropy change \( \Delta S_{1a} \) of the system (including the atmosphere) during the process. | ||
490 | THERMODYNAMICS | 1 mol of ideal gas undergoes a certain quasi-static process:
\[ P = 6P_0 + \frac{P_0}{V_0} V \]
where \( P_0 \) and \( V_0 \) are positive constants. For this 1 mol of ideal gas, find a new quasi-static process expression \( P'(V') \) so that its state curve passes through the state point \( (4P_0, 4V_0) \), and has the same heat capacity as the aforementioned process at each temperature. | ||
500 | MODERN | The following systems are located in a vacuum, with the known vacuum permittivity \(\varepsilon_{0}\) and vacuum permeability \(\mu_{0}\). Ignore relativistic effects and electromagnetic radiation. A large number of small magnetic needles are arranged in a row with a spacing of \(r\), and the length of the magnetic needles is much smaller than \(r\). The positions of each small magnetic needle are fixed, and the magnitude of the magnetic moments is \(m\), with the direction of the magnetic moments able to rotate within the plane. Each small magnetic needle has a moment of inertia \(I\) with respect to its own fixed axis. To simplify the analysis, we agree that only adjacent small magnetic needles interact with each other. Initially, all small magnetic needles are in a stable equilibrium state, and their magnetic moment directions are all parallel to each other. We apply a disturbance to the system, causing the direction of the magnetic moments of the small magnetic needles to slightly deviate from the direction at stable equilibrium. Under certain conditions, waves will form and propagate in the array of small magnetic needles (which can be regarded as harmonic waves). Let the angular frequency of the wave be \(\omega\). If \(\omega\) is known and satisfies the conditions for stable wave propagation, please provide the phase velocity \(v(\omega)\) of the waves in the array of small magnetic needles. | ||
263 | MECHANICS | The geometric shape uses an equilateral triangle with a side length of $L$ as the main body, with three sides extending outward into curves. Using the bottom-left vertex of the main equilateral triangle as the pole, establish a polar axis $x$ vertically upward. The three curves form a continuous function defined on $[0, 2\pi)$ in polar coordinates, with exactly three possible non-differentiable points, which are the three vertices of the original main triangle. In order for the extended shape to become a Reuleaux triangle, it must satisfy the following geometric properties: for any set of parallel lines, when at least one of the parallel lines is just tangentially "clamping" the geometric shape, the distance between the lines is the same. One special solution is, of course, a circular shape (that is, three 1/3 circle arcs joined together). In fact, there should be countless such curves of constant width, but we impose the following restrictions on a "Reuleaux triangle": ⃝1 The curves extending from each side should be congruent, and the shape should have rotational symmetry ⃝2 At the vertex positions of the main body, the curve is non-differentiable, meaning when drawing equidistant parallel lines, one parallel line passes through the vertex, and another is tangent to the opposite curve ⃝3 Composed of arc or straight line segments.
A Reuleaux triangle, when translated perpendicular to the plane of the paper, forms a cylinder in space called a Reuleaux prism. Consider a homogeneous Reuleaux prism with mass $m$, placed on a horizontal surface with its lateral side down, with a known gravitational acceleration $g$. Due to the excellent geometric properties of the Reuleaux triangle, it has inspired attempts to build a "cart" using Reuleaux triangles as wheels.
$N=10$ homogeneous Reuleaux prisms of mass $m$ are placed at equal intervals, with the horizontal distance between adjacent centroids being $L$. A large flat board with mass $m$ (long enough, not considering the prism detachment in this question) is placed horizontally as the cart body. The coefficient of friction at the contact surface is sufficient to ensure all contacts maintain pure rolling throughout the process. The initial configuration is a stable equilibrium. At this point, an initial velocity is provided to the system to the right, with the initial speed of the chassis being $v$. To ensure that the cart moves forward, solve for the minimum value of $v$, denoted as $v_{01}$. (Keep five significant digits in the decimal part.) | ||
491 | MODERN | In the inertial frame $S$, there is a point source of light with an equation of motion ${\pmb r}=(v t,0,0)$. The emitted light is monochromatic with a frequency $f$, and the space is a vacuum. At time $t$, there is an observer $P$ on the $y$-axis located at $(0,y_{0},0)$ and moving with a velocity of $(0,\dot{y}_{0},0)$. Calculate the signal frequency $f^{\prime}(y_0, \dot y_0, t)$ received by the observer $P$. | ||
632 | OPTICS | A simple telescope is composed of a thin convex objective lens $L_{o}$ and a thick convex eyepiece $L_{E}$ installed in a tube. The leftmost end of the tube overlaps with the left side of the eyepiece, and the rightmost end is the objective lens. On the right side of the eyepiece, a reference line is engraved to assist in measuring the angular size of the target object. The entire system has an angular magnification $M$. To ensure that all light that strikes the right side of the eyepiece eventually propagates to the left side without any loss of light energy, the telescope design requires the following: if a small paraxial object is placed close to the left side of $L_{o}$, it should just pass through the right side of the eyepiece and form an image on the left side. Determine the ratio of the absolute values of the radii of curvature for both sides of the eyepiece, expressed in terms of $M$. | ||
639 | THERMODYNAMICS | A thin circular disk with mass $M$, area $S$, and heat capacity $C$ is initially at a temperature $T_{1}$. It is placed in an environment filled with a monoatomic gas that maintains a constant temperature $T_{0}$. The gas molecules have a mass $m$ and a number density $n$. Gravity is neglected. The properties of the two surfaces of the disk are different. The upper surface is ideally adiabatic, meaning gas molecules reflect off this surface like balls colliding on a smooth plane. The lower surface is ideally conductive, meaning when gas molecules collide with this surface in the disk's frame, they are re-emitted as if leaking from a small hole at temperature $T$, which is the temperature of the disk, with the emission directions being randomly distributed. Note: The problem uses the following approximations: The relative temperature difference between the initial disk temperature and the gas temperature is very small, so the speed of the disk is always much less than the average speed of the gas; also, $M \gg m$. The Maxwell speed distribution law of the gas is known: $$ {\mathrm{f}}({\vec{v}})={\Big(}{\frac{m}{2\pi k T}}{\Big)}^{\frac{3}{2}}\mathrm{exp}\left(-{\frac{m}{2k T}}v^{2}\right) $$ Assume that at a certain moment, the temperature of the disk is $T$, and its speed perpendicular to the plane of the disk is $u$ (positive upwards, negative downwards). Neglect gravity. Determine the rate of change of the disk's speed $\frac{\mathrm{d} u}{\mathrm{d} t}$. | ||
765 | ELECTRICITY | A lightweight insulated plastic thin disc with a radius of $R$ is placed horizontally and can freely rotate without friction around a vertical fixed axis through the center of the disc. A lightweight small circular coil with a radius of $a$ $(a \ll R)$ is fixed on the disc's surface, coaxial with the disc. Four positively charged metal balls, each with mass $m$, are fixed at equal intervals on the edge of the disc, and each metal ball carries a charge of $q$. This device is in a uniform strong magnetic field with a magnetic induction intensity of $B_{0}$, directed vertically upward. Initially, the disc is at rest, and a constant current $I$ flows through the small circular coil in a clockwise direction (as viewed from above). If the current in the small circular coil is cut off, the disc will start to rotate. Find the magnitude of the force in the horizontal direction acting on each metal ball after the thin disc reaches stable rotation. Assume the metal balls can be considered as point masses, neglect the self-inductance of the small circular coil and the magnetic field generated by the motion of the charged metal balls. It is known that a circular coil with a radius of $a$ fixed on the disc's surface carrying a current $I$ generates a magnetic induction intensity of $B = k_{m} \frac{2 \pi a^{2}I}{r^{3}}$ at a distance $r$ $(r \gg a)$ from the center of the coil. In this formula, $k_m$ is a known constant, and the direction of the magnetic field is vertical and upward when the current in the coil flows clockwise. The electrostatic force constant is $k_e$. | ||
325 | OPTICS | There are $N$ identical, uniform, transparent dielectric spheres arranged coaxially and equidistantly from left to right. Each sphere has a radius $r$ and a refractive index of $n=3/2$. The distance between the centers of two adjacent spheres is $8r$. To the left of the first sphere, at a distance $u_1$ from its center, there is a small luminous object (located near the axis). The light emitted by this object undergoes refraction successively through the $N$ dielectric spheres, forming images. The final image distance (distance from the last image to the right vertex of the $N$-th sphere) is $\nu_{N}$. It can be proven that, as $N$ becomes very large, $\nu_{N}$ approaches a constant value. Determine this constant value, $\nu_{\infty}$.Note that this constant value should be stable against small perturbations in the initial conditions. | ||
518 | THERMODYNAMICS | Before the discovery of radioactivity (and the mass-energy equation), Kelvin (1899) suggested estimating the age of the Earth based on the process of decay in the Earth's internal heat flow. He assumed that the Earth initially had a molten crust, and the exothermic process of solidification from the outside to the inside led to a gradual increase in the thickness of the solid crust. The thickness of the solid crust simultaneously determined the rate of heat dissipation outward (according to the classical heat conduction and heat radiation theorem). This allowed him to calculate the relationship between the rate of heat dissipation and time. Given the latent heat of solid-liquid phase transition per unit mass of the crust as $L$, the density of the solid as $\rho$, the temperature of the molten material as $T$, and the current surface temperature approximately as $T_0$, as well as the Stefan constant $\sigma$ and Fourier's thermal conductivity coefficient $\kappa$, estimate the age of the Earth. | ||
422 | MECHANICS | A triangular wedge with an angle of inclination $\theta$ and mass $M$ is placed on a rough horizontal surface. A homogeneous sphere with mass $m$ and radius $r$ is placed on the inclined plane of the wedge. The sphere begins to move freely downward from rest, and the wedge does not rotate during the entire process. The gravitational acceleration is $g$. It is assumed that the static and kinetic coefficients of friction between the sphere and the wedge are both $\mu_{1}$, and the static and kinetic coefficients of friction between the wedge and the ground are both $\mu_{2}$, where $(\mu_{1},\mu_{2}>0)$.
If the sphere rolls down without slipping and there is relative sliding between the wedge and the ground, find the minimum value of $\mu_1$, expressed as a function of $\mu_2$. | ||
82 | MECHANICS | Consider the problem of the motion of an ideal rigid body under the influence of gravitational force. A stationary rigid homogeneous sphere has a mass $M$, its surface is smooth, and a homogeneous rod of length $L$ has a mass $m$, with $M \gg m$. The rod is tangent to the sphere, and at the initial moment, the point of tangency between the rod and the sphere is at a distance $\pmb{\mathcal{x}_0}$ from the center of the rod. The rod is released from rest. The gravitational constant is known to be $G$, and the radius of the sphere is $R$.
