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292 | ELECTRICITY | In space, there are two homogeneous concentric spherical shells, each with its center of mass at rest. The inner shell has a mass of $m_1$, radius of $R_1$, and is uniformly charged with a charge of $q_1$. The outer shell has a mass of $m_2$, radius of $R_2$, and is uniformly charged with a charge of $q_2$. The relationship $\frac{q_i^2}{4\pi\varepsilon_0 R_i c^2}\ll m_i$ holds. After consideration, we find that the motion of the two shells should satisfy the following manner: both the inner and outer shells undergo precession around a common "axis." Given that initially both centers of mass are at rest and the angular velocities are $\vec{\omega_1} = \omega_1 \hat{x}$ and $\vec{\omega_2} = \omega_2 \hat{y}$, we seek to determine the subsequent evolution of $\vec{\omega_1}$ for the inner shell. | ||
513 | MECHANICS | Consider an arrangement with $n$ ($n \geq 3$) identical springs, where one end of each spring is attached to a small ball of mass $m$, and the other end is evenly distributed along the circumference of a rigid circular ring with radius $b$ (i.e., the angular spacing between adjacent spring endpoints is $\frac{2\pi}{n}$). The small ball is initially located at the center of the circular ring. The natural length of each spring is $a$ ($a \ll b$), with a spring constant of $k$. Gravity is neglected, and the rigid circular ring remains stationary. When the small ball moves within the plane of the circular ring, determine the angular frequency $\omega$ of the small ball’s small oscillations near its equilibrium position. | ||
156 | THERMODYNAMICS | When a disturbance propagates in a medium at a speed faster than the speed of sound in the medium, a shock wave forms.
The Mach number $M$ of a disturbance is defined as the ratio of the propagation speed of the disturbance in the medium to the speed of sound in the medium.
Due to the motion of the shock wave, there is a discontinuity in physical quantities on either side of the shock wave front.
Choose the shock wave reference frame. Let the upstream gas have a velocity magnitude $v_{1}$ (positive when directed opposite to the wave front), density $\rho_{1}$, temperature $T_{1}$, and specific internal energy $\boldsymbol{u}_{1}$. Let the downstream gas have a velocity magnitude $v_{2}$ (positive when directed opposite to the wave front), density $\rho_{2}$, temperature $T_{2}$, and specific internal energy $u_{2}$. You can derive the conservation laws on both sides of the shock wave front.
Treat the gas as an ideal gas and assume the adiabatic index is $\gamma$.
In the ground reference frame, the upstream gas is stationary ($w_{1}=0$), the velocity magnitude of the shock wave front is $w$, and the downstream gas velocity magnitude is $w_{2}$ (positive in the same direction as the wave front's motion).
The speed of sound in an ideal gas is given as $v_{s} = \sqrt{\frac{\gamma RT}{\mu}}$, and the Mach number of the shock wave is $M$.
Find $w_{2}$ in terms of $w$, $\gamma$, and $M$. | ||
377 | THERMODYNAMICS | At $t=0$, an adiabatic ,thin and lightweight piston with a cross-sectional area of $S$ divides an adiabatic cylinder into two regions, each with an initial volume $V_0$. The left side contains an ideal gas with an adiabatic index $\gamma=3/2$ and initial pressure $p_0$. The cylinder walls are coated with a specific optional viscous material of negligible heat capacity and volume, characterized as follows: when the piston moves in a direction at a speed $v$, the material exerts a resistive force $F=-k v$ on the piston. The heat generated due to friction can be transmitted in two distinct ways:
- For forward-transmitting viscous material, the heat is conducted forward in the direction of the piston’s motion and absorbed by the gas on that side (or conducted externally if no gas is present).
- For reverse-transmitting viscous material, the heat is conducted backward against the direction of the piston’s motion and absorbed by the gas on that side (or conducted externally if no gas is present).
Consider the following problem (retain the results in analytic form, without converting to numerical values):
The region on the right side of the container is vacuum, and reverse-transmitting viscous material is used. Calculate the time $t$ required for the piston to move to the far-right position. | ||
441 | ELECTRICITY | In a vacuum, two infinitely long, uniformly charged thin-wall coaxial cylinders are placed. The radii of the cylinders are $r_{1}$ and $r_{2}$, respectively. In the direction parallel to the cylinder axis, the mass per unit length of both the inner and outer cylinders is $m$, and the charge per unit length of the inner and outer cylinders is $q$ $(q > 0)$ and $-q$, respectively. Both cylinders can freely rotate around their respective central axes. The permittivity and permeability of the vacuum are given as $\varepsilon_{0}$ and $\mu_{\mathrm{0}}$, respectively. $r_1 < r_2$
Initially, both the inner and outer cylinders are stationary. If a torque is applied to the outer cylinder, causing it to begin rotating, calculate the total mechanical angular momentum $L$ along the axis per unit length of the inner and outer cylinders when the angular velocity of the outer cylinder reaches $\varOmega$. | ||
530 | MECHANICS | 2. Frictional Transmission
A large disk with a sufficient radius acts as the driving wheel and rotates counterclockwise around a fixed axis $o$ with a constant angular velocity $\Omega$. There is another small disk (with a radius of $r$) that can freely rotate around a fixed axis $O_{1}$, which acts as the driven wheel. This small disk is pressed against the large disk with a force of magnitude $N$ by the axis $O_{1}$. If the large and small disks are coaxial, the small disk will be driven by the large disk and rotate with the same angular velocity. Now we consider the situation where the distance between the two axes is $d \neq 0$. It is assumed that the pressure between the two disks is uniformly distributed. We can also add lubricant between the two disks, turning the friction between them into "wet friction." The characteristic of this is: for two mutually contacting surface elements of size $\Delta\sigma$, the wet friction force on one of them can be expressed as
$$
\Delta f = -\gamma v_{r} \Delta\sigma
$$
where $v_{r}$ is the magnitude of the relative velocity between the two surface elements, $\gamma$ is the viscosity coefficient, and the negative sign indicates that the force opposes the direction of relative motion. When the angular velocity of the small disk reaches $\omega$ (but has not necessarily reached stable rotation yet), find the heating power at the lubricant $P_{Q}$. | ||
274 | MECHANICS | A sufficiently rough conical surface with a half-apex angle of $\alpha$ is fixed vertically, with the cone's sharp angle pointing straight down. A homogeneous small ball with a radius of $R$ is placed on it, and the distance from the ball's center to point $O^{'}$ is denoted as $r$. The gravitational acceleration $g$ is known. It is assumed that the small ball will never leave the cone surface. A line parallel to the cone surface passes through the ball's center and intersects the central axis of the cone surface at point $O'$, with the spin angular velocity and the line connecting the ball's center and $O'$ forming an angle $\phi$. It is known that $\begin{array}{r}{\alpha=\frac{\pi}{4}.}\end{array}$ When the ball is in stable precession, it satisfies $r_{0}=4R$. Let the angle between the spin angular velocity and the cone surface be $\varphi$. If a radial perturbation is applied to the ball, causing its center to oscillate back and forth near $r_{0}$, find the oscillation period and the range of values for $\varphi$. Note: The angular velocity vector of the small ball in the ground frame is the sum of the precession angular velocity and the spin angular velocity. | ||
610 | THERMODYNAMICS | The Earth is round, and the street is also a circle with a length of $L$. There are 2024 point masses (completely elastic) on the street, and their tangential velocity distribution along the street follows the Maxwell velocity distribution. The probability that the velocity is between $v$ and $v+\mathrm{d}v$ is: $$ \mathrm{d}p={\frac{1}{\sqrt{2\pi}\sigma}}\mathrm{e}^{-{\frac{v^{2}}{2\sigma^{2}}}}\mathrm{d}v $$ Neglecting the possibility of three-body collisions, calculate the average frequency of elastic collisions between the point masses in the system. | ||
111 | MECHANICS | Under zero gravity, a soft rope with a mass line density of $\lambda$ is used for rope skipping. The rope is in the same plane, and an orthogonal coordinate system $xOy$ is set up in this plane. The two endpoints of the rope are symmetrical about the $y$-axis. The ends of the rope are connected to handles, and the direction of the handles is perpendicular to the tangent direction of the rope ends. The angular velocity $\omega$ of the rope rotation is along the positive direction of the ${\pmb x}$-axis. The lowest point of the rope is at a distance $h$ from point $O$, and the tension at the lowest point is $T_{0}$. If $\lambda\omega^{2}h^{2}=2T_{0}$, find the angle between the handle on the left side and the positive direction of the $x$-axis. | ||
344 | MECHANICS | The dynamics of the following unique system are considered: A homogeneous cylindrical shell with a mass of $M$ and a radius of $R$ can freely rotate around its center $O$. Inside it, there is another smaller homogeneous cylinder with a mass of $m$ and an undetermined radius $r$. The two cylinders are completely rough, maintaining no relative sliding either during continuous motion or during short impact time intervals. The gravitational acceleration $g$ is vertically downward. Initially, the small cylinder is at the lowest point of the large cylinder, and both cylinders are at rest. Suddenly, a very brief impulse $I$ is applied at the center $O^{\prime}$ of the small cylinder. The various angles of rotation during the impact process are negligible.