If the offset value ${\mathcal{x_0}}$ is small, the rod will perform small harmonic oscillations. Find the angular frequency of the small oscillations. | ||
668 | ELECTRICITY | A conducting sphere will generate a current under the influence of an external alternating electric field, thereby producing Joule heat. As shown in the figure, a metal conducting sphere with a radius of $R$ is placed in an electric field $E=E_{0}\cos\omega t$. Given the vacuum permittivity $\epsilon_{0}$ and the sphere's resistivity $\rho$, ignore all electromagnetic effects other than the electrostatic effect, and find the average power of heat generation $P$ when $\omega \neq 0$. | ||
729 | MECHANICS | A bomb explodes on the ground, and after the explosion, all fragments are ejected at the same speed $u$, with their directions confined to a narrow cone with an angle not exceeding \(\alpha\) to the vertical (the angles are uniformly distributed within this range). All fragments eventually land, and the landing is completely inelastic. Let the mass of the bomb be $M$. Define the radial density distribution \(\rho(r)\) such that \(\rho(r)2\pi r\,dr\) is the mass of debris within the range of radius \(r\) to \(r+dr\) from the center. Find \(\rho(r)\) within the maximum range \(R\) (retaining terms up to \(r^2\) in the approximation.
| ||
444 | THERMODYNAMICS | Given that the ambient temperature is $T_{a}$ and the atmospheric pressure is $p_{a}$. The initial temperature in a tire is also $T_{a}$, with the pressure being $p_{0}$. The maximum pressure in the tire right after inflation is noted as $p_{m}$, and the temperature at this time is denoted as $T_{m}$. Afterward, it slowly cools down to the ambient temperature $T_{a}$, and the pressure decreases to the final target pressure $p_{f}$. Assuming the air is an ideal gas with a constant heat capacity ratio $C_{p}/C_{\nu} = \gamma$, the tire volume $V_{0}$ is considered constant, and the inflation process is adiabatic with all frictional dissipation neglected. The universal gas constant is $R$.
Using a large air compressor to inflate the tire, assume the pressure of the air in the compressor remains constant during inflation, and its temperature is $T_{a}$. What is the minimum air pressure $p_{c}$ in the compressor needed to achieve the final pressure $p_{f}$? (The answer to the above question should only contain $\gamma$, $p_{0}$, and $p_{f}$.) | ||
337 | MECHANICS | Small balls $a$ and $b$, with masses $m_{a}$ and $m_{b}$ respectively, are placed on a smooth insulated horizontal surface. The two balls are connected by an insulating light spring with an original length of $l_{0}$ and a spring constant of $k_{0}$. If the two small balls are given equal charges of the same sign, the spring's length is $L_0$ when the system is in equilibrium. Let $K$ be the electrostatic force constant. Find the frequency of small oscillations of the system comprising the two balls and the spring (neglect the effects of induced charges). The final answer should not include any physical quantities not provided in the problem statement. | ||
151 | MODERN | The so-called photon rocket refers to a spacecraft that can achieve propulsion by converting its own rest energy into photons for emission. Consider the following theoretical problem:
A photon rocket undergoes two stages of acceleration in a vacuum from an initial state to a final state. It is known that the photon rocket is initially at rest with a rest mass of $m$. In the final state, the photon rocket reaches a velocity of $0.8c$, and its rest mass decreases to $0.25m$. Here, $c$ is the speed of light. It is also known that if we take the instantaneous inertial frame of the spacecraft at the end of the first stage of acceleration (i.e., an inertial frame in which the spacecraft is momentarily at rest), the direction of acceleration of the spacecraft remains fixed in both stages, but the magnitude of acceleration is unspecified. Additionally, in this frame, the angle between the directions of acceleration in the two stages is $\alpha$. Find the minimum value of $\alpha$. | ||
637 | MODERN | A railroad is moving at a high speed with velocity \( v \) perpendicular to the direction of the railroad relative to the observer. In the railroad's reference frame, there is a rectangular train with the same width as the railroad, moving with velocity \( 2v \) parallel to the railroad with respect to the railroad. Considering relativistic effects, calculate the angle between each of the four sides of the train and the railroad as seen in the observer's reference frame. It is known that all four angles between the sides and the railroad are equal. Please express the result in terms of \( v \) and \( c \), where \( c \) is the speed of light. | ||
468 | MECHANICS | You are requested to solve the following physics problem and present the final answer using \boxed{}: Four uniform-density slender rods $M A, A P, B N, P B$ are smoothly hinged together to form an "M"-shaped structure, with $M$ and $N$ constrained to move along horizontal rails, and $P$ constrained to move along a vertical rail. The width of the rails and the diameter of the rods are negligible. The rails do not affect the rotation of the rods. An $O x y$ coordinate system is established, and at the initial moment, the coordinates of the endpoints of the rods are given as: $M(-2l,0)$, $N(2l,0)$, $P(0,l)$, $A(-l,2l)$, $B(l,2l)$.
Starting from the initial moment, motors control the movement of rod ends $M$ and $N$ along the negative and positive directions of the $x$ axis, respectively, with velocity $v$ and acceleration $a$. Meanwhile, rod end $P$ moves along the positive direction of the $y$ axis with velocity $2v$ and acceleration $2a$. The mass per unit length of the rods is $\lambda$, and the gravitational acceleration is $g$. Ignoring all losses, determine the power exerted by the motors at the initial moment. | ||
577 | THERMODYNAMICS | One mole of a certain substance is a simple $p,v,T$ system. The coefficient of volumetric expansion under any circumstance is
$$
\alpha=3/T
$$
The adiabatic equation under any circumstance is
$$
p v^{2}=C
$$
And the constant pressure heat capacity of the system under any circumstance is exactly:
$$
c_{p}=\lambda{\frac{p v}{T}}
$$
Here, $\lambda$ is a dimensionless constant independent of the state. Find all possible values of $\lambda$.
Hint 1: According to the third law of thermodynamics, the absolute entropy should be zero at a temperature of zero for any volume.
Hint 2: Entropy $S$ can be expressed as a function of $T,p$, i.e., $S=S(T,p)$. In this case, we have $\begin{array}{r}{\mathrm{d}S=\left(\frac{\partial S}{\partial p}\right)_{T}\mathrm{d}p+\left(\frac{\partial S}{\partial T}\right)_{p}\mathrm{d}T}\end{array}$, where the subscript T indicates the partial derivative is taken at constant T, and $\begin{array}{r}{\left(\frac{\partial S}{\partial p}\right)_{T}=-\left(\frac{\partial v}{\partial T}\right)_{p}}\end{array}$. If $\mathrm{d}f=A\mathrm{d}x+$ $B\mathrm{d}y$, then $\begin{array}{r}{\left(\frac{\partial A}{\partial y}\right)_{x}=\left(\frac{\partial B}{\partial x}\right)_{y}.}\end{array}$ | ||
417 | MODERN | There is a rocket in space with an initial rest mass of $m_{1}$, moving at a very high velocity $\mathbf{v}$ (v can be compared with the speed of light c). Starting from a certain moment, the rocket is ejecting gas at a constant high speed $\mathbf{u}$ in the direction perpendicular to its own velocity as observed in its reference frame. The mass of the gas ejected per unit time in its reference frame is $\lambda$. Considering relativistic effects, find the angle turned by the rocket in its direction of motion after time t in the ground reference frame. | ||
682 | MECHANICS | A homogeneous disk with mass \(m\) and radius \(R\) is suspended by three light strings of length \(L\). In the equilibrium state, the three strings are equally spaced and remain vertical. If the disk is rotated through a small angle \(\theta\) around its axis and then released, find the period of the disk's motion.",
| ||
159 | MODERN | A sphere in vacuum has a radius of $R$ in its rest frame, and its surface is uniformly charged with a charge $Q$. It is moving with a constant velocity $v=\beta c$ relative to the ground frame, where $c$ is the speed of light in vacuum. Given the vacuum permittivity $\varepsilon_0$, find the electromagnetic field energy in all space in the ground frame. | ||
147 | MECHANICS | A light thin wire of length $L$ is wound around a rough cylinder with radius $R$ to form a coil. The coil is subjected to a pulling force at one point, parallel to the axis of the cylinder. Due to the friction between the coil and the cylinder, with the coefficient of friction $\mu$, there exists a critical length $L_{0}$ beyond which the coil cannot remain stationary on the surface of the cylinder no matter what. Determine this critical length $L_{0}$. | ||
239 | MECHANICS | The influence of the gyro effect: In the 20th century, Arnold Sommerfeld, Felix Klein, and Fritz Noether jointly proposed an explanation—the gyro effect. The gyro effect is essentially a manifestation of the angular momentum theorem. Under the action of external torque, the angular momentum of a rotating object changes, leading to the generation of precession angular momentum. In other words, under the influence of the gyro effect, a rapidly rotating object will not fall down directly but will instead experience a change in its rotational direction.