We find that if, at the initial moment, a point $P$ fixed to the small cylinder is located at a distance $l$ directly beneath $O^{\prime}$ (possibly outside the cylinder), then, under appropriate initial conditions, this point will keep moving up and down vertically, and the small cylinder will never detach from the large cylinder. Find the value of impulse $I$, expressed solely in terms of $R, r, M$. | ||
593 | ELECTRICITY | A proton (charge \(+e\)) moves with a speed \(v\) parallel to an infinitely long, thin wire at a distance \(r\) from the wire's axis. The wire carries both a current \(I\) and a positive charge per unit length \(\lambda\). If the net force on the proton due to the electric and magnetic fields is zero, allowing it to move in uniform linear motion parallel to the wire, find the relationship between the proton's speed \(v\) and the other parameters. Provide the final answer as a clear expression. The speed of light is given as \(c\). | ||
616 | OPTICS | In recent years, with the discovery of numerous planets orbiting stars, observing exoplanets at astronomical distances has become a challenge for scientists. Gravitational microlensing is one of the detection methods. It utilizes a principle discovered by Einstein in general relativity: when light passes by a spherically symmetric object with mass \( M \), its direction is deflected toward the object by a small angle \(\alpha = \frac{4GM}{rc^2}\), where \( G \) is the gravitational constant, \( c \) is the speed of light, and \( r \) is the shortest distance between the light and the object. In this problem, we will study the principle of detecting exoplanets through the microlensing effect. Consider a distant star \( S \), at a distance \( D_s \) from Earth \( E \), as the source star. Another star \( L \), with mass \( M \), is at a distance \( D_l \) from Earth (\( D_l < D_s \)), serving as the lens. Consider the situation where the lens star and the source star are aligned (the source star, lens star, and the observer on Earth are on the same straight line). To the observer on Earth, the image of \( S \) appears as a ring, known as the Einstein ring. Derive the expression for the angular radius of the Einstein ring using the parameters given in the problem. | ||
393 | OPTICS | A point light source is placed at the center of a spherical screen with a radius of $R$. A planar object is placed inside the sphere, and the distance from the light source to this plane is $d$. Establish a rectangular coordinate system $xOy$ on the plane, with the origin at the projection point of the light source on the plane. Calculate the relationship between the magnification of the image projected onto the spherical screen and the coordinates of the object plane. Here, magnification is defined as the ratio of the differential area of the projection on the screen to the differential area of the projection on the object plane before projection. | ||
109 | THERMODYNAMICS | A rigid adiabatic rectangular container with length $L$ (the thickness of the container walls can be ignored) is filled with an ideal gas composed of monoatomic molecules, where the mass of each gas molecule is $m$, and the initial temperature is $T_0$. Now, the container suddenly accelerates uniformly with an acceleration $a$ in the direction of $L$ ($maL \ll kT_0$, where $k$ is the Boltzmann constant). Try to find the temperature $T$ of the system after reaching a stable state. | ||
784 | THERMODYNAMICS | Due to the sudden drop in temperature, several small ice crystals have appeared in the cylinder, severely affecting operational safety. Now, an external force $F$ is applied to slowly compress and melt all the ice into water, which is then expelled by centrifugal force. The atmospheric pressure is $p_0$. Given that the area of the piston in the cylinder is $S$ (the mass of the piston can be neglected), it can slide smoothly and freely within the cylinder. Initially, the cylinder is in equilibrium, the gas volume is $V_0$, the temperature is $T_0$, and the total mass of the ice crystals is $m$. The gas is known to be a diatomic ideal gas, the specific heat capacity of ice is $c$, the latent heat of fusion is $l$, and the melting point is $T_{ice}$ (approximately constant). Ignore the volume of the solid and liquid; the cylinder is considered adiabatic. By slowly compressing the substances in the cylinder, all the ice is melted. Find the final state gas pressure. | ||
209 | THERMODYNAMICS | Consider an adiabatic bubble wall with a constant surface tension coefficient $\sigma$. The external pressure is much smaller than the additional pressure and can be regarded as 0. Let us examine the process of first injecting gas into the bubble and then expelling the gas from the bubble. This problem focuses on the gas injection process. Initially, the bubble's radius is denoted as 0, and the temperature of all the gas prior to injection is uniform (to be determined). After injection, a bubble with a radius of $r_{0}$ is formed, and the temperature of the gas is $T_{0}$. It is known that the adiabatic index of the gas is $\gamma$. Neglecting heat exchange between the liquid and the gas, find the temperature of the gas before injection, denoted as $T_{1}$. | ||
201 | ELECTRICITY | Two smooth sliders with charges $Q$ and $4Q$, respectively, can freely slide on a horizontal straight insulated rail. The volume of the sliders can be ignored, and they are connected by two light, insulating strings of equal length $l$ to a massive particle with mass $m$. When the system is in static equilibrium, the angle between the strings and the horizontal direction is $ \theta_{0}=60^{ \circ}$. Now, introduce another slider with charge $2Q$ between the two existing sliders and keep it fixed in position. Due to factors such as air resistance, the system will settle into a new equilibrium configuration after a dynamic process. Calculate the angle $ \theta$ between the strings and the horizontal direction at this new equilibrium. | ||
338 | ELECTRICITY | In the spatial rectangular coordinate system, point charges each with a charge amount of $+Q$ are fixed at $(0,0,-a)$ and $(0,0,a)$, and there is a free point charge with a charge amount of $+q$ at the origin. $(Q, q > 0)$. Electrostatic force alone cannot make the charge be in stable equilibrium. Therefore, to stabilize the charge (restrict the range of charge motion), a magnetic field will also be applied. If a uniform magnetic field is applied in the $z$ direction, with the magnitude $B$ (sufficiently large such that the charge is in stable equilibrium at the origin). If the $+q$ point charge leaves the origin with a very small initial velocity (the direction is arbitrary), try to discuss the condition under which the point charge can pass through the origin again in terms of the magnetic field's magnitude (limited to the meaning of small oscillations). Hint: You may use two unknown integers in your answer, denoted by $n_{1}$, $n_{2}$. | ||
411 | MODERN | A particle with rest mass $m$ is projected with an initial velocity of magnitude $v_0$ at a horizontal angle of elevation $\theta$. During its motion, the particle is subject to a constant gravitational force $F$. Considering relativistic effects, strictly solve for its trajectory equation (express the result as $y(x)$, ensuring that the answer contains only the known quantities $u_{0}= \frac {v_{0}}{\sqrt {1- v_{0}^{2}/ c^{2}}} , a_{0}= \frac Fm,\theta$). | ||
787 | THERMODYNAMICS | The mechanical properties of the rubber membrane are similar to the surface layer in thermodynamics. In the first approximation, the tension within the rubber membrane can be described using the surface tension coefficient (a layer of rubber membrane is equivalent to a layer of surface layer). (The refractive index of the membrane is considered the same as that of water, and the membrane is very thin.) The tension formula is as follows: \[ F = \sigma l \] --- A hollow cylinder with height \(L \) and radius \(R \) has its two ends covered with rubber membranes of the same surface tension coefficient. An ideal gas with pressure \(p_0 \) is sealed inside, exerting pressure when the membrane is taut. The cylinder is submerged in a pool to a depth \(H \) (with \(L \ll H \) assumed, the liquid additional pressure on both membranes is the same), and when the cylinder is at a distance of \(H/3 \) from the water surface, the bottom of the pool is precisely imaged at the surface. The temperature of the water is independent of depth, the cylinder has good thermal conductivity, the refractive index of water is \(n \), its density is \( \rho \), the gravitational acceleration is \(g \), and the atmospheric pressure is \(p_0 \). If the cylinder in water can be considered as a **thin lens**, and the pressure of the gas inside can be considered unchanged (i.e., the membrane is relatively rigid and the air volume is approximately unchanged). Continue to submerge the cylinder, and the pool bottom can still be precisely imaged at the water surface. Determine the distance from the water surface to the cylinder when it images again. | ||
240 | MECHANICS | It might seem that the gyroscopic effect is the reason why bicycles don't fall over, but in fact, the mass of the spinning wheels is not very large, and the gyroscopic effect it generates is not sufficient. Moreover, when people equipped bicycle wheels with counter-rotating wheels that provided equal but opposite angular momentum, it was found that bicycles still maintained stability. These facts are enough to show that the gyroscopic effect is not the absolute reason for a bicycle's stability. In fact, the decisive reason that bicycles don't fall is "centrifugal force." (For ease of calculation, this question does not need to consider the influence of the gyroscopic effect or Coriolis torque in rotating systems.)
A bicycle can be regarded as an inverted pendulum (unstable from side to side), requiring additional restorative forces to maintain balance, and the restorative force provided is precisely the "centrifugal force" when the bicycle is steering.
Centrifugal force is a function of speed and handlebar steering angle. At a fixed speed, it can be considered that controlling the handlebar steering angle means controlling the restorative force.
First, please imagine yourself riding a bicycle. At some moment, you notice that you have a tendency to tip over to the right. To prevent yourself from falling, which way will you turn the handlebars?
Then please calculate the model below.
The bicycle is divided into four parts as rigid bodies: front wheel, rear wheel, rear frame and rider, front frame and handlebars. Both the front and rear wheels can be considered homogeneous rings with mass $m$ and radius $r$. The distance between the centers of the front and rear wheels is $L$. The mass of the rear frame and rider is $M_{1}$, with the center of mass at the coordinates $(x_{1},z_{1})$. The mass of the front frame and handlebars is $M_{2}$, with the center of mass at the coordinates $(x_{2},z_{2})$.