$AB$ is fixed to the ceiling. A lightweight rod $BO$ with length $L$ is rigidly connected to the center of a uniform disk with mass $m$ and radius $R$. The rod $BO$ is connected to $AB$ by a ball joint, allowing it to rotate freely in all directions. The disk rotates with an angular velocity $\omega$ around $BO$. Solve for the precession angular velocity $\Omega$ required to keep $BO$ horizontal. | ||
472 | ELECTRICITY | Consider a uniform thin spherical shell whose center of mass is fixed but is free to rotate about the center of mass. The shell is uniformly charged, with a total charge of $Q$, mass $m$, and radius $R$. Suppose that there are many such thin spherical shells distributed in space, with their centers of mass fixed but free to rotate about the center of mass. The radius of each shell is $R$, their number density is $n$, and it satisfies $n R^{3} \ll 1$. Considering the difference between the macroscopic average magnetic field $\vec{B}$ of the system and the effective external magnetic field experienced by each thin shell, and accounting for the effects of this difference, determine the relative magnetic permeability $\mu_{r}$ of the system. The vacuum permeability $\mu_0$ is given. | ||
170 | OPTICS | The Nobel Prize-winning "optical tweezer" technology actually uses the single-beam gradient force to form an energy potential well for controlling small molecules. We are attempting to use a similar principle to design a small robot driven by light. We assume this micro-robot can be considered as a disk with a radius $a$, which is much larger than the wavelength of light used. We use a beam of parallel light with a radiation energy density of $u_{0}$ to uniformly illuminate the surface of the robot perpendicularly. The entire disk is within the illumination range of the parallel light. Assume that the exterior protective layer of the micro-robot is a medium whose refractive index continuously changes according to $n=\alpha/\left(z+\beta\right)$. Here, $\alpha$ and $\beta$ are undetermined constants, and $z$ is the depth into the surface. The refractive index changes continuously from $n_{1}$ at the outer surface $(z=0)$ through a transition layer of thickness $l_{0}$ to $n_{2}$ at the inner surface. The radius of the robot is much larger than the wavelength $\lambda$ of the incident wave. Calculate the reflection coefficient $R$ of this protective layer, assuming the layer is extremely thin. Express $R$ in terms of $n_1$ and $n_2$, without using any approximations not provided in the problem statement. | ||
471 | ELECTRICITY | A rigid insulating cubic frame with side length $a$, labeled $A B C D - E F G H$, has charges fixed at each vertex with a magnitude of $q$. The charges at vertices $B$ and $H$ are negative, while the charges at vertices $A$, $C$, $D$, $E$, $F$, and $G$ are positive. A rigid and smoothly lined fixed track (whose inner diameter is much smaller than $a$) passes through vertices $B$ and $H$. At the midpoint $O$ of track $\vec{B H}$ (which is also the center of the cube), a point charge with a negative charge $-Q$ and mass $m$ is placed. Neglect relativistic effects, with vacuum permittivity $\varepsilon_{0}$ and vacuum permeability $\mu_{0}$, and disregard gravity. When the cubic frame is fixed, discuss the period $T_r$ of the small oscillations of the point charge near equilibrium at $O$ due to a slight disturbance. | ||
169 | MECHANICS | The inhabitants of Blue Planet have always fantasized about one day getting rid of the inefficient rocket as a means of transportation and directly creating a "Sky Ladder" from the Earth to the heavens. Some call it the "Space Elevator," while others refer to it as the "Orbital Lift." However, they do not realize that due to the limitations imposed by physical laws, building a space elevator on Blue Planet is unrealistic. In contrast, years later, on Mars, which is further from the Sun, space elevators can "stand tall."
As early as in the Book of Genesis in the Bible, humans hoped to jointly construct a giant tower reaching from the ground to the sky—the Tower of Babel—to proclaim their fame. The person who first proposed the concept of a space elevator should be the Russian "father of rockets," Tsiolkovsky. The space elevator he envisioned would extend from the ground to the geosynchronous orbit tens of thousands of kilometers high. As the elevator ascends, the internal gravity gradually decreases. When cargo reaches the endpoint via the elevator, its speed already allows it to maintain synchronous orbital motion around Earth. Therefore, the synchronous orbit station is in a state of complete weightlessness.
In 1979, Arthur C. Clarke's science fiction novel "The Fountains of Paradise" introduced the revolutionary concept of the space elevator to the public's attention. Since then, space elevators have been widely present in various sci-fi works. Whether it is the space elevator made of advanced nanomaterials in "The Three-Body Problem" or the three orbital lifts operated by Union, AEU, and Human Reform League in "Mobile Suit Gundam 00," the basic concepts and principles are the same.
Nowadays, we occasionally hear some related news reports, either from Japan's company "Obayashi Corporation" planning to build the first space elevator by 2050, or from a Canadian company proposing new space elevator schemes. Regardless of whether these are commercial companies creating hype, such news indeed prompts 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 needed to build a space elevator? We will conduct a detailed analysis in this matter (the space elevators discussed here are all constructed at the equator, with no consideration for any curvature).
In Tsiolkovsky's vision, the space elevator following Earth's rotation would reach a certain height, after which a satellite released freely would automatically become a geostationary satellite. Calculate the height $l_{1}$ of the space elevator in this model. Given are: gravitational acceleration $g$, Earth's radius $R$, and Earth's rotation period $T_{day}$. | ||
678 | ELECTRICITY | Calculate the magnetic field produced in space by a moving electric dipole (with dipole moment \(\vec{p}\) and velocity \(\vec{v}\)) under the following conditions: the movement is parallel to the dipole moment direction (\(\vec{v} // \vec{p}\)), and the velocity \(v\) is much smaller than the speed of light \(c\). Use the dipole as the origin, designate the direction of \(\vec{v}\) as the polar axis, and represent the system in spherical coordinates \((r,\theta,\phi)\). Only the \(\phi\) component of the magnetic field \(B\), denoted as \(B_{\phi}\), needs to be provided. | ||
16 | MECHANICS | Consider a child with mass \( m \) sitting on a swing, the child can be regarded as a point mass with the mass concentrated at the seat plank. Ignore the mass of the other parts of the system. The distance from the swing seat plank to the pivot is \( l \). At this time, consider the frictional torque \( M_f = a \) (where \( a \) is a constant) at the swing's suspension point. There is someone behind who applies an impulsive torque \( J_0 \) to the swing every time it reaches the furthest back position. Find the difference in speed rates of the child after passing the lowest point twice successively when the motion reaches a steady state (with gravitational acceleration \( g \) and assuming the swing angle is relatively small). | ||
515 | MECHANICS | Collisions are one of the simplest mechanical interactions between objects, and they are the process of energy and momentum exchange between objects. The cascade of one-dimensional collisions resulting in chain events provides a simple model for a series of natural phenomena dominated by continuous collisions. When studying such systems, the physical quantities we are interested in are the proportion of energy or momentum transmitted by the collision cascade and its dependence on the coefficient of restitution \( e \) and the collision mass ratio. We arrange \( n(n\gg1) \) balls (with masses \( m_{i}(i=1,2,...,n) \), adjustable) between an initial ball with mass \( M \) (fixed) and an end ball with mass \( m \) (fixed). Only the first collision between adjacent balls is considered. All collisions are inelastic with a coefficient of restitution \( e \) (very close to 1).
Recently, Ricardo and Lee proved that when the mass distribution of the cascade ball chain is such that the mass \( m_{i} \) of each ball is the geometric mean of the masses of the two adjacent balls \( \sqrt{m_{i-1}m_{i+1}} \), the efficiency of kinetic energy and momentum transfer between the initial and final balls is maximized. For a system of \( n \) balls, the maximum transfer efficiencies of kinetic energy and momentum between the initial and final balls can be expressed as \( r_{K n},r_{p n} \).
We aim to optimize the efficiency of kinetic energy and momentum transfer between the initial and final balls in this chain event. Provide the number of balls \( n \) that should be arranged between the initial and final balls for maximum transfer efficiency. (Expand to the lowest order in \(\delta=1-e\), with the answer expressed in terms of \( m \), \( M \), and \(\delta\)). | ||
176 | MECHANICS | A cylindrical thin-walled water bucket is fixed on a horizontal surface, with a cross-sectional area of \( S_0 \). Inside, there is water at a height of \( a + b \). Now, a small hole is suddenly opened at a distance \( b \) from the bottom, extending in a horizontal direction (the hole's area \( S_1 \ll S_0 \)), causing the water to start leaking. The height \( h \) of the water above the hole gradually decreases from \( a \) to 0. The density of the water is \( \rho \), and the gravitational acceleration is \( g \). The process goes from the beginning until all the water has flowed out, treating the water as an ideal fluid and ignoring the higher-order effects of internal flow and the bucket's shape and size. Find the functional relationship between the linear density of water distribution on the ground \(\lambda\) and the horizontal distance \( x \) from the bucket wall (approximating the water trajectory as a thin line). | ||
626 | OPTICS | After parallel light enters a Michelson interferometer, it is split into two beams by the beam splitter. These two beams travel different paths and interfere at the observation screen. Mirrors $M_{1}$ and $M_{2}$ are plane mirrors with different distances to the beam splitter, causing the two beams that eventually reach the observation screen to have an optical path difference of $\Delta L$. If the incident light consists of two wavelengths that are very close together, with an average wavelength of $\lambda$ and a wavelength difference of $\Delta\lambda{(\Delta\lambda\ll\lambda\bigr)}$, and both beams have the same intensity upon reaching the observation screen, calculate the relationship of the interference contrast $M$ with the change in $\Delta\lambda$, $M(\Delta\lambda)$. | ||
108 | OPTICS | When electromagnetic waves undergo total internal reflection at an interface, it has been observed that under certain circumstances, the reflected wave experiences a lateral shift along the interface relative to the point of incidence. This phenomenon is historically known as the "Goos-Hänchen shift." The core concept is that during total internal reflection, the reflected wave acquires an additional phase $Φ(\theta)$, which depends on the angle of incidence $\theta$ (typically decreasing as the angle of incidence increases, becoming zero at the critical angle $\theta$ for total internal reflection, with its derivative with respect to the angle of incidence becoming infinite at the critical angle). When the incident wave is not a perfect plane wave, electromagnetic waves with closely spaced but different angles of incidence can interfere. The condition for this interference to occur is that the reflected waves from different angles of incidence must achieve equal phase by undergoing a lateral shift $d$ along the interface. Given that the wavelength of the electromagnetic wave is $\lambda$, and the phase shift in the "Goos-Hänchen shift" is $Φ(\theta)$, determine the lateral shift $d$. | ||
363 | THERMODYNAMICS | The reversible Carnot cycle is an ideal cycle process; actual processes, due to irreversible factors such as thermal resistance, heat leakage, and friction, are always irreversible. Now, by solely considering the presence of thermal resistance (ignoring factors such as heat leakage and friction), an ideal cycle process—the internally irreversible Carnot cycle—is introduced. The internally irreversible Carnot cycle satisfies the following three conditions:
(i) The working substance undergoes a quasi-static Carnot cycle;
(ii) Due to the thermal resistance between the working substance and the heat source, the temperatures $T_{1},T_{2}$ of the two isothermal processes that the working substance undergoes are different from the heat source temperatures $T_{H},T_{L}$, specifically: $T_{H} > T_{1} > T_{2} > T_{L}$. In this way, there is a finite temperature difference between the heat source and the working substance, allowing heat conduction to occur over a finite period, resulting in power output from the cycle;
(iii) Heat conduction satisfies Fourier's law of heat conduction: $\frac{\mathrm{d}Q}{\mathrm{d}t}=\kappa\Delta T$, where $\pmb{\kappa}$ is a known proportionality coefficient. It is assumed that the adiabatic processes proceed very quickly.
Compared to the ideal Carnot cycle, the internally irreversible Carnot cycle is of more practical significance. Using this model:
Find the total entropy increase of the system per unit time when the power output of the internally irreversible Carnot cycle is maximized, using $\kappa,\:T_{H},\:T_{L}$ for representation, assuming the heat source temperatures are fixed. | ||
754 | THERMODYNAMICS | Consider a closed system with a constant volume, initially containing liquid and vapor phases of the same substance in equal molar amounts. The initial pressure and temperature of the vapor are $P_{1}$ and $T_{1}$, respectively. Then, under the condition of keeping the total volume of the system unchanged, the system is heated to $T_{2}$. Assuming that the vapor satisfies the ideal gas condition during this process, the molar volume of the liquid phase is $v_{l}$, the constant pressure molar heat capacities of the gas and liquid are $C_{p}^{g}$ and $C_{p}^{l}$ respectively, and the molar latent heat of phase transition for the liquid is $\Delta h$, which is independent of temperature and pressure. Find the ratio $f_{l}$ of the molar number of the liquid phase to the total molar number at the temperature $T_{2}$. | ||
664 | OPTICS | The flow of the medium will affect the additional bending of the light. Below the $x$-axis is a vacuum, and above the $x$-axis is a uniform medium moving at a constant speed of $\beta c$ in the positive $x$-axis direction, with a refractive index of $n$. Now consider a beam of light obliquely incident into the medium from below. The speed of light in a vacuum is known to be $c$, and the angle of incidence is $i$. Considering relativistic effects, find the angle of refraction $t$. | ||
782 | ELECTRICITY | To consider the motion of an electron in a hydrogen atom as uniform circular motion around the hydrogen nucleus, when studying the magnetic effect of the electron's motion, the electron's motion can be equivalently considered as a loop current, with the radius of the loop equal to the orbital radius of the electron $r$. Now, apply an external magnetic field to a hydrogen atom, with the magnetic field having a magnetic induction strength of $B$, and a direction perpendicular to the orbital plane of the electron. At this time, the equivalent current of the electron's motion is represented by $I_{1}$. Now reverse the external magnetic field, but the strength of the magnetic induction remains unchanged, still $B$. At this time, the equivalent current of the electron's motion is represented by $I_{2}$. Assume that when the magnetic field is applied and when it is reversed, the position of the hydrogen nucleus, the orbital plane of the electron's motion, and the orbital radius all remain unchanged. Calculate the difference in the equivalent current of the electron's motion before and after the external magnetic field is reversed, that is, how much is $ \mid I_{1}-I_{2} \mid$? Use $m$ and $e$ to represent the mass and charge of the electron, respectively. | ||
217 | MECHANICS | An infinitely long uniformly negatively charged line with a charge linear density of $-\lambda$ is fixed in space and vertically connected through two light strings of length $b$ to the ends of an insulating rod of length $a$. The rod has a mass $m$, is uniform, and carries a uniformly distributed charge with a linear density of $+\lambda$. The gravitational acceleration is $g$. The rod can be normally suspended below a critical mass $m_0$. Under the condition that $m > m_0$, if the ends of the rod are subjected to equal amplitude perturbations in the vertical direction perpendicular to the plane, the system can undergo small oscillations. Calculate the angular frequency $\omega$ of these oscillations. | ||
300 | MECHANICS | On a horizontal surface stands a homogeneous ring with a radius of $R$ and a mass of $m$. The coefficient of friction between the ring and the ground is infinite. A disk with the same mass $m$, and mass evenly distributed, can roll without slipping inside the ring. The disk has a radius of $r$. The gravitational acceleration is $g$. Initially, the system is at rest, with the disk in contact with the inside of the ring, and the centers of the disk and the ring are at the same horizontal height. Release the system and determine: the speed $v$ of the ring when the disk reaches the lowest point. | ||
736 | THERMODYNAMICS | There is a cone-shaped cylinder with internal gas separated from external gas by a special piston. The mass of the piston is not considered, but its area can change. Its edge is always fixed on the inner surface of the cone, moving without sliding friction, and the piston remains in a single plane during motion (always a cross-section of the cone), without bulging or sinking. The piston can be viewed as a thin film with variable area stretched by a ring-shaped rubber band, with the film not providing any force. The original length of the rubber band is $0$, and the tension is proportional to its length, where: ${F}_{T}=k L$. The angle between the cone's slant height and the axis of symmetry is $\theta$. Assume initially there are $n\ \mathrm{mol}$ of monoatomic ideal gas in the cylinder with temperature $T$, and a heat quantity $Q$ is given to these gases. Find the pressure of the gas at this time. The ideal gas constant $R$ is known. Atmospheric pressure is ignored. | ||
527 | MECHANICS | The surface tension coefficient of a soap bubble is $\sigma$. A soap bubble is placed in a vacuum, and the molar specific heat at constant volume of the internal gas is $C_v = \frac{3}{2}R$. Given the mass of the soap bubble $m$, and the equilibrium radius of the bubble after change as $R_{0}$, determine the vibration period $T$ of the soap bubble after applying a disturbance by adding a charge $Q$, where the bubble maintains its spherical shape. The vacuum permittivity is $\epsilon_0$. | ||
746 | MECHANICS | There is a sufficiently high smooth cylinder with a radius of $R$, vertically fixed in place. There is a vertical smooth track on the surface of the cylinder. Now we have an elastic thick rope (it can bend, but it is straight when not under force) with a radius of $r$, smooth surface, Young's modulus of $E$, negligible mass, and an original length of $l_{0}$. It is assumed that the thick rope can only elongate in the length direction, while changes in thickness are to be ignored. Now, one end of the thick rope is fixed at the top of the track (it cannot slide up and down), and then the rope is wrapped around the cylinder in a counterclockwise direction from top to bottom, making a total of $N$ turns. Throughout the process, due to the smooth surface of the thick rope, its shear deformation can be ignored. Additionally, because $l_{0}$ is much larger than $r$, the complex situation at the ends of the rope can be ignored. Suppose the angle between the midpoint of the rope at any point and the vertical direction is $\theta$. Find the elastic potential energy $V_{\mathrm{elastic}}$ of the rope. (Note: Consider the energy change caused by the bending of the rope)"
| ||
537 | OPTICS | Sometimes we see a slightly upward-curved colorful halo in the sky. This is not a rainbow. This halo is caused by the refraction of light in ice crystals, so it is called an ice halo. There are many types of ice halos, and one of them is called the circumhorizontal arc. Let's explore this phenomenon.
In the phenomenon of the circumhorizontal arc, the ice crystals involved have flat hexagonal shapes. Due to the resistance encountered while falling, these ice crystals tend to align their top and bottom faces horizontally. When light enters from the side of a hexagonal ice crystal and refracts twice before exiting from the bottom face, a circumhorizontal arc is formed. Assume the altitude angle of the sun at this time is $\theta$, and the refractive index of ice is denoted as $n$.
Find the lower bound of the central angle of the circumhorizontal arc for different altitude angles $\theta$, providing the result in an expression containing $n$. | ||
778 | OPTICS | |||
679 | ELECTRICITY | A uniformly charged homogeneous rigid small sphere has a radius of $R$, electric charge $Q$, mass $m$, and a moment of inertia of $s m R^{2}$. Now, suppose the small sphere is placed on a sufficiently rough turntable which rotates at a constant angular velocity $\Omega$. A uniform magnetic field $B$ is perpendicular to the plane of the turntable and is in the same direction as the turntable's angular velocity. Assuming there is no gravity in the space and the small sphere will not detach from the turntable. It is known that the differential equation for the motion of the small sphere's center of mass can be written in the form $$ m\dot{\vec{v}}=Q\vec{v}\times\vec{B_{e}}-k_{e}\vec{r} $$ where $k_{e}$ and $\vec{B_{e}}$ are effective parameters to be determined. You only need to find the product $k_{e}|\vec{B_{e}}|$. | ||
661 | MECHANICS | In movies, there are often scenes like this: diving underwater immediately when there's an explosion on water, or surfacing immediately when there's an explosion underwater, seemingly to avoid the damage caused by the explosion. Let's study this problem below. In space, below the $y$-axis is water, and above the $y$-axis is air. A sound (with small amplitude) is transmitted from the water, where the sound speed and density are $c_{1}, \rho_{1}$ respectively, to the air, where the sound speed and density are $c_{2}, \rho_{2}$ respectively. The incident angle is $\theta_{1}$. Find the ratio of the transmitted sound power to the incident sound power (expressed in terms of $c_{1}, \rho_{1}, c_{2}, \rho_{2}, \theta_{1}$) (Hint: Both water and air are fluids, and their shear modulus is 0). | ||
154 | MODERN | In approximately flat spacetime, far away from any celestial bodies, four identical spaceships, A, B, C, and D, each have a rest length of $l_0 = 1.25 c t_0$ (where $c$ is the speed of light in vacuum, and $t_0$ is a known constant) and move uniformly in a straight line. Each spaceship is equipped with two fixed positioning rods $x_i$ and $y_i$ (where $i = A, B, C, D$, hereafter referred to as "positioning rods"), with rod $x_i$ along the direction of the spaceship.
Initially, the tails of spaceships A, B, C, and D (denoted as $a_i$) coincide.
The observed motion of the spaceships and directions of the positioning rods in different reference frames are:
1. In the reference frame of spaceship A, rod $y_A$ is perpendicular to rod $x_A$.
2. In the reference frame of spaceship A, spaceship B moves at a velocity $v$ along the direction of rod $x_A$, and the two positioning rods of B are parallel to those of A.
3. In the reference frame of spaceship B, spaceship C moves at a velocity $v$ along the direction of rod $y_B$, and the two positioning rods of C are parallel to those of B.
4. In the reference frame of spaceship C, spaceship D remains stationary relative to spaceship A, and the two positioning rods of D are parallel to those of C.