Try to solve the minimum angle $\alpha (\ll 1)$ within the horizontal plane that must be rotated by the handlebars when the bicycle-rider system is moving forward at a constant speed $v$ and has a tilt angle $\theta$ from the vertical balanced position. | ||
783 | ELECTRICITY | Consider a **lightweight solid ball** rotating around a fixed axis in a vacuum. The radius of the ball is \( R \), it is **uniformly charged**, and the volume charge density is \( \rho \). Let the axis be the \( z \) axis, and establish a spherical coordinate system \( (r, \theta, \varphi) \), not considering relativistic effects. The ball is initially stationary, and a **positive external torque along the \( z \) axis** is applied to the ball, causing it to rotate with an **angular acceleration** \( \beta \) along the \( +z \) direction, and at some moment the angular velocity is \( \omega \). The vortex electric field in space is only in the \( \varphi \) direction. Based on this, calculate the magnitude of the required external torque \( M \). Appendix: Conclusions that might be used in this problem: - The **energy density** of the electromagnetic field: \[ w = \frac{1}{2} \varepsilon_0 E^2 + \frac{1}{2 \mu_0} B^2 \] - Current density (energy flux density): \[ \vec{S} = \vec{E} \times \vec{H} \] - Momentum density: \[ \vec{g} = \vec{D} \times \vec{B} \] Where: - \( \vec{E} \): Electric field strength - \( \vec{B} \): Magnetic induction strength - \( \vec{D} \): Electric displacement vector - \( \vec{H} \): Magnetic field strength | ||
386 | THERMODYNAMICS | At time $t=0$, an adiabatic, thin and lightweight piston divides an adiabatic cylinder with a cross-sectional area $S$ into two equal parts, each having a volume $V_0$. On the left side, the cylinder contains an ideal gas with an adiabatic index $\gamma = 3/2$ and an initial pressure $p_0$. The cylinder's walls are equipped with a type of viscous material possessing negligible heat capacity and volume. The viscous material has the following properties: when the piston moves at a velocity $v$ in a certain direction, it exerts a resistive force on the piston given by $F = -k v$. At the same time, the heat generated by friction has two possible behaviors. For **pro-directional viscous material**, the heat is conducted forward along the original velocity direction and absorbed by the gas on that side (or transferred out of the system if no gas is present). For **anti-directional viscous material**, the heat is conducted backward opposite to the original velocity direction and absorbed by the gas on that side (or transferred out of the system if no gas is present). Consider the following problem (retain results in analytical form without converting them to numerical values):
The right side of the container holds the same type of gas with a pressure of $2p_0$, and an anti-directional viscous material is used. Calculate the displacement $l$ of the piston after it comes to a stop. | ||
119 | ELECTRICITY | On an infinite superconductor plate placed horizontally on a surface with a friction coefficient of $\mu$, there is a uniformly charged insulating spherical shell with total charge $Q$, radius $R$, and mass $M$. Fixed smooth baffles are placed on both sides of the spherical shell to prevent its center of mass from moving horizontally. It is known that in the subsequent motion, the sphere will not leave the superconducting plate. Now, a sphere is given an initial angular velocity of $\pmb{\omega}$. The magnetic permeability in a vacuum is $\mu_0$, and the direction of $\omega$ is vertical. Try to determine the relationship between the angular displacement and the angular velocity of the sphere afterward (considering the sphere will not leave the superconducting plate). | ||
780 | MECHANICS | In this problem, we are given a 0-length elastic string, with a spring constant \(k \) that is constant. The original length is 0, and the mass distribution is uniform (it can be considered that the original length is almost 0). The total mass is \(m \), and it is hanging in Earth's gravitational field. We need to find its shape under the following circumstances. If the string is far from the Earth's surface, and the stretched length is close to Earth's size, with Earth's mass being \(M \), the gravitational constant \(G \), the suspension points are at a distance \(R \) from Earth's center, and the angle between the line connecting to Earth's center is \( \varphi \), we need to find the shape of the string. It is not necessary to consider the contact between the string and the Earth's surface. (This question uses polar coordinates \((r , \theta) \), the string is symmetric about the polar axis, passing through \((R, \frac{ \varphi}{2}) \) and \((R,- \frac{ \varphi}{2}) \)). Due to the difficulty in providing internal force and deformation boundary conditions for this problem, we directly give an additional geometric boundary condition. That is, the angle between the string tangent at the suspension point and the line to Earth's center is \( \alpha \). The final equation of the curve should be expressed in terms of \( \alpha \), \( \varphi \), and \(R \). | ||
438 | MECHANICS | There is a table with six legs placed on a horizontal surface. The length of the table is 2a, with three legs on the longer side: two located at the corners and one located at the midpoint of the long side. The width of the table is a. The tabletop is rigid, and the legs are elastic with very small elastic deformation. The self-weight of the table is neglected. Now, apply a concentrated force P at a certain point on the tabletop. Please try to calculate the total area on the table where the force P can be applied when only 4 out of the 6 table legs have internal pressure. | ||
461 | MECHANICS | On a rainy day, the process of raindrops falling is quite complex, involving knowledge of dynamics, fluid mechanics, phase transition, and other fields. By combining some actual phenomena and experience, we can establish a simple model to simulate the falling of droplets. Gravity causes a condensation nucleus to fall in uniform still water vapor, and it can absorb a mass of water vapor equal to \( a \) per unit distance fallen. The initial velocity and mass of the condensation nucleus, as well as the viscous force during the raindrop's fall, can be ignored. Find the speed \( v \) of the raindrop after falling vertically for \( t \) seconds (before reaching the ground). \( g \) is the constant gravitational acceleration.
| ||
357 | MECHANICS | A rope with length $L$ and mass $m$ is lying on a rough horizontal surface (the rope can be considered as concentrated at one point), and the coefficient of friction between the rope and the horizontal surface is $\mu$. Now, a horizontal constant force $F$ is applied to one end of the rope, gradually pulling the rope straight. Given that the acceleration due to gravity is $g$, find the velocity $v$ of the rope at the moment it is completely straightened. | ||
735 | OPTICS | The design of the laser targeting and strike system needs to consider changes in the air's refractive index. Due to the influence of various factors such as surface conditions, altitude, temperature, humidity, and air density, the distribution of the air's refractive index in the atmosphere is uneven, and hence the propagation path of the laser is not a straight line. For simplicity, assume that the variation of the air's refractive index with height \( y \) at a certain location is as follows: $$ n^2 = n_0^2 + \alpha^2 y $$ Where: \( n_0 \) is the refractive index of air at \( y = 0 \) (ground level); \( n_0 \) and \( \alpha \) are known constants greater than zero; The propagation time of the laser itself can be ignored; The laser emitter is located at the coordinate origin. If the emission direction of the laser forms an angle \( \theta_0 (0 \leq \theta_0 \leq \frac{\pi}{2}) \) with the vertical \( y \)-axis, find the expression for the height \( y \) when the horizontal propagation distance is \( x \). Express the result using the following physical quantities: Refractive index constant \( n_0 \), \( \alpha \), emission angle \( \theta_0 \), and horizontal propagation distance \( x \). | ||
358 | THERMODYNAMICS | A steel chain is composed of N oval-shaped steel rings connected together, with the rings having a long axis length of \( l+a \) and a short axis length of \( l-a \). The mass of the steel rings can be ignored. The upper end of the steel chain is fixed to the ceiling, and the lower end is connected to a weight with mass \( m \). It is known that the steel rings can only be in "State 1" (long axis oriented vertically) and "State 2" (long axis oriented horizontally). If there are \( n \) steel rings in State 1 and \( N-n \) steel rings in State 2, the energy of the system at this time is denoted as \( E_{n} \), with the potential energy corresponding to \( n=0 \) as the reference point for potential energy. Let \( \varOmega_{n} \) be the number of states of the system with energy \( E_{n} \) (i.e. \( E_{i}=E_{n} \)). Calculate the partition function \( Z=\sum_{n}\varOmega_{n}e^{-\beta E_{n}} \). | ||
390 | MECHANICS | A small ball with mass $m$ (considered as a point mass) is tied to one end of a very long but inextensible light string and placed on a smooth horizontal surface. The other end of the string is fixed to a vertical, stationary cylinder with radius $R$. Initially, the string wraps around the cylinder exactly once, and the ball is snug against the cylinder at point $P$. Let the center of the cylinder on this horizontal cross-section be O. At time $t=0$, an external force imparts an impulse to the ball in the radial direction, giving the ball an initial speed $v_0$, causing the ball and string to begin unwinding from the cylinder surface. Label the instant contact point where the string begins unwinding from the cylinder as $Q$. $\theta$ denotes the angle between OP and OQ (i.e., the angle the segment OQ has rotated relative to OP on the cylinder).
For the case where the cylinder is completely free: Assume the cylinder has a mass $M$, radius $R$, and its mass is concentrated in a thin layer on the surface. Now, the cylinder is not constrained and can move freely on the smooth horizontal surface (the cylinder does not tip over). At time $t=0$, an external force imparts an impulse to the ball in the radial direction, giving the ball an initial speed $v_0$, causing the ball and string to begin unwinding from the cylinder surface. Use the angle $\varphi$ to represent the angle PO has rotated relative to the horizontal plane. Determine $\theta$ as a function of time under this condition. | ||
621 | THERMODYNAMICS | Consider a submarine that discharges a bubble of oxygen (molecular weight $32$, diatomic gas) at a temperature of ${T}_{1}$ and volume ${{V}}_{0}$ from a depth $h$ on the seabed, with an initial velocity of 0. The atmospheric pressure at the sea surface is ${\mathrm{p}}_{0}$, and the temperature distribution below the sea surface is given by $\begin{array}{r}{{T}_{{x}}={T}_{0}-\frac{a}{h}x}\end{array}$, where $x$ represents the depth at that point. The density of water is $\rho$. Establish a simple model in which the underwater expansion and contraction of the gas, as well as heat equilibrium, occur quickly in a quasi-static process, and determine the amount of heat absorbed (released) throughout the process $\Delta Q$. The acceleration due to gravity is a constant $g$, and the oxygen is assumed to be an ideal gas. Express your answer using $T_0,T_1,p,\rho,g,h,a$. | ||
635 | MECHANICS | A person riding a bicycle can be simplified into the following model: A homogeneous rod of length 2L and mass 2m, with its two ends rigidly connected to two homogeneous rods of length L and mass m, respectively. The short rods are perpendicular to the long rod, and their ends are connected to wheels which have negligible volume and mass. The wheels are in contact with the ground. The person is considered as a point mass located at the center of the long rod, with a mass of m. If at a certain moment, the front wheel suddenly hits a small stone (with negligible size), causing the front wheel's speed to instantly become zero, what should be the constant speed V of the bicycle for the person and the bicycle to just avoid contact at this instant? | ||
459 | ELECTRICITY | The magnetoresistance effect refers to the phenomenon where the resistance value changes when a magnetic field is applied to a metal or semiconductor through which current is flowing. This phenomenon is called the magneto-resistance effect, also known as magnetoresistance (MR). Let the resistivity of the sample be $\rho_0$ when no magnetic field is applied, and $\rho_B$ when an external magnetic field is present. The magnitude of the magnetoresistance effect can be expressed as: $MR = \frac{\Delta\rho}{\rho_0}=\frac{\rho_B - \rho_0}{\rho_0}$.
The magnetoresistance effect in metals can usually be explained using a two-carrier model: In metals, there are two types of charge carriers, with mobilities $\mu_1$ and $\mu_2$ (mobility $=\frac{\text{drift velocity of the carrier}}{\text{magnitude of the field}}$), and corresponding conductivities $\sigma_1$ and $\sigma_2$. Derive the relationship between the magnitude of the magnetoresistance effect and the external magnetic field strength $B$. It is known that the external magnetic field is perpendicular to the current direction, and the change in resistivity $\Delta\rho$ is much smaller than $\rho_0$. | ||
34 | THERMODYNAMICS | Consider a cylinder with its axis along the vertical direction, with the base fixed on a horizontal surface and a piston sealing the top. The seal between the cylinder and the piston is perfect, without friction, and both are adiabatic. Inside the cylinder, there is a monoatomic gas composed of molecules with molecular mass $m$ and total molecule count $N_{0}$, at a temperature of $T_{0}$. The exterior of the cylinder is vacuum, and at this moment, the piston remains stationary. Let the gravitational acceleration be $g$, the bottom area of the cylinder be $S$, and the initial distance between the piston and the horizontal surface be $h$. The gas follows the Boltzmann distribution, where the probability distribution is proportional to $e^{\frac{E_{p}}{k T}}$, with $E_{p}$ representing gravitational potential energy and $k$ being the Boltzmann constant.
In the initial state, a certain mass of sand is slowly placed on top of the piston. Determine the temperature $\scriptstyle{T_{1}}$ inside the cylinder when the piston descends by $h/2$. For calculation purposes, assume $frac{mgh}{kT_{0}} = 1$, and express the result in terms of $T_{0}$, retaining four significant figures.
Possible formula that may be used:
$$
\mathsf{d}\left(\frac{x}{e^{x}-1}\right)=x\mathsf{d}\left(\frac{1}{e^{x}-1}\right)+\mathsf{d}\left(\ln\left(1-e^{-x}\right)\right)
$$ | ||
51 | MECHANICS | $n$ uniform thin rods, each with a length of $a$ and mass $m$, are sequentially connected by lightweight smooth hinges. The gravitational acceleration is known to be $g$. The head of the first rod is suspended at a fixed point O (the origin in the diagram), and the end of the last rod is subjected to a horizontal force $F$. Find the tangent value tan $\theta_{i}$ of the angle $\theta_{i}(i=1,2,...,n)$ between each rod in the equilibrium state and the vertical direction (the $y$-axis direction in the diagram). | ||
157 | MECHANICS | In fact, resonance does not necessarily need to be caused by an external force. A system may exhibit approximate resonance even if its own parameters are stable. Below is the study of a spring pendulum.
A light spring with original length $L$ and spring constant $k$ is suspended from the ceiling, with a mass $m$ hanging below. The gravitational acceleration is $g$. Assume the spring always remains as a straight line, and the system moves only within a two-dimensional plane. Initially, the spring is vertical, and the system reaches equilibrium.
Introduce the spring's inclination angle $\theta$ and the extension amount $x$ relative to the vertically suspended equilibrium to describe the system's motion.
Here we define $\delta=\frac{x}{L}$, $\frac{mg}{kL}=\eta$, $\omega_0^2=\frac{g}{l}$ to simplify your expression.
First expand to first-order approximation, with initial conditions as
$$
\delta\vert_{t=0}=0\\\partial_t\delta\vert_{t=0}=0\\
\theta\vert_{t=0}=0\\\partial_t\theta\vert_{t=0}=\omega
$$
After calculating the result, consider the second-order approximation. Please provide the resonance condition by calculating the equation of motion of $\delta(t)$ under second-order approximation, using only $\eta$ to express the answer. | ||
494 | ELECTRICITY | Within an infinitely long cylindrical region centered at point $O$, there exists a uniform magnetic field that varies with time. The magnetic field is directed into the paper, and its magnetic flux is proportional to time, satisfying $\varPhi=K t$. From the center $O$, a light, inextensible rod of length $l_{1}$ is extended, with its end connected to the center of another light rod of length $2l_{2}$ through a smooth hinge. At the two ends of the rod of length $2l_{2}$, there are fixed one positive and one negative point charge, each with mass $m$ and charge $\pm q$. Considering only the motion of the entire system in the plane of the paper, the charges do not enter the magnetic field region and are only subject to the vortex electric field forces caused by the changing magnetic field.
Under suitable initial conditions, the two rods precisely rotate uniformly about $O$ with a constant angular velocity, while maintaining a perpendicular orientation throughout the motion, and the tension in the rod of length $l_{1}$ remains zero at all times.
Based on the above motion, if the system is subjected to a small perturbation, the angle between $l_{1}$ and $l_{2}$ will no longer remain $90^{\circ}$ but will instead become $90^{\circ}+\delta$. Under suitable conditions, $\delta$ can oscillate harmonically with time. Determine the angular frequency $\omega$ of this oscillation. | ||
694 | ELECTRICITY | In a certain medium, electrons can be viewed as particles with an effective mass of \( m \), subject to a force \( F = -m\omega_{0}^{2}x - m\gamma\dot{x} \) from the surroundings. The first term is a linear restoring force where \( \omega_{0} \) is the natural frequency of the medium, and the second term is the damping force. Assume the electron has a charge \( q \) and an electron number density of \( N \) in the medium. Ignoring the magnetic part of the incident light, the electric field is \( E = E_{0} \text{e}^{\text{i} \omega t} \). The vacuum permittivity is \( \epsilon_0 \). By solving the electron motion, calculate the equivalent refractive index squared (which can be a complex number) of the medium. | ||
379 | MECHANICS | Uranus is a spherical planet with a uniformly distributed mass density of $\rho_{1}$ and a radius of $R$. In the cosmic space from a distance of $R$ to $2R$ from the center of the sphere, there is uniformly distributed cosmic dust with a density of $\rho_{2}$.
A space probe is launched from the surface of the planet at an angle of 45 degrees to the horizontal plane. Determine the critical velocity $v_{1}$ required for the probe to escape the range of the cosmic dust. | ||
706 | MECHANICS | Before a new model of car leaves the factory, it has to undergo destructive testing. During one such test, one of the three spokes of a car wheel was knocked off, leaving two spokes with an angle of 120° between them. The wheel can be considered as consisting of a disk with inner and outer radii of $R_{1}{=}4R_{0}/5$ and $R_{0}$, respectively, and two spokes (assuming the disk can be viewed as a uniform annular disk and the spokes as uniform thin rods). The angle between the two spokes is 120°, each spoke has a mass of $m$, and the disk has a mass of $M = 8m$. The wheel starts from rest and is released such that the bisector of the angle between the spokes is horizontal to the ground. Subsequently, the wheel undergoes pure rolling on a horizontal surface. Find: The angular acceleration of the wheel upon release.
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598 | ELECTRICITY | As shown in the figure, four smooth particles, each with mass $m$ and charge $+q$, are initially at rest on an infinitely large smooth table, forming a square with side length $L$. Ignore the electrostatic force between the charges. There is a uniform magnetic field $B$ perpendicular to the table surface and directed upwards. A lightweight elastic rope with a spring constant $k$ and original length zero, whose thickness can be neglected, wraps around the four particles to form a closed loop. The distance from the center $O$ to each particle is denoted as the radial distance $r$. Suddenly, each of the four particles is given an initial velocity $v_{0}$ in a clockwise direction perpendicular to the radial distance within the plane. Analyze the subsequent motion of the particles. At a certain initial velocity $v_{0}$, the particles will be able to reach the center point $O$ during their subsequent motion. By deriving the expression for $v_{0}$, ultimately find the relationship between the radial distance $r$ and the angle $\theta$ through which the particles rotate around $O$. | ||
356 | MECHANICS | A fine circular ring with a circular cross-section is subject to a uniformly distributed couple (whose vectors are all along the tangential direction of the ring). Experiments tell us that when the couple density \( m \) increases, the rotation angle \( \phi \) of the ring’s cross-section also increases; when \( m \) reaches a certain critical value, the ring will suddenly flip. Both the uniformly distributed couple and couple density \( m \) refer to the couple per unit length, \( m = \frac{dM}{ds} \); the flipping of the ring refers to the part that was originally on the inner side of the ring suddenly flipping to the outer side on the wire forming the ring. The wire forming the ring (radius \( R \)) has an elastic modulus \( E \) and a radius \( r \). Determine the critical value of the couple intensity at which this flipping occurs. | ||
758 | MECHANICS | The relationship between the energy and momentum of a quasiparticle is expressed as $H=H(p_{x},p_{y})$, which is significantly meaningful. Additionally, the velocity of the quasiparticle is given by the following equations: $$ v_{x}=\frac{\partial H}{\partial p_{x}}\quad,\quad v_{y}=\frac{\partial H}{\partial p_{y}} $$ Suppose in a certain crystal, the energy-momentum relationship of the quasiparticle is: $$ H=\frac{p_{x}^{2}+p_{y}^{2}}{2m}+\frac{p_{x}p_{y}}{m^{\prime}}\quad,\quad m<m^{\prime} $$ If there exists an orthogonal uniform electromagnetic field in space, with an electric field strength $E$ along the $+y$ direction, a magnetic induction strength $B$ along the $+z$ direction, and the quasiparticle carrying a charge of $q$, the quasiparticle initially at the origin and at rest, find the equation of motion in the $y$ direction $y(t)$.