The spaceships communicate with each other via electromagnetic waves, with each spaceship receiving electromagnetic waves at its bow (denoted as $b_i$) and emitting electromagnetic waves from its tail. In the reference frame of spaceship A, the initial moment is marked as $t=0$, and it is known that $v=0.6c$. Starting from $t=0$, the spaceships take turns to emit contact signals with electromagnetic waves in the order A→B→C→D→A→⋯, and in the reference frames of each spaceship, receiving the signal emitted by the previous spaceship and starting to emit a signal to the next spaceship occurs simultaneously. Find the moment when spaceship A receives the signal emitted by D for the first time (keep 4 significant digits in the coefficients in the result). | ||
467 | MODERN | Solve the following physics problem, and frame the final answer using \boxed{}:
Consider the following problem, which requires taking relativistic effects into account. A particle with a rest mass of $m$ starts from the origin and undergoes one-dimensional motion under the influence of a linear restoring force $F = -k x$. Its initial kinetic energy is $E_{k}$, and the speed of light in a vacuum is $c$. Calculate the period of motion $T^{\prime}$ when $E_{k}$ is maximized. | ||
770 | ELECTRICITY | Consider a photonic crystal in the shape of a cube with side length 𝑎 and a relative permittivity of $ \varepsilon_{r1}$. Inside, there are impurities with a radius of 𝑏, a relative permittivity of $ \varepsilon_{r2}$, and a number density of $n$. Industrially, sparse pores are dug vertically through the photonic crystal on one of its faces, with a number density of $N$ and a radius of 𝑐. During problem-solving, edge effects do not need to be considered, and the relative permittivity of air is 1. Furthermore, $a \gg b, a \gg c$. Determine the effective relative permittivity $ \varepsilon_{3}$ obtained in the direction along the pores. | ||
186 | MECHANICS | In daily life, due to various influences, a chandelier will vibrate near its equilibrium position, and we establish a model to study this phenomenon. A homogeneous disk with mass $\pmb{m}$ and radius $\pmb R$ is suspended by three light strings of length $I$. The suspension points above form an equilateral triangle with side length $\sqrt{3}R$, and the points where the strings attach on the lower disk also form an equilateral triangle with side length $\sqrt{3}R$. In the equilibrium state, the three strings are equally spaced and remain vertical. A homogeneous disk with mass $\pmb{m}$ and radius $2R$ is suspended in a similar manner using three light strings below the above disk. The suspension points form an equilateral triangle with side length $\sqrt{3}R$, and the points of attachment on the lower disk form an equilateral triangle with side length $2\sqrt{3}R$. The height difference between the two disks is $l$. Find the maximum angular frequency among the possible vibration modes of the two disks.
| ||
534 | ELECTRICITY | A positive charge \( Q \) is fixed at the center of a uniform magnetic field, which is distributed in a circular area. The direction of the magnetic field is perpendicular to the plane of the paper, pointing inward. The magnetic induction intensity is \( B \). A particle with mass \( m \), carrying a negative charge (charge quantity \( -q \)), moves in a clockwise circular motion at a constant rate around the fixed positive charge \( Q \) within the plane of the paper. The radius of the circular motion is \( R \). Ignore the effects of gravity and friction. Find:
If the magnetic field suddenly vanishes and the time taken for the magnetic field to disappear is very short (much less than the period of the circular motion), what is the period of the particle's motion if it continues to move periodically after the magnetic field disappears? | ||
514 | MECHANICS | A solid sphere with mass \(m\) and radius \(a\) is placed on a spherical surface with radius \(R\), starting from the initial angle \(\theta_0\) and undergoing pure rolling without slipping. When the sphere rolls to a certain angle \(\theta\), find the expression for the frictional force \(f\) on the sphere. Only consider scenarios where separation has not occurred. | ||
40 | MECHANICS | A rectangular wooden block with height $H$ and density $\rho_{1}$ is gently placed on the water surface, where the density of water is $\rho_{2}$, and $\rho_{1} {<} \rho_{2}$. The magnitude of gravitational acceleration is $g$. Only consider the vertical translational motion of the wooden block, neglect all resistance in the direction of motion, and assume that the height of the water surface remains unchanged. When the block is slowly pressed down to a certain depth and then released, the block's top edge enters the water (i.e., the block is fully submerged). After this, if the block does not leap out of the water surface, find the maximum distance $h$ by which the top of the block descends from its original equilibrium position. | ||
645 | MECHANICS | This problem discusses the situation of a snowball rolling down a slope under pure rolling conditions. Suppose at a certain moment there is a spherical snowball rolling down an inclined slope with an angle of inclination $\theta$. It is assumed that as the snowball rolls downward, the increase in the radius of the snowball is the same each time it completes a full rotation, and during this process, the density of the snowball remains constant. It is known that at the initial moment the radius of the snowball approaches $0$. Assuming the snowball and the snow are thermally insulated from the external environment, calculate how much the temperature of the snowball increases after sliding a distance $L$ along the slope. It is known that the specific heat capacity of the snow is $c$, the acceleration due to gravity is $g$, and no phase change occurs when the temperature of the snow rises.
| ||
255 | OPTICS | A massless rod can rotate without friction about a pivot along the $z$-axis at its center. Plane light waves propagate along the $x$-axis from left to right. The light's electric field is given by the equation
$$
\vec{E}(x,t)=E_{0}\hat{y}\cos(k x-\omega t)
$$
At the ends of the rod are disks parallel to the rod and the $z$-axis. One side of each disk is a 100% reflective perfect mirror, while the other side is 100% absorptive. The initial orientation of the disks is such that light always hits the absorptive side at the top of the rod (above the pivot), and hits the reflective side at the bottom. The mass of each disk is $m$, and the radius is $r\circ$. Assume the distance from the pivot to the center of each disk, $R$, satisfies $R\gg r$. Find the average angular acceleration of the rod for one full rotation. (In the answer, represent physical constants using $\epsilon_0$.) | ||
618 | ELECTRICITY | A uniform sphere with a mass of $m$ and a radius of $R$, uniformly carrying a charge $Q$, rotates about the $z$ axis (through the sphere center) with a constant angular velocity $\omega$. The calculation formula for the magnetic moment is known to be $\sum_{i}I_{i}S_{i}$, where $I_{i}$ is the current of the current loop and $S_{i}$ is the area vector of the current loop. The sphere is placed on an infinitely large superconducting plane with the $z$ axis vertical and perpendicular to the plane surface, with gravitational acceleration $g$. Assuming that the selection of parameters ensures such a balanced state exists (i.e., no need to discuss the reasonableness of the parameters), find the distance $h$ from the sphere center to the plane surface. The vacuum permittivity $\varepsilon_0$ and the vacuum permeability $\mu_0$ are known. | ||
328 | MECHANICS | Consider the process of a step ladder collapsing and rebounding after hitting a wall. The step ladder can be viewed as two uniform slender rods of length $L$, hinged together at the top $A$ with a smooth, lightweight hinge. The mass of the left rod is $m$, while the mass of the right rod can be ignored. The coefficient of static friction and kinetic friction between the bottom end $B$ of the left rod and the ground is $\mu$. The bottom end $C$ of the right rod makes smooth contact with the ground. Initially, the two rods are almost aligned, forming an angle of zero. The ground is horizontal, and the end $C$ is at a distance of $\sqrt{2}L$ from a vertical wall corner $O$. The entire rod is released from rest, with gravitational acceleration $g$. The coefficient of friction satisfies $15 \sqrt{10}/128<\mu<1$. Solve for the energy lost at the moment of collision. | ||
17 | MECHANICS | During winter break, a student named Xiaoliang discovered an interesting phenomenon while playing with a slingshot. He noticed that when the projectiles had almost the same volume, overly light projectiles and overly heavy projectiles both failed to travel very far. Being thoughtful, Xiaoliang decided to build a model to analyze this process.
Assume the slingshot consists of a rubber band with a spring constant $k$, fixed at both ends, and has an original length of $d$. The fixed ends are also $d$ apart. When firing, a projectile is placed in the middle of the rubber band, and the rubber band is pulled back, generating a displacement $l$ for the projectile,and the direction of the displacement is perpendicular to the initial direction of the rubber band,letting go completes the firing. Assume that the elastic potential energy of the rubber band is fully converted into the kinetic energy of the projectile during firing. The projectile is a small sphere with radius $a$ and uniform density $\rho$. During its flight, the projectile is subject to air resistance given by $f=-\gamma\vec{v}$ (where $\gamma$ is very small, and the approximation is retained to the lowest order of $\gamma$). Assume the projectile is launched at an angle $\theta$ with respect to the horizontal ground.
Find the density of the projectile $\rho$ that maximizes its range, given that the angle $\theta$ remains constant. The acceleration due to gravity is given as $g$. | ||
675 | THERMODYNAMICS | From a microscopic perspective, the surface of a liquid is not a geometric surface, but rather a thin layer with a certain thickness, known as the surface layer. If a certain mass of liquid is stored in a closed container that is adiabatic to the outside and has a volume slightly larger than the liquid's volume at a certain temperature, a unit liquid equilibrium system consisting of the liquid's internal region, liquid surface layer region, and saturated vapor region will form. During the process where liquid molecules transform into vapor molecules while passing through the liquid surface layer region, they must overcome the conservative forces pointing inward from the surface layer to perform work and increase the surface free energy. The density of a certain liquid is given by $\rho$, and the molecular molar mass is $\mu$. Avogadro's constant is denoted as $N_{A}$, and the Boltzmann constant as $k$. At a certain temperature, the particle number density of components within the surface layer, as a percentage of the particle number density within the liquid interior, is $\beta$ (understood as the number of bonds formed by surface layer molecules is $\beta$ times that of internal particles). Its molar enthalpy of vaporization $L_{\mathrm{m}}$ can be expressed by the interaction potential energy $\varepsilon$ between its component particles and the number of bonds $n$ between each particle and adjacent particles as: $$ L_{m}=N_{\mathrm{A}}\cdot\frac{\left|\varepsilon\right|}{2}n $$ From a macroscopic perspective, when considering the liquid surface free energy, under dynamic equilibrium conditions, the molecular density distribution of the liquid internal region and saturated vapor region can be regarded as uniformly distributed, while the molecules within the liquid surface layer region are in the conservative force field of the liquid surface, satisfying the Boltzmann distribution law. Given the temperature $T$ of a certain substance's liquid, the molar volume of the liquid is $V_{i m}$, and the corresponding saturated vapor pressure is $P_{g}$. Assuming the liquid can be approximated as a quasi-crystal lattice model, the density at the liquid-gas surface layer interface is $\frac{\rho}{2}$, and the ratio of the number of molecules with surface free energy $\varepsilon$ per unit area at the liquid-gas interface to the total number of molecules is $\delta$. Find the expression for the surface tension coefficient of this liquid. The answer should be expressed in terms of $\delta$, $N_{A}$, $\rho$, $\mu$, $k$, $T$, and $P_{g}$ | ||
718 | MODERN | Protons are composed of smaller entities called \"partons.\" The Large Hadron Collider (LHC) is a high-energy proton-proton collider, where the energy of a single proton in the proton beam is $E=7.0\mathrm{TeV}$ ( $1\mathrm{TeV}=10^{3}\mathrm{GeV}{=}10^{12}\mathrm{eV}$ ). Two proton beams with the same energy collide head-on, and the two colliding partons a and b interact and annihilate to produce a new particle. Let the ratio of the kinetic energy of partons a and b to the proton energy be $x_{\mathfrak{a}}$ and $x_{\mathfrak{b}}$ respectively, and assume these ratios are much greater than their rest energy. Assuming that the two partons a and b collide and annihilate to produce a new particle S with a rest mass of $m_{\mathrm{s}}=1.0\mathrm{TeV}/\mathrm{c}^{2}$, find the product $x_{\mathfrak{a}} x_{\mathfrak{b}}$, and express the answer numerically. | ||
113 | THERMODYNAMICS | The slogan of condensed matter theory is "More is different". Physics is not only the study of the composition and forms of interaction of matter, but also how new physical concepts are constructed at different theoretical levels, and more remarkably, how profound connections between physical concepts at different levels are revealed. Based on this idea, let's look at the following questions concerning connections, which are themselves interconnected.