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425 | MECHANICS | Generally speaking, the potential field we handle is a central force field, namely $V=V(r)$. Using the conservation of angular momentum: $m r^{2}{\dot{\theta}}=L$, eliminating $\dot{\theta}$ yields the effective potential energy: $\begin{array}{r}{V_{e f f}=V(r)+\frac{L^{2}}{2m r^{2}}}\end{array}$, we can transform the equation of motion into a radial one-dimensional motion equation. However, there are still a few non-central force fields that can yield similar conclusions using similar procedures. Following the aforementioned procedure, try to find the effective potential energy $V_{eff}$ of a particle with mass $m$ moving in the following potential field:
$$
V(r,\theta,\varphi)=f(r)+{\frac{F(\theta)}{r^{2}}}+{\frac{G(\varphi)}{r^{2}\sin\theta^{2}}}
$$
Leave an indefinite constant $C$ given by initial conditions, and express the final answer as a formula in terms of the constant $C$ and $r$. | ||
751 | THERMODYNAMICS | There is a horizontally placed disc-shaped box, filled with gas, rotating at a constant angular velocity $\omega$. The volume of the gas molecules themselves cannot be neglected, but for simplification, assume there are no interactions other than elastic collisions. The radius of the box is known to be $R$, and the pressure at the center is $p_{0}$. The volume \"actually occupied\" by each mole of gas molecules is $v$ (i.e., the parameter $b$ in the Van der Waals equation), the molar mass is $\mu$, and the temperature of the gas is $T$. Please calculate the distribution of the pressure $p$ as a function of the radius $r$ (express it in the form $r = f(p)$). | ||
615 | OPTICS | The gravitational field near a massive object can have a lens-like effect on passing light rays, known as gravitational lensing. Here we consider manufacturing an optical lens out of acrylic to simulate the gravitational lensing phenomenon. According to General Relativity, when a light ray passes a spherical symmetric object with mass M, it will experience deflection. When the deflection angle is small and the closest distance between the light ray and the object's center is r, the angle can be derived using the following formula: $$\varepsilon=4GM/rc^2$$ In the above formula, G is the gravitational constant and c is the speed of light. Therefore, we require the simulated lens to deflect light at an angle given by the formula: $$\varepsilon=R/r$$ In the above formula, R is a known constant. We can assume the front surface of the simulated lens is flat, and parallel incident light is perpendicular to the plane of the front surface of the simulated lens. Our goal is to design the shape of the rear surface of the simulated lens. $\varepsilon$ represents the deflection angle of the light ray. Assume in the axial cross-section of the lens, the equation for the rear surface is \( x(r) \), with \( r > 0 \). Let \( x(r_1)=0 \) at radius \( r_1 \), meaning \( x \) represents the relative thickness of the lens with respect to \( r_1 \). All angles are assumed to be small angles. The refractive index of the simulated lens is \( n \). Try deriving the expression for the thickness of the lens \( x(r) \). The answer should be expressed in terms of \( n \), \( R \), \( r \), and \( r_1 \). | ||
622 | THERMODYNAMICS | In a cylindrical cylinder with a cross-sectional area of ${A}$, there is a certain special gas. This gas has a molar constant-volume heat capacity $c_{ u}=\frac{3}{2}R$ when the temperature is below $2T_{0}$, and a molar constant-volume heat capacity $c_{ u}=\frac{5}{2}R$ when the temperature is above $2T_{0}$. The top of the cylinder is a piston with a mass of $m$, which separates the internal gas from the external vacuum. There is a small hole with an area of $S(S\ll A)$ on the piston. The piston can slide up and down in the cylinder without friction. The relaxation time of the gas is much shorter than the characteristic time of the piston movement. Initially, the distance from the piston to the bottom of the cylinder is $L$, the initial temperature of the gas inside is $T_{0}$, the gravitational acceleration is $g$, the Boltzmann constant is $k$, and the mass of the gas molecules is $\mu$. Do not consider the effect of gravity on the distribution of the gas. Calculate the time $t$ it takes for the gas to completely leak out starting from the initial moment. Express the answer using $L$, $k$, $\mu$, and $T_{0}$, and please provide a precise numerical expression without accepting decimal results. | ||
38 | ELECTRICITY | In this problem, we consider an anisotropic linear medium. Assume $D_{x}=\varepsilon_{x}E_{x}$, $\mathit{D_{y}}\overset{\widehat{}}{=}\varepsilon_{y}E_{y}$, $D_{z}=\varepsilon_{z}E_{z}$. Suppose there is a conducting sphere with a radius of $R$ in space, and find its capacitance. In this question, assume $\varepsilon_{x}>\varepsilon_{y}=\varepsilon_{z}$. | ||
656 | ELECTRICITY | A magnetic field exists in space with central symmetry, perpendicular to the paper and directed inward. The magnetic field changes with distance from the center, given by the formula \( B(r) = B_0 \left( \frac{r}{R} \right)^n \). A particle with charge \( q \) and mass \( m \) moves in uniform circular motion with a radius \( R \), velocity \( v \), and the center of the circle as the symmetry center. Now, if a small radial disturbance is applied to the particle, it will undergo small oscillations along the radial direction. Find the period of these small oscillations. | ||
747 | MECHANICS | Above a horizontal ground at a height of $h$, there is a ball launcher that can shoot elastic balls in various directions in the upper half-plane. It is known that the launching directions are uniformly distributed on the sphere (i.e., the probability of launching within each unit solid angle in various directions is the same). The initial velocity of the launched elastic balls is $v = \sqrt{g h}$, and the launched balls do not affect each other (i.e., disregarding collisions between balls). A single ball can be regarded as a solid sphere with uniform mass distribution, with a known mass of $m$. The launcher is activated at $t = 0$, and within an extremely short time (i.e., the launch time can be neglected, and it can be considered that the balls are launched simultaneously at $t = 0$), $N$ balls are launched ($N \gg 1$). If each elastic ball can be considered as a rigid sphere, undergoing completely inelastic collisions with the ground, and the friction coefficient between the elastic ball and the ground is $\mu = 0$. After all the launched balls have landed, try to find the number density distribution of the balls at a radial distance $r$ from the point directly below the launcher on the ground, $n(r, t)$. | ||
416 | THERMODYNAMICS | The left end of an adiabatic cylinder is connected to a large atmospheric reservoir via a magical semipermeable membrane. The atmospheric reservoir contains a monoatomic gas with constant pressure and temperature \(p_0\) and \(T_0\). Assume this semipermeable membrane has the following magical properties: diatomic molecules cannot pass through at all; when the partial pressure of monoatomic molecules inside the cylinder exceeds the partial pressure of monoatomic gas in the reservoir, monoatomic molecules can freely enter the reservoir, but molecules inside the reservoir can never enter the cylinder under any circumstances. Apart from allowing monoatomic molecules to pass through unidirectionally, this semipermeable membrane is considered adiabatic (heat exchange of diatomic gas cannot occur through the membrane, and monoatomic gas undergoes isobaric release when passing through the membrane).
The right side of the cylinder is an adiabatic piston. Initially, the cylinder is filled with equal molar amounts of monoatomic and diatomic molecules, with a total pressure of \(2p_0\), a temperature of \(T_0\), and a volume of \(V_0\). The piston is now slowly compressed to the left. Using the cylinder's temperature \(T\), volume \(V\), and their differentials \(dT\) and \(dV\) as variables at a given moment, derive a differential equation and solve to express \(V\) as a function of \(T\), \(V(T)\). | ||
565 | ELECTRICITY | A regular octahedral frame, with each edge having a resistance of $R$, an edge length of $l$, and negligible mass. Metal spheres with a mass of $m$ and no resistance are welded at each vertex. The frame is now placed in a uniform magnetic field $\boldsymbol{B}$, initially ($t=0$) rotating relative to its geometric center at an angular velocity of $\boldsymbol{\omega}_{0}$.