In fact, diffusion is also a typical transport process. We can consider a gaseous medium, where a dilute component (thus the mechanical effects caused by its diffusion can be neglected) is composed of (quasi)particles with effective charge and mass $q, m$. Before discussing diffusion, let's consider the following question: if the system maintains a temperature $T$, and a uniform magnetic field $\pmb{B}$ is applied to the gas. Then, when scattering of particles is not considered $(\gamma\rightarrow0)$, the particles originally conforming to the Maxwell distribution in the medium will now rotate around the magnetic field. Next, we consider the evolution caused by the diffusion of particles in the external magnetic field. We let a large number of particles start at rest from the origin. Its dynamics equation can be phenomenologically written as:
$$
m{\ddot{r}}=f-\gamma{\dot{r}}+q{\dot{r}}\times B
$$
where $\pmb{f}=-\gamma\pmb{v}$ reflects the scattering (or damping) of the particles, and $q\dot{\pmb r}\times\pmb B$ is the force caused by the magnetic field, with the magnetic field taken as ${\boldsymbol{B}}= B e_{z}$. Meanwhile, $\pmb{f}$ reflects the forces caused by thermal factors, which is a function that randomly changes over time and is independent of the particle's location, with an average value of zero. Compute the average of the angular momentum in the $z$ direction $J=\langle m(x{\dot{y}}-y{\dot{x}})\rangle$.
Hint: use the position vector perpendicular to the $z$ direction ${\pmb r}^{\prime}=({ x},{ y})$ to dot the components of the dynamics equation perpendicular to the $z$ direction, and assume $\dot{R}=0$ at $t=0$. Boltzmann constant is $k$. | ||
305 | MECHANICS | A uniform spring with a spring constant $k$, an original length $L$, and a mass $m$, is connected at one end to a wall and at the other end to a block with mass $M$. It undergoes simple harmonic motion on a smooth horizontal surface. It is known that $m << M$.It can be approximately considered that the velocity of the spring varies linearly with the position.
The spring undergoes corrosion over a long period of use, causing the spring constant to decrease and the mass to increase. It is approximately assumed that the spring constant and mass change uniformly throughout the spring.
A new spring initially has an amplitude $A_{0}$. After a period of time, the mass of the spring becomes $\gamma_{1}$ times the original, and the spring constant becomes $\left(1-\gamma_{2}\frac{m}{M}\right)$ times the original. Assuming the mass added during the corrosion process moves with the spring, that is, the added mass has the same velocity as the point where it is added, find the change in amplitude $\Delta A$. | ||
573 | ELECTRICITY | As we all know, the electric field generated by a point charge \(q\) in a vacuum at any point is described as:
$$
\overrightarrow{E} = \frac{q}{4\pi\varepsilon_{0}r^{3}}\overrightarrow{r}
$$
However, this form presents the difficulty of the point charge's self-energy being infinitely large. To address this issue, Born and Infeld proposed a hypothesis in 1934: the vacuum also exhibits "polarization" and "magnetization" effects. Specifically, the vacuum's permittivity \(\varepsilon\) and permeability \(\mu\) are dependent on the electric field strength \(\vec{E}\) and magnetic flux density \(\vec{B}\) at that point:
$$
\frac{\varepsilon}{\varepsilon_{0}} = \frac{\mu_{0}}{\mu} = \left[1 + \frac{1}{b^{2}}\left(c^{2}\overrightarrow{B}^{2} - \overrightarrow{E}^{2}\right)\right]^{-\frac{1}{2}}
$$
Here, \(\varepsilon_{0}\) and \(\mu_{0}\) represent the permittivity and permeability of free space when the electromagnetic field approaches 0, \(b\) is a known parameter, \(c\) is the speed of light in vacuum, and \(c^{2} = 1 / {\varepsilon_{0}\mu_{0}}\). This problem takes relativistic effects into account.
In the inertial reference frame \(S\), an infinitely long uniformly charged straight wire, fixed along the \(z\) axis, moves in the positive \(z\) axis direction with a velocity \({v}\). The charge per unit length of the wire is measured as \(\lambda\), where \(\lambda > 0\). At a distance \(r\) from the wire, a point charge with charge \(-q\) (\(q > 0\)) is released with an initial radial velocity \(v_{0}\) relative to the \(S\) frame. The rest mass of the point charge is \(m\). Under the conditions specified in the problem, derive the expression for the maximum distance \(r_{0}\) that the point charge can reach from the \(z\) axis. | ||
91 | MECHANICS | An ellipsoid rigid body has a semi-major axis \(a\), semi-minor axis \(b\), and is formed by rotating about the major axis. Its mass distribution is rotationally symmetric about the major axis, with a total mass \(m\). The center of mass \(C\) is located at a focal point. The moment of inertia about the principal axis perpendicular to the major axis passing through the center of mass is \(mr_1^2\), and the moment of inertia about the principal axis parallel to the major axis passing through the center of mass is \(mr_2^2\), where \(r_1, r_2\) are called radii of gyration. The ellipsoid is placed on a horizontal ground with its major axis aligned vertically and the center of mass located on the upper half of the major axis. The ellipsoid is released from rest in this position and begins to roll after a slight disturbance. It is known that the ellipsoid maintains pure rolling throughout. Find the magnitude of the angular acceleration \(\beta\) after the ellipsoid has rotated by 270°. | ||
426 | MECHANICS | The mass of the homogeneous elastic rope is $m$, and its length is $l_{0}$ when freely placed horizontally. The stiffness coefficient of the rope is $k$. Now place the elastic rope on a rough horizontal surface, where the coefficient of friction between the rope and the ground is $\mu$.
Find the maximum length at which the elastic rope can lie flat and stationary on the horizontal surface. | ||
487 | MODERN | Mesons are hadronic particles consisting of a quark–antiquark pair, bound by the strong force. For light meson systems composed of light quark-antiquark pairs (with rest mass of several \(\mathrm{MeV}/c^{2}\)), both experimental measurements and theoretical calculations indicate that: In the static approximation, the strong interaction potential energy between quark-antiquark pairs is proportional to their distance \(r\), i.e., \(U(r)=\sigma r\), where \(\sigma\) is a positive constant. The linear behavior of the potential energy distribution described above prompted physicists to construct a semi-classical model known as the “string model” to describe the energy spectrum distribution of light mesons observed in experiments. As shown in the figure, light mesons are envisioned as rapidly rotating strings of length \(r\), with rest mass \(m_0=\sigma r/c^{2}\) uniformly distributed along the length, where the mass (energy) of light quarks is negligible. Their kinematic effects are equivalently replaced by boundary conditions such as "the speeds at the ends of the string are equal to the speed of light in vacuum" in the center of mass frame, i.e., \(\omega r/2 = c\). The quantization condition applied to the above strings is:
\[
L_n=\frac{nh}{2\pi}, \quad n = 1,2,3,\cdots
\]
where \(L_n\) is the angular momentum corresponding to the \(n\)-th energy level of the system in the center of mass frame. Try to provide the expression for the energy level \(E_n\). | ||
506 | ELECTRICITY | There are two infinitely long cylindrical ideal conductors forming a parallel transmission line pair. The conductor radius is $a$, and the distance between the axes is $2d$. Assume the conductors are perpendicular to the plane of the paper, one on the left side with the current direction perpendicular to the paper coming out, and one on the right side with the current direction perpendicular to the paper going in. The vacuum permeability $\mu_0$ is known. Find the inductance per unit length $L$ between the wires. | ||
270 | MECHANICS | Made of the same metal material, two hollow spheres \( A \) and \( B \) are constructed, with the density of the metal material being \( \rho_1 \). The mass of sphere \( A \) is \( m_A \), and the mass of sphere \( B \) is \( m_B \). The outer diameters are \( R_A \) and \( R_B \), respectively. The average density of sphere \( B \) is equal to \( \rho_0 \). The cavities of the two spheres are connected by a thin metallic conduit. Due to the small diameter of the conduit, applying an excessive tensile force can cause breakage. This setup can be used to monitor the density of a liquid.
Water with a mass of \( m_w \) is injected into the cavity of sphere \( A \), and wires are connected to both \( A \) and \( B \). After integrating the setup into an alarm circuit, it is placed horizontally on the surface of the liquid to be monitored. The density detector requires the liquid's density to not be lower than \( \rho_0 \). The circuit triggers an alarm when a significant reduction in current is detected.
When the liquid's density is just slightly lower than \( \rho_0 \), how much does the volume of sphere \( A \) above the liquid surface increase compared to before the alarm was triggered? (Assume that after the alarm is triggered, no water leaks out from the cavity, and the acceleration due to gravity is \( g \)). | ||
132 | MECHANICS | Three individuals, ABC, simultaneously start from the vertices $A_{0}, B_{0}, C_{0}$ of an equilateral triangle with side length $L$, each moving at the same constant speed $v$. Throughout the motion, A's velocity is always directed towards B, B's velocity towards C, and C's velocity towards A. Determine the time it takes for the three individuals to meet. | ||
208 | ELECTRICITY | In a zero-gravity space, infinitely far above a conductive plate with thickness $D$ and conductivity $\sigma$, there is a small magnet with its pole facing the plate. The magnet is incident from infinity with an initial velocity $v_0$. The magnet has a magnetic moment $\mu$, mass $m$, and the vacuum permeability is $\mu_0$. Neglect the magnetic fields generated by changing electric fields and conductive currents. Find the final stop position of the small magnet and the distance from the plate $h_e (h_e \gg D)$. | ||
163 | ADVANCED | Compute the commutator of the total angular momentum squared operator $L^2$ with the raising operator of the vector $V$, $V_{+} = V_x + i V_y$. Express the result using $V_z, V_{+}, L_z, L_{+}$, where $L_{+} = L_x + i L_y$. Use $\hbar = 1$ for the answer. Moreover, if there is a product of $V$ and $L$, please place $L$ at the end. | ||
572 | ELECTRICITY | Inside an insulating pipe with a cross-sectional area $S$, there are two frictionless pistons, each with mass $m$. The pistons seal a certain amount of monatomic ideal gas with a total number of molecules $N$. The initial distance between the pistons is $L$, and the initial gas temperature is $T$. The two plates carry positive and negative charges $\pm Q$ respectively. It is known that if the molecules are in an external electric field $E$, they will induce a dipole moment $p=\alpha E$ in the direction of the external electric field $E$, where $\alpha$ is a temperature-independent constant. The dielectric constant in vacuum is $\varepsilon_{0}$, and the Boltzmann constant is $k$. For convenience in expressing the answer, introduce a dimensionless constant $x=N\alpha/\varepsilon_{0}S L$, so $\alpha$ no longer appears in the answer. Assume that the piston area is very large, allowing the use of a capacitor model. If the pistons are initially in static equilibrium and the outside of the pistons is a vacuum, calculate the quantity of positive and negative charges $\pm Q$. | ||
516 | ELECTRICITY | Consider an L-shaped grounded conductor formed by the \(xz\) and \(yz\) planes, with its cross-section located in the region \(x>0\) and \(y>0\). Outside this region, there is a line charge (with line charge density \(\lambda\)) located at the point \((a,b)\), where \(b>a>0\). Find the expression for the electric potential \(V(x,y,z)\) in the region. | ||
68 | ELECTRICITY | Establish a rectangular coordinate system with an infinitely large insulating plate as the $xy$ plane. There is a charge distribution on the plane:
$$
\sigma(x,y)=\sigma_{0}\cos\frac{x}{d}\quad,\quad(\sigma_{0}>0,d>0)
$$
Taking infinity as the zero point of potential, find the potential distribution $\varphi(x,y,z)$. The vacuum permittivity is known to be $\epsilon_0$.
Hint:
$$
\int_{-\infty}^{\infty}{\frac{\cos(\beta\zeta)}{\zeta^{2}+a^{2}}}\mathrm{d}\zeta=\frac{\pi}{a}e^{-a b}
$$
$$
\int_{-\infty}^{\infty}\frac{\zeta\sin(b\zeta)}{\zeta^{2}+a^{2}}\mathrm{d}\zeta=-\frac{\partial(\frac{\pi}{a}e^{-a b})}{\partial b}=\pi e^{-a b}
$$ | ||
107 | MECHANICS | A child with a mass of \( m \) is swinging on a swing. When the swing is swinging stably, the distance from the child's center of mass to the swing's fixed point is \( l \). Then, each time the swing passes through the lowest point of oscillation, the child raises their center of mass by a small distance \( b \) (\( b << l \)). Thus, the distance from the center of mass to the fixed point becomes \( l-b \). And at each swing's highest point, the child lowers their center of mass by a small distance \( b \), and the distance from the center of mass to the fixed point becomes \( l+b \). 1) Assuming the oscillation amplitude is very small, calculate the work \(\Delta E\) done by the child in each oscillation cycle. Express it in terms of the initial mechanical energy \( E_{i} \) at the beginning of one cycle, using the lowest point of the swing as the zero point for gravitational potential energy. \(\begin{array}{r}{(\cos\theta\approx1-\frac{\theta^{2}}{2})}\end{array}\) | ||
347 | MECHANICS | There are two thin rods $AB$ and $CD$ near a vertical wall. The $A$ end of the $AB$ rod is in contact with the ground and is located on the right side of the vertical wall, while the $B$ end is in contact with the vertical wall. The $D$ end of the $CD$ rod is in contact with the vertical wall and is positioned above the $B$ end of the $AB$ rod, and the $C$ end is located on the $AB$ rod and slides along it. The $A$ end moves to the right along the ground at a constant speed $v$, and the $C$ end slides downward relative to the $AB$ rod with a relative velocity of $v'$. At a certain moment, the $C$ end is located at the midpoint of $AB$, the angle between the $AB$ rod and the ground is $\alpha$, the angle between the $CD$ rod and the vertical wall is $\beta$, and $\alpha + \beta < \pi/2$. Find the velocity $\vec{v}_E$ of the midpoint $E$ of the $CD$ rod at this moment. (Use basis vectors $\hat{i}, \hat{j}$.) | ||
31 | MECHANICS | In the study of classical mechanics, the research on the properties of central force fields is the most representative and common. To facilitate the study of central force fields, for the central potential field with magnitude ${V}({r})$, assuming the particle mass is ${m}$ and its angular momentum in motion is L, the effective potential energy ${U}_{eff}$ in which the particle moves can be defined as:
$$ {{U}}_{{eff}} = V(r) + {\frac{L^{2}}{2m r^{2}}} $$
Assuming a particle with mass m is moving in a central force field, and the potential energy $V(r) = \frac{kr^2}{2} - \frac{\alpha}{r^2}$. Under the condition that the particle can avoid falling into the center, find the period of the particle's radial motion. | ||
216 | ELECTRICITY | At the initial time $t=0$, there are 100 metal plates with an area of $S$ positioned parallel to each other, with an equal spacing of $D\ll\sqrt{S}$. These plates are charged sequentially with $Q$, $2Q$, $3Q\cdots 100Q$, such that the metal plate numbered $n$ carries a charge of $nQ$. Between all metal plates, a lossy dielectric with an absolute permittivity of $\varepsilon$ and a conductivity of $\sigma$ is used to fill the space. Throughout the process, edge effects are neglected. Determine how the voltage between plates 1 and 100 changes with time across the entire system. | ||
317 | MODERN | In space, there is a coordinate axis. In the $S$ system, $N$ particles numbered $1, 2, 3 \cdots N$ are static at positions $x_{1}, x_{2}, x_{3}\cdots x_{N}$ to the right of the origin, with distances $x_{1}, x_{2}, x_{3} \cdots x_{N}$, respectively. All particles have a mass of $m$. At time $t = -t_{n}$, a constant leftward force $F_{n} = n m a$ is suddenly applied to particle number $n$. At $t = 0$, all particles arrive at point $O$ simultaneously and undergo a completely inelastic collision. Furthermore, during the acceleration process, the proper time interval for each particle (i.e., the time interval measured by the clock carried by the particle; or, equivalently, the sum of the time intervals of infinitesimal sub-processes when each sub-process is approximately at the same speed as the particle) is $\tau$.
Find the instantaneous velocity $V$ of the resulting single particle after all particles have undergone a completely inelastic collision and combined into one particle in the final state. | ||
601 | MECHANICS | A semi-cylinder of radius R and mass M is placed on a horizontal table with the plane of the cylinder facing up. Initially the cylinder is at rest and then a small perturbation is given to the cylinder. Consider only the direction of the axis of the half-column under the perturbation, i.e., the angular velocity of the half-column is always in the direction of the axis.
There is friction on the table, and assuming that the initial inclination of the half-column (the angle between the rectangular surface and the horizontal) is $\theta_0$ and it is at rest, find the minimum value of the coefficient of static friction of the table so that the half-column can do pure rolling. | ||
388 | MECHANICS | Two spacecraft will pass through a region densely packed with stationary dust, where it is known that there is no friction between the dust and the spacecraft, and the dust will scatter tangentially upon collision with the spacecraft. Spaceship 1 is cylindrical with a cross-sectional radius of $R$, with its base facing the direction of motion; spaceship 2 is spherical with a radius of $R$. What is the ratio of the resistance experienced by the two when they are moving at the same speed? Ignore the effects of gravity.
| ||
319 | ELECTRICITY | This problem does not consider relativistic effects.
An infinitely long leaky dielectric cylindrical object, with a radius of $R$, an absolute permittivity $\varepsilon$, initially stationary and uncharged as a whole. The charge carriers have an extremely small mass $m \to 0$, charge magnitude $q$, and a mobility $\mu$ (i.e., the drift velocity of the charge carriers under a uniform electric field $E$ is $v = \mu E$). The number density of carriers is $n$. A cylindrical coordinate system $(z,r,\theta)$ is established inside the cylinder, with the origin being the center of the circular top surface, the $z$ axis pointing axially outward along the cylinder, and $r$ pointing radially outward. During the process, the charge volume density $\rho$ and current densities $j_{r}, j_{\theta}$ inside the cylinder will vary with time $t$, and they are also functions of spatial coordinates $r$ and $\theta$. Due to the translational symmetry along the $z$ axis, the current density in the $z$ direction, $j_{z}$, will always be zero, and all non-zero physical quantities are independent of the $z$ coordinate.
Firstly, keep the cylinder stationary, and slowly apply a uniform magnetic field $B$ along the cylinder axis. Then, maintain the uniform magnetic field and slowly rotate the cylinder steadily around the axis in the positive $\theta$ direction with an angular velocity $\omega$. Ignore the centripetal force required for the circular motion of the charge carriers (due to $m \to 0$), and also ignore the magnetic field generated by the rotation of the charge carriers. In the steady state, the charge carriers will redistribute in the radial direction, resulting in a non-zero charge and current distribution $\rho, j_{r}, j_{\theta}$ in the ground frame. Solve for the total current vector $\vec{j}$.
Hint: Although $\rho \neq 0$, assume that the number density of charge carriers is approximately $n$. | ||
484 | MECHANICS | Two steel balls, each with a mass of $m$, are connected by a string of length $2L$ and placed on a smooth horizontal surface. A constant force $F$ is applied at point $O$, the midpoint of the string. The force acts horizontally and is perpendicular to the initial direction of the string. The string is very soft, non-stretchable, and has negligible mass. Assume the two balls are rigid small spheres, undergo elastic collisions, and the collision time is negligible. Taking the initial midpoint of the two balls as the origin and the direction of $F$ as the positive $x$-axis, derive the function for the $x$-component of the trajectory vector $\mathbf{r}$ for the upper ball on the horizontal plane as a function of time $t$, assuming $t=0$ at the start. | ||
584 | MECHANICS | A special material elastic ring is placed on a frictionless horizontal table. The average value of the inner and outer radii is $R$, the difference between the inner and outer radii is $b\left(b\ll R\right)$, and the height is $h$. Initially, it is a stress-free circular ring. Assume that the material's Young's modulus is $E$ and the density is $\rho_{\mathrm{0}}$, and there is no energy loss during the vibration.
As a common form of vibration for metallic rings, the ring vibrates between a circle and an ellipse with a particularly small eccentricity, with the centers of both coinciding. If the amplitude of the vibration is very small, taking the center of the ring as the origin, the polar equation of the ellipse with a particularly small eccentricity can be approximately written as: $r=R{\left(1+\varepsilon\mathrm{cos}2\theta\right)}$, where the change in the shape of the ring can be described by the change in the parameter $\varepsilon$ (where $\varepsilon$ is very small). During vibration, it can be approximately assumed that each micro-segment vibrates only radially, being compressed or stretched tangentially, meaning only radial kinetic energy is considered. (In fact, the tangential velocity is even smaller by an order of magnitude than the radial one.)