Find the angular velocity $\boldsymbol{\omega}(t)$ of the frame at any subsequent time. | ||
659 | ELECTRICITY | There is a smooth elliptical track, and the equation of the track satisfies $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 (a>b>0)$. On the track, there is a small charged object that can move freely along the track. The mass is $m$, and the charge is $q$, located at $(a,0)$. Point charge $Q$ is located at the left focus of the elliptical track. Find the period of the small charged object's small oscillations around its stable equilibrium position. | ||
320 | MECHANICS | Three homogeneous rods OA, AB, and AC, each with a length of \( L \) and mass \( m \), are hinged together at point A. Between point O and point B, as well as between point C and point O, there is a spring with natural length 0 and stiffness coefficient \( \mathrm{k} \). The rod OA is hinged at point O on a smooth surface, and the entire system is placed on a smooth surface, allowing it to rotate around an instantaneous axis through point O without being subjected to any additional forces. Initially, the system rotates uniformly at a certain angular velocity, and the rods AB and AC are perpendicular to rod OA. Now, a symmetrical disturbance is given to rods AB and AC, causing a small deviation of the angle between each rod and OA from \( \frac{\pi}{2} \). The system will then undergo small vibrations near the equilibrium state of uniform rotation. Find the period \( T \) of these small vibrations. | ||
760 | ELECTRICITY | Given that the angular momentum density of the electromagnetic field is $ \pmb{l}= \pmb{r} \times( \varepsilon_{0} \pmb{E} \times \pmb{B})$, there is a lightweight cylinder with a radius of $R$ and a length of $ L$($L\gg R$), carrying a uniform charge density $ \rho$, rotating about its vertical central axis with an angular velocity $ \omega$. To stop the cylinder from rotating, we apply a resistive torque on it that is proportional to the angular velocity, $ \pmb{M}=- \alpha \pmb{ \omega}$. Given that at $ \scriptstyle t=0$ the angular velocity of the cylinder is ${{ \omega}_{0}}$, determine the relationship between the angular velocity of the cylinder and time $t$. | ||
352 | ADVANCED | Here is a structure composed of three elastic beams. The left and right beams are both vertically suspended relative to the horizontal plane (with connection points at both ends), with lengths of 2l each. The upper connection points are marked as F and S, and the lower connection points are marked as G and T. The connections are hinged, meaning no additional constraint moments are generated; now a horizontal beam with length l is rigidly connected at the midpoints of both vertical beams, meaning any deformation here is continuous, and the two beams always maintain a perpendicular relationship. The bending stiffness of each beam is EI, that is $M=EI\frac{d^2y}{dx^2}$. Now, a concentrated downward vertical force P is applied at the midpoint of the horizontal beam AB, disregarding gravity. Calculate the downward displacement at the midpoint of the beam. ```markdown A structure is composed of three elastic beams. The left and right beams are vertically suspended relative to the horizontal plane with lengths 2l and connection points F, S on top and G, T at the bottom, respectively. The connections are hinged (without any additional constraint moments). A horizontal beam with length l is rigidly connected at the midpoints of these vertical beams, maintaining a perpendicular relationship under deformation and keeping it continuous. Bending stiffness of each beam is EI, expressed as $M=EI\frac{d^2y}{dx^2}$. A concentrated downward force P is applied at the midpoint of the horizontal beam AB. Ignoring gravity, the task is to determine the downward displacement at this midpoint. | ||
712 | THERMODYNAMICS | There is a closed container filled with water vapor, and the pressure inside the container remains constant. When we continuously cool the water vapor to reach the critical temperature of phase change, many small water droplets will condense inside the water vapor and suspend within it. Assume that the total mass of these small water droplets is much less than the total mass of the water vapor. Find the speed of sound propagation within it. To simplify the problem, assume that the wavelength of the sound wave is much greater than the scale of the system's inhomogeneity, and we can use the ideal gas state equation to describe the water vapor. The molar mass of water is $M$, the latent heat of vaporization is $L$, the temperature is $T$, the ideal gas constant is $R$, and the specific heat capacity at constant pressure of the gas is $c_{p}$ | ||
180 | ELECTRICITY | Place a thick conductive spherical shell (with inner and outer radii $R_{1}$ and $R_{2}$, respectively) in a uniform electric field (with a field strength of $E_{0}$), and allow it to reach electrostatic equilibrium. The dielectric constant of the conductive sphere is the vacuum permittivity $\varepsilon_{0}$. At time $t = 0$, the external electric field is suddenly removed, ignoring any induced effects caused by the removal of the field. The resistivity of the conductor is given as $\rho$. Determine how the charge distribution on the outer surface of the conductive spherical shell changes over time. | ||
502 | MECHANICS | The Magnus effect is a common fluid dynamics phenomenon in our daily lives. In real situations, the Magnus effect is very complex. This problem uses a simplified model without viscosity derived from the Kutta–Joukowski theorem to deduce a formula for the Magnus effect. The Kutta–Joukowski theorem describes a scenario where a cylinder and air are in relative motion, and the cylinder rotates around an axis perpendicular to its base. The cylinder will experience a force perpendicular to the angular velocity vector and the direction of relative motion velocity. The magnitude of the force per unit length along the cylinder's axis is $f=\rho V_{air} \Gamma$ (the direction of relative motion to air must be perpendicular to the angular velocity direction), where $\rho$ is the air density (known), $V_{air}$ is the relative motion speed of air and the cylinder as a whole, unaffected by the cylinder’s movement at infinity, and $\Gamma$ is the circulation satisfying the loop integral $\Gamma=|\int \vec{v}\cdot d\vec{l}|$, where the integral path is the circumference of the base of the cylinder. In this problem, you can consider $\vec{v}$ as the speed generated at the edge points due to the rotation of the cylinder. Air resistance is not considered in this problem, and the air is considered stationary relative to the ground. Now there is a sphere rotating around a horizontal axis passing through its center at an angular velocity $\omega$, while moving horizontally at a speed of $V$ perpendicular to the rotation axis relative to the ground. The sphere has a radius of $R$. Try to find the lift force magnitude caused by this motion of the sphere (the problem assures that the directions of angular velocity and motion velocity result in an upward lift on the sphere). | ||
440 | MECHANICS | A homogeneous hypersphere of radius $r$ is launched between two fixed parallel hard plates above and below, colliding with both plates 3 times in succession, nearly returning to the original position. The $x$-axis is directed horizontally to the right, the $y$-axis vertically downward, and the positive $z$-axis is determined by the right-hand rule. Initially, the horizontal component of the sphere's velocity is $\nu_{0x}$, the component in the $z$ direction is 0, and the angular velocity about the sphere's axis (parallel to the $z$-axis) is $\omega_{0z}$ $(\omega_{0z}<\nu_{0x}/r)$. Gravity is neglected.
Find the horizontal component of the velocity of the sphere’s center $\nu_{3x}$ and the angular velocity of the sphere $\omega_{3z}$ after the third collision with the plates.
Hint: It is known that the moment of inertia of a homogeneous sphere with mass $m$ and radius $r$ about its axis through the center is $J=2m r^{2}/5$. A hypersphere is a hard rubber ball, whose rebound on hard plate surfaces can be considered completely elastic; in other words, there is no sliding at the contact point. The tangential deformation and normal deformation that occur at the contact point when subjected to static friction and normal pressure are considered to be elastic. For simplification, assume these two types of deformation are independent of each other (hence the corresponding elastic forces are conservative forces). | ||
739 | ELECTRICITY | Given a circuit, a power source connects in series with an inductor $L_1$, followed by two parallel branches. Branch one consists of an inductor $L_2$ and a capacitor $C$ in series, and branch two contains inductor $L_3$, after which it returns to the power source. The power source is a constant source with an electromotive force of $E$. At the initial moment, there is no current anywhere in the circuit, and there is no charge on the capacitor plates. Determine the current $I(t)$ flowing through $L_1$ at any subsequent time $t$. | ||
581 | MECHANICS | In the $y O z$ plane, a thin steel wire is fixed, and the shape of the wire satisfies $y^{2}+z^{2}=a^{2}$. The wire is smooth. There is a uniform thin rod with mass $m$, one end of which is smoothly hinged at $A(a,0,0)$, where $a>0$. The length of the rod is $l>\sqrt{3}a$. The thin rod rests on the wire, and the contact point with the wire is $B$. The gravitational acceleration is known to be $g$.
Assume that initially, the contact point is $B(0,0,-a)$ and within the wire. After giving the rod a tangential perturbation, calculate the angular frequency of the rod's oscillatory motion. | ||
684 | MODERN | Atoms confined in a fixed-volume box have an average energy \[\epsilon\]. In a short time period \(\Delta \tau\), a small number of atoms \(\Delta N\) (where \(\Delta N<0\)) leave the box with an average energy of \((1+\beta)\epsilon\) (where \(\beta>0\)), resulting in a slight change in the average energy of the remaining atoms \(\Delta\epsilon\) (where \(\Delta\epsilon<0\)). Assume \[ \left|\frac{\Delta\epsilon}{\epsilon}\right|\ll 1,\quad \left|\frac{\Delta N}{N}\right|\ll 1, \] and the term \(\frac{\Delta\epsilon}{\epsilon}\frac{\Delta N}{N}\) can be neglected. Please derive the relationship between the change in average energy of the atoms remaining in the box and the change in the number of atoms, and provide the formula expressing \(\frac{\Delta\epsilon}{\epsilon}\) in terms of \(\frac{\Delta N}{N}\) and \(\beta\). | ||
174 | MECHANICS | A smooth insulated cone with a half-angle of $\theta$ is placed vertically with its vertex pointing downwards (the axis is vertical). At the vertex 0 of the cone, there is a fixed electric dipole with a dipole moment $p$, and the dipole moment direction is vertically upward along the cone's axis. Now, a small charged object (charge amount q>0, mass m) is released inside the cone at a vertical distance $h_{0}$ from the vertex. The initial speed of the small object is $v_{0}$, along the horizontal direction and tangent to the cone surface. Find the minimum vertical height from the vertex that the object can reach while moving on the cone surface. (Ignore the volume of the object)
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306 | MECHANICS | In a certain rigid vertical plane, a non-elastic light string of length \( l \) has one end fixed at point \( O \) and the other end connected to a small ball with mass \( m \). Initially, the string is straightened, forming an angle \(\theta\) (less than \(\frac{\pi}{4}\)) with the horizontal direction.And at this time, the position where the small ball is is higher than point O.
. At a certain moment, the small ball is released freely. Once the string is taut again, the small ball immediately starts moving along an arc orbit. Eventually, the ball will leave the arc orbit. Relative to point \( O \), find the maximum height \( h_m \) that the ball can reach after leaving the arc orbit. | ||
672 | OPTICS | A circular plate with radius $R$ and thickness $b$ is made of a material whose refractive index $n$ varies radially. The refractive index at the center is $n_{0}$, and at the edge is $n_{R}$. The refractive index inside the circular plate follows a certain distribution with respect to the radius, such that rays parallel to the principal axis can be perfectly focused. Please provide the focal length $f$ of this circular plate lens. | ||
456 | THERMODYNAMICS | Scientists are attempting to eradicate cancer cells using thermal therapy. We will model this process with a simple model. Assume a cancer cell is approximately a perfect sphere with a radius $R_{T}$, sharing the same thermal conductivity $k$ as the surrounding human body environment (which can be considered as a constant). The ambient temperature of the body is constant at $T_{0}$. The cancer cell contains a specific protein that, when irradiated by a certain laser, generates heat at a rate of $p$ per unit volume per unit time within the cancer cell. The effect of the laser on the human body environment outside the cancer cell can be neglected. Determine the temperature difference $\Delta T$ between the highest temperature point within the cancer cell in steady state and the body temperature $T_{0}$ (since the cancer cell is quite small and has a low number density, it can be considered isolated, and for simplification, thermal expansion of the cancer cell can be ignored, assuming that the temperature at $r=\infty$ is $T_0$). | ||
773 | MECHANICS | A right-angled isosceles triangular plate with mass M and uniformly distributed mass hangs from a hinge. Let the three vertices of the triangular plate be A, B, and C, where ∠A = 90°. Vertex B of the triangular plate is fixed on the hinge. Assume the right-angled sides of the isosceles triangular plate have a length of 𝑎, and the hinge is smooth. The triangular plate is in a stable equilibrium state. A particle with mass m enters the sloping side of the triangular plate at a horizontal velocity v. After entering, the particle embeds into the sloping side of the triangular plate. If the entire incident process is very short and the axis of rotation does not experience a horizontal impulse, the point where the particle enters is called the strike center. Try to give the larger of the two solutions for the vertical distance between the strike center and the hinge. Only the case within the plane is considered in this problem. Do not consider situations where the strike center does not exist.
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213 | ELECTRICITY |
A long straight coaxial insulating cylindrical shell is placed vertically, with inner and outer radii of $a$ and $b(b>a)$, respectively. Two sufficiently long ideal electrode sheets are fixed between the inner and outer shells, each with a width of $l=b-a$. The planes containing the two metal sheets both pass through the cylindrical axis, and the angle between the two planes is $\theta_{0}{\left(<2\pi\right)}$. A third metal sheet identical to the aforementioned metal sheets is inserted between the two metal sheets (within the dihedral angle of $\theta_{0}$). The mass surface density of the metal sheet is $\sigma$, and the angle between the plane of this sheet and one of the fixed sheets is $\theta\left(0\le\theta\le\theta_{0}\right)$.
A constant voltage DC power supply with an electromotive force of $V$ and negligible internal resistance is connected between the two electrodes (with the high-potential pole connected to the plate at $\theta=0$). The plane of the movable sheet always passes through the cylindrical axis. The collision between the movable sheet and the fixed sheet is completely inelastic, and during separation after colliding with the fixed sheet, the movable sheet can acquire $\chi(\chi<1)$ times the initial charge of the fixed sheet. The vacuum permittivity is given as $\varepsilon_{0}$.
When the sheet becomes positively charged due to collision with the plates and is at an angular position $\theta$, find the electric field torque $M_{+}(\theta)$ on the sheet relative to the axis of the cylinder. | ||
591 | ELECTRICITY | There are $n$ terminals, and there is a resistor $R$ between any two terminals. Find the resistance between any two terminals. | ||
651 | ELECTRICITY | Consider a tightly wound solenoid of finite length. The radius of the solenoid is $R$, the number of turns per unit length is $n$, and the current $I$ is known. However, the solenoid is only located between $-l<x<l$, and we need to calculate the x-components of the magnetic field $B_{x}$ away from the solenoid at the point $(x, y)$. It is known that $R << l$ and the permeability of vacuum is $\mu_{0}$.
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526 | ADVANCED | Calculate the strictly analytical solution for the average photon number of the cubic phase state
\[
|\psi\rangle = \exp\Bigl(i \gamma \,q^3\Bigr) \exp\Bigl(i \frac{P}{2}\,q\Bigr) \exp\Bigl(-\frac{s}{2}\,(a^2 - a^{\dagger\,2})\Bigr) |0\rangle
\]
where:
- \( \gamma ,\, P,\, s \) are parameters,
- \( q = a + a^\dagger \) (position operator),
- \( a \) is the annihilation operator, and \( a^\dagger \) is the creation operator. | ||
763 | ELECTRICITY | There is an infinitely long conducting thin cylindrical shell with a thickness of 𝛿 and a radius of 𝑅, and it has a resistivity of $ \rho$. A small permanent magnet with a magnetic moment 𝑚 is released along the central axis. The magnet is known to be small enough that its geometric dimensions can be ignored. It is made to fall at a constant speed 𝑣 using an external force. At a certain moment, the angle between the magnetic moment direction and the axis of the cylinder (which is oriented vertically downward) is $ \alpha$. It is known that $ \alpha$ is small (but do not make small-angle approximations), and the change in magnetic flux parallel to the axial direction can be neglected. During the descent, the magnetic moment undergoes slight oscillations, causing changes in the angle $ \alpha$. The rate of change is given by $ \dot{ \alpha} = \omega$. Calculate the magnitude of the torque exerted on the small permanent magnet by the copper pipe. | ||
538 | ELECTRICITY | In the center of an infinite grounded cylindrical metal surface with a radius of $R$, there is a thin, coaxial infinite metal wire with a radius of $r_0$. The wire is at a potential of $U_0 > 0$. The region beyond the metal wire and within the cylindrical metal surface is filled with a certain dielectric medium. Due to the influence of the electric field, the atoms of the medium nearby the exterior of the wire are ionized into free electrons and cations, where the free electrons are adsorbed by the wire upon ionization, while the cations move radially away from the wire. Assume the radial mobility of the cations (defined as the ratio of radial speed to electric field intensity) is a constant $w$, and during the migration process, the cations always form a uniform cylindrical thin layer surrounding the wire.
Assume the total electric charge of all the cations is $Q$. To keep the potential of the wire unchanged at the original $U_0$, it is necessary to supply an additional electric charge $Q^{*}$ to the wire. Determine the relationship between $Q^{*}$ and time $t$. | ||
680 | MECHANICS | Three homogeneous rods 1, 2, and 3, each with length \(2r\) and mass \(m\), are connected by smooth hinges \(A\) and \(B\). The bottom rod 1 is connected to a fixed point \(O\) on the ground through a smooth hinge. Initially, the lower two rods 1 and 2 are vertical, while the top rod 3 deviates to the right from the vertical direction by a small angle \(\varepsilon \ll 1\). The system is then released from rest. The gravitational acceleration is \(g\). Find the horizontal component \(F_x\) of the force \(\vec F\) at support point \(O\) at this moment, where the \(+x\) direction is horizontally to the right.
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400 | MODERN | In the $S$ frame, there is a clock positioned in the $_{\mathrm{X-Y}}$ coordinate system, with the 12 o'clock direction pointing along the positive $y$-axis and the 3 o'clock direction pointing along the positive $\mathbf{X}$-axis. The $S_{1}$ frame moves relative to the $S$ frame with velocity $\nu_{0}$ along the $\mathbf{X}$-axis. The $S_{2}$ frame moves relative to the $S_{1}$ frame with velocity $\nu_{0}$ along the $y$-axis of the $S_{1}$ frame. The $S_{3}$ frame moves relative to the $S_{2}$ frame with velocity $-\nu_{0}$ along the $\mathbf{X}$-axis of the $S_{2}$ frame. The $S_{4}$ frame moves relative to the $S_{3}$ frame with velocity $-\nu_{0}$ along the $y$-axis of the $S_{3}$ frame. Assuming $\nu_{_0}/c \ll 1$, keep only the first non-zero term in the result.
Find the angle of the clockwise deflection of the 12 o'clock direction of the clock as seen in the $S_{4}$ frame. | ||
696 | MECHANICS | Objects with masses $M_1$ and $M_2$ move in a circular orbit around the center of mass, with a distance $R$, angular velocity $\Omega$, and gravitational constant $G$. The center of mass is taken as the origin $O$ of the coordinate system, and the direction from $M_2$ to $M_1$ is the positive direction of the $x$-axis. We establish a rectangular coordinate system $xOy$ in the plane of the orbit, with angular velocity $\Omega$ in the $+z$ direction. A mass $m$ ($m\ll M_1, M_2$) is placed near the Lagrange point $L_4$, which is not located on the $x$-axis and has $y>0$, to study its small amplitude motion deviating from the equilibrium position. The initial deviation is known to be $(\delta x_0, \delta y_0) = (\epsilon, \delta y_0)$, and the initial velocity is $(\delta v_{x0}, \delta v_{y0}) = (-\frac{1}{2}\Omega\epsilon, \delta v_{y0})$, where $\delta y_0$ and $\delta v_{y0}$ are unknowns. Given that the masses satisfy $\frac{M_1 - M_2}{M_1 + M_2} = \sqrt{\frac{11}{27}}$, and it is required that the mass can perform stable motion near the equilibrium position, find $\delta v_{y0}$, expressed solely in terms of $\epsilon$ and $\Omega$.",
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452 | OPTICS | There is a hemispherical uniform isotropic medium, which is a light-emitting diode made of semiconductor material gallium arsenide (GaAs). Its core ($AB$) is the circular light-emitting region, with a diameter $d$, and is concentric with the circle on the ground. To avoid total internal reflection, the upper part of the light-emitting diode is polished into a hemispherical shape, so that the light emitted from the core has the maximum transmittance when emitted outward. If the light emitted from the edges of the emitting region at points $A$ and $B$ is required not to undergo total internal reflection, what should the radius $r$ of the hemisphere be? It is given that the refractive index of GaAs is $n$. | ||
660 | MECHANICS | A spring with an original length of $L_{0}$ and total mass $m$ has a total elastic coefficient of $k$ when it is uniformly stretched, and the mass distribution of the spring is also uniform at this time. Now, the spring is placed inside a horizontal, smooth-walled cylindrical tube and rotates with the tube around a vertical axis at a constant angular velocity $\omega$. One end of the spring is fixed at point $P$, which is a distance $r_{0}$ from the axis of rotation, and the spring remains stationary relative to the cylindrical tube. Try to find the length $l$ of the spring (you may introduce parameters $\omega_{0} = \sqrt{\frac{k}{m}}, \alpha = \frac{\omega}{\omega_{0}}$ to simplify the expression).