Find the angular frequency of this vibration. | ||
327 | MECHANICS | A uniform rod of length $l$ is suspended by a spring above a liquid surface. When the system is at equilibrium, the lower end of the rod just touches the liquid surface, and the spring's elongation is exactly $l$. Now, the rod is now pulled downward vertically into the liquid, so that the distance between its lower end and the liquid surface becomes $2l$, and then the rod is released from rest. It is known that the rod and the liquid have the same density $\rho$, and it is assumed that the rod moves only along the vertical direction. Determine the time elapsed from when the rod is released to when it rises to its maximum height. | ||
245 | MECHANICS | A flat pivot can be regarded as a cylinder, and a bearing can be treated as a plane. They are in contact and exert pressure on each other. It is known that the pressure $F$ between the pivot and the bearing is uniformly distributed over a circular cross-section with a radius $R$, and the coefficient of friction is $\mu$. Find the torque of the frictional force. | ||
269 | ADVANCED | Using the natural units system \[c=\hbar=1\], consider a particle with mass \(\frac{1}{2}\) moving in an external field with the potential energy function \(x^2+\alpha \delta(x)\). Use non-relativistic quantum mechanics to solve for the energy spectrum of the particle (\(\alpha\) is a known parameter). | ||
316 | ELECTRICITY | An ion with mass $m$ and charge $Q$ is projected from a very distant point towards a neutral conducting sphere with mass $M$ and radius $a$. Due to the ion's electric field, the neutral conducting sphere becomes polarized. When the polarization reaches equilibrium, the electric dipole moment of the neutral conducting sphere is $\vec{p} = \gamma \vec{E}_0$, where $\vec{E}_0$ is the electric field strength created by the ion at the location of the neutral conducting sphere, and $\gamma$ is a constant related to the structure of the neutral conducting sphere. Assume the initial speed of the ion is $v_0$, and the impact parameter (the perpendicular distance from the neutral conducting sphere to the direction of incidence) is $b$. Neglect the effect of gravity, do not consider relativistic effects, and assume that at every moment during the ion's movement, the neutral conducting sphere can reach polarization equilibrium. Find the minimum distance between the ion and the center of the conducting sphere (assuming the minimum distance is still much larger than the radius $a$ of the sphere). | ||
173 | THERMODYNAMICS | In a cylinder, a certain amount of water vapor is enclosed, and the water vapor can be considered an ideal gas. The heat capacities of the liquid phase and gas phase are constant. The constant-pressure molar heat capacities of 1 mol of vapor and liquid are $C_{g p}$ and $C_{l p}$, respectively. At ${T}{=}0{K}$, the heat of vaporization of 1 mol of water is L(0). The saturated vapor pressure of water satisfies the Clausius equation:
$$
p=A{e}^{-L(0)/k T}T^{( C_{g p}-C_{l p})/R}
$$
Due to the high fluidity and expansiveness, water and water vapor are excellent choices for thermodynamic working substances. If water and water vapor undergo the following cycle, A-B-F-C-D-G-A is a complete cycle process, A and B are pure water, F and G are the transition points from mixed water and vapor to pure vapor. The pressure at point D is $P_0$, and the volumes at points C and D are both $V_0$. Find the cycle efficiency of this cycle. Express the final result using $L(T1), L(T2), A, T_1, T_2, V_0, k, C_{gp}, C_{lp}$, where $C_{gp}$ is the molar heat capacity of water vapor and $C_{lp}$ is the molar heat capacity of water. (The volume of water is negligible, the volume at points AB is considered 0, and B-F-C and D-G-A are isothermal processes at temperatures T1 and T2, respectively). | ||
536 | MECHANICS | A carriage of length $L$ is at rest on a smooth horizontal track. A launcher $A$, fixed at the center of the carriage floor, fires a particle of mass $m_1$ with an initial velocity $u_0$ relative to the horizontal track from the middle of the carriage along the smooth floor inside the carriage. The particle $m_1$ then collides and instantly sticks with another particle of mass $m_2$ placed on the carriage. At this moment, $m_2$ just comes into contact but does not adhere to a lightweight spring fixed at one end to the carriage at a horizontal position. The spring has a stiffness constant $k$ and its natural length is $l$. The total mass of the carriage and the launcher $A$ is $M$.
Find the expression for the displacement of the carriage from rest until the spring reaches its maximum compression.
| ||
533 | ELECTRICITY | A particle with mass $\mathfrak{m}$ and positive charge $q$ moves in an electromagnetic field. The electric field is a uniform field with field strength $\pmb{E}$, directed along the positive y-axis. The magnetic field is directed along the z-axis (perpendicular to the xy-plane), and the magnetic flux density varies only with y, specifically given by $B=\alpha\sqrt{|y|}$. The particle initially starts at the origin and is released from rest. During the subsequent motion, the particle's gravitational force can be neglected, and it comes to a stop for the first time at $t=T$. How far is the particle from the starting point when $\scriptstyle t={\frac{3}{2}}T$? | ||
545 | THERMODYNAMICS | A hemisphere of ice with a radius $R$ and refractive index $n$ is situated on a warm, flat table and melts slowly. The rate of heat transfer between the table and the ice is proportional to their contact area. The coordinate system is defined with the center of the semicircle as the origin; the $x$ and $y$ axes are horizontal, and the $z$ axis is vertical. It is known that the ice hemisphere completely melts in time $T_{0}$. Throughout the process, a laser beam shines on the ice from above. The beam is incident perpendicular to the plane along the line $x=R/2$, $y=0$.
Assume the temperatures of the ice and the surrounding atmosphere are both $0^\circ C$, and remain unchanged during the melting process. The laser beam does not transfer energy to the ice. The melted water immediately flows off the table, and the ice does not move along the table.
For $t \geq 0$, where is the location of the point $x(t)$ on the table hit by the beam? Express the answer in terms of $n, R, T_{0}, t$ (and should not include trigonometric functions). | ||
13 | MECHANICS | 10 identical wooden blocks are placed one next to the other on a horizontal floor. Each wooden block has a mass of $m=0.40kg$ and a length of $l=0.50m$ . The static and kinetic friction coefficients between the block and the ground are both $\mu_{2}=0.10$. Initially, the blocks are at rest. A small lead block with mass $M=1.0$ kg is placed above the left end of the first wooden block on the left. The static and kinetic friction coefficients between the lead block and the wooden block are both $\mu_{\mathrm{i}}=0.20$. Now, the lead block is suddenly given an initial velocity to the right of $V_{0}=4.3m/s$ , causing it to slide over the large wooden blocks. Determine on which block the lead block finally stops. Take gravity acceleration $g=10m/s^2$ . Assume the dimensions of the lead block can be ignored compared to $l$. You need to give an integer between 1-10 as the answer. | ||
324 | ELECTRICITY | Two coaxial thin cylinders have radii of $R$ and $2R$, respectively, with both having a length of $L (L >> R)$, and a mass of $m$, with uniform distribution. The inner and outer surfaces are uniformly charged, carrying charges of $Q$ and $-Q$, respectively, and the charges do not move across the surfaces. Both cylinders can rotate about the central axis and are initially at rest. Now, due to the action of an external force (assuming the action time is very short), the inner cylinder gains an initial angular speed of $\omega_{10}$. Assume the outer cylinder experiences a small frictional torque, while the frictional torque acting on the inner cylinder can be neglected. Find the work done by the frictional torque from the initial state to the final state. | ||
406 | ADVANCED | Given the form of the electromagnetic field momentum conservation theorem:
\[\vec{f_m} + \frac{\partial \vec{g}_{em}}{\partial t} + \nabla \bm{T}_{em}\]
where $\vec{g}_{em}$ is the electromagnetic field momentum density, and $\bm{T}_{em}$ is the electromagnetic field momentum flow density.
Now, consider a given point charge $q$ moving at velocity $\vec{v}$ in a constant external magnetic field $\vec{B}_0$. Consider only the momentum flow density of the electromagnetic field and ignore the momentum density. Provide the force on the charge, that is, provide the mechanical force density \[\vec{f_m}\] (do not calculate \[\vec{g}_{em}\]). To simplify the expression, assume the particle starts from the origin and moves in uniform linear motion. | ||
762 | MECHANICS | The spinning top is a children's toy with a long history, and its principles can also be applied in various fields such as navigation. Now we consider a uniform mass distribution conical top, with a half-apex angle of $\alpha$, height $h$, and mass $m$. The top spins around its apex on a horizontal surface, with its apex fixed at the origin $O$. Possible resistance is not considered. Now let the axis of the spinning top maintain a constant angle $\beta$ with the vertical direction while spinning. Find the expression for the minimum total angular velocity magnitude $\omega_{0}$ of the top's rotation.
| ||
686 | MODERN | In a three-dimensional harmonic trap, atoms oscillate approximately with frequency \(\omega_i\) in the \(i\) direction (\(i=x,y,z\)). According to the kinetic energy distribution of a classical gas, the characteristic kinetic energy in the \(x\) direction is \[ \frac{1}{2}k_BT, \] with the corresponding harmonic potential energy \[ \frac{1}{2}m\omega_x^2R_x^2, \] When these two are equal, the maximum displacement \(R_x\) of the atom in the \(x\) direction can be obtained. Please write out the characteristic confined volume \(V\) of the atom (expressed in terms of temperature \(T\) and the harmonic frequencies \(\omega_x,\omega_y,\omega_z\)). | ||
225 | MECHANICS | In the fifth century AD, the Saxons designed a timekeeping device. They placed a bowl with a hole in the bottom in water and timed it by observing the bowl sinking. This device was called the "Saxon Bowl." The bowl is a hollow open cylinder with a mass of $M$, the thickness of the cylinder wall and bottom is $D$, the internal radius is $R$, the height is $H$, and there is a small hole at the center of the bottom with a radius of $r$. Assume that $H$ and $h$ are much larger than $D$, but $R$ is of the same order of magnitude as $D$. The liquid density is $\rho$. The critical condition for sinking: When the bowl is gently placed on the water surface, it is found that when $r$ meets certain conditions, the bowl remains balanced on the water surface without water flowing into the bowl. Given the surface tension coefficient of water is $\sigma$, and the acceleration due to gravity is $g$, find the critical radius $r_{m}$? |
Subsets and Splits
No saved queries yet
Save your SQL queries to embed, download, and access them later. Queries will appear here once saved.