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767 | MECHANICS | A uniform cylindrical shell with an inner radius of $\pmb{R}_{1}$, an outer radius of $\pmb{R_{2}}$, a total length of $\pmb{L}$, and a mass of $M$ is placed vertically on a frictionless horizontal surface. Inside, there are $\pmb{n}$ turns of a screw with a single-layer width of $\pmb{t}$ and a single-layer height of $\pmb{d}$. Both the screw and the cylindrical shell are made of the same material with a density of $ρ$. A homogeneous small spherical shell with mass $m$ and radius $r$ is fixed on the screw by a light partition, and the small spherical shell is in close contact with the cylinder. We assume the following: $R_{1}, R_{2}, L \gg R_{2}-R_{1}, t, d, r; R_{1} \approx R_{2} \approx R$. Now suppose the inner wall of the cylinder is smooth, and the screw is also smooth. Remove the partition and release the small ball from the top (where the contact point of the small ball with the screw is on the same horizontal plane as the upper surface of the cylinder). Initially, the small ball is at rest and does not rotate. Calculate the speed of the ball when it falls to the bottom (where the contact point of the small ball with the screw is on the same horizontal plane as the lower surface of the cylinder). Note: the acceleration due to gravity is $g$. You can simplify the final expression using $$ \tan\theta={\frac{L}{2n\pi R}} $$ to ensure the final expression does not include $n$ and $R$.
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279 | ELECTRICITY | Consider a simple circuit that contains two parallel plate capacitors connected in parallel, with capacitances $C_{1}$ and $C_{2}$, connected via wires and a switch. Capacitor $C_{1}$ is initially charged to a voltage $V_{0}$, while $C_{2}$ is completely uncharged. Throughout the problem, the circuit remains within a square loop with side length $l$, and the conductive wire has a diameter of $D$.
Feynman made the following assumption: The missing energy is converted into the kinetic energy of charge carriers moving from $C_{1}$ to $C_{2}$. Assume the average free path $\lambda>2l$ of the carriers between collisions. Find the total kinetic energy $\Delta K$ acquired by the carriers during the transfer of total charge from $C_{1}$ to $C_{2}$. | ||
613 | MECHANICS | In this problem, we will explore how modifying the law of gravity changes the orbit. Suppose the gravitational potential energy $$ U=-k{\frac{e^{-r/a}}{r}} $$ is modified from $U=-{\frac{k}{r}}$, where $k=G M m$, and $a$ is a known constant with units of length. When $a\rightarrow\infty$, the closest and farthest distances of the Earth to the Sun are $r_{close}$ and $r_{far}$, respectively. For the modified potential energy $U=-k\frac{e^{-r/a}}{r}\Big(a\gg r_{far}\Big)$, if the Earth has the same energy and angular momentum and the orbit is still approximately elliptical, and the Earth's orbital equation is $$ r={\frac{b}{1+e \cos\theta}} $$ where $e$ is the eccentricity of the orbit. Try to find the expression for the change in eccentricity $\delta e$ of the Earth’s new eccentricity $e^{\prime}$. The answer should be expressed in terms of $r_{far},\quad r_{close}$, and $a$, retaining the first-order term of ${\frac{1}{a}}$.",
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585 | ELECTRICITY | This problem considers the role of symmetry and scaling transformations in physics for solving practical problems. Given the dielectric constant in vacuum being $\varepsilon_{0}$.
Consider a logarithmic spiral, whose curve equation is $r=a\mathrm{e}^{k\theta}$. Place $n$ equal point charges $q$ at $\theta=0, \alpha, 2\alpha, \cdots, (n-1)\alpha$. Find the electric potential $\varphi$ at the origin. | ||
395 | ELECTRICITY | There is a metallic sphere shell with a radius of $R$, and inside, there is a spherical metallic electrode with a radius of $r$. The space in between is filled with a medium with conductivity $\rho$. Assume the electrode is displaced from the center by a distance $l$, and that $r << l$, $r << R$, and the electrode is not near the boundary. Estimate the resistance between the electrode and the metallic sphere shell, accurate to the first-order approximation. | ||
215 | MECHANICS | A smooth horizontal shaft at a height $h$ from the ground is smoothly connected to a lightweight board of length $2L$. The shaft is fixed at the center of the board. A long, thin string is placed on the board, with a mass per unit length of $\lambda$. The lightweight board is sufficiently rough such that the string does not slip on its surface. The remaining part of the string is piled on the ground, and the suspended string does not experience any tension from the string pile. Initially, the system is in equilibrium.
Let $\theta$ be the angle of rotation counterclockwise from the equilibrium position. At time $t=0$, a small impulse is applied to the right side of the board, giving it an initial angular velocity $\omega_{0} = \sqrt{\alpha g/L} \left(\alpha \ll 1\right)$ in the counterclockwise direction, after which the board begins to oscillate. Since $\alpha \ll 1$ and $|\theta| \ll 1$, it can be approximated that the strings on both the left and right sides remain vertical. Let the tensions exerted on the endpoints of the board by the hanging strings on the left and right be $T_{l}$ and $T_{r}$, respectively. The gravitational acceleration is given as $g$.
Given $h = L/3$, find the time $T_{1/2}$ required for the average amplitude of the board to decay to half of its original value. | ||
466 | ELECTRICITY | On a smooth horizontal plane, there is a fixed regular $n$-gon $A_{1}A_{2}\dotsm A_{n}(n>3)$ with a center $O$. Each vertex has a point charge $Q(Q>0)$, and there is a point charge $q$ with mass $m$ at the center. Assume the side length of the regular $n$-gon is $a$. If the central charge is $-q(q>0)$, and the permittivity of vacuum is $\varepsilon_{0}$, a uniform magnetic field perpendicular to the plane is needed to constrain the point charge so it moves within a small range. Find the minimum value of the magnetic field. | ||
528 | ADVANCED | Consider the two-mode squeezing operator:
\[
S(\lambda, \theta) = \exp\left(\frac{\lambda}{2} \Big[ (a_1^{\dagger\,2} - a_2^{\dagger\,2})\sin\theta + 2a_1^\dagger a_2^\dagger \cos\theta - (a_1^2 - a_2^2)\sin\theta - 2 a_1 a_2 \cos\theta \Big] \right).
\]
where:
- \(\lambda\) and \(\theta\) are parameters,
- \(a_1, a_2\) are the annihilation operators for the two modes.
The task is to calculate:
\[
F = |\langle 00 | S^\dagger(\lambda, 0) S(\lambda, \theta) | 00 \rangle|^2,
\]
which is the fidelity between the squeezed states with different parameters, \(S(\lambda, \theta)|00\rangle\) and \(S(\lambda, 0)|00\rangle\). | ||
728 | MECHANICS | A bomb explodes on the ground, and after the explosion, all fragments are ejected at the same speed $u$, with directions confined within a narrow angle not exceeding \(\alpha \ll 1\) from the vertical upward direction (angles are uniformly distributed within this range). All fragments eventually land, and upon landing, undergo a completely inelastic collision. Let the mass of the bomb be $M$. Find the maximum radius \(R\) of the fragment distribution (retaining the first-order term of \(\alpha\)).
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405 | ADVANCED | The known form of the electromagnetic field momentum conservation theorem is expressed as:
\[\vec{f_m}+\frac{\partial \vec{g}_{em}}{\partial t}+\nabla \bm{T}_{em}\]
where $\vec{g}_{em}$ is the momentum density of the electromagnetic field, and $\bm{T}_{em}$ is the momentum flux density of the electromagnetic field.
Now, consider a given point charge $q$ moving with velocity $\vec{v}$ in a constant external magnetic field $\vec{B}_0$. By taking into account only the momentum density of the electromagnetic field while ignoring the momentum flux density, derive the force on the charge, i.e., provide the mechanical force density \[\vec{f_m}\]. The trajectory of the particle $\vec{r}_q(t)$ may be retained. | ||
220 | ELECTRICITY | There is a ring with radius \( R \) and uniform charge \( Q \), with the center of the ring at the origin. The ring is located in the \( oxy \) plane. Try to find the electric potential at the spherical coordinates \( (r, \theta) \) where \( r \gg R \) (require keeping the lowest order correction term up to \(\frac{R}{r}\)). | ||
658 | ELECTRICITY | A uniform metal ring with radius $a$ has 3 metal spokes connecting the center of the ring to the ring itself. The spokes and the ring are made of the same type of metal with an identical cross-sectional area. Each spoke has a resistance of $R_{1}$. The angles between the spokes are equal. The connections at the center and between the spokes and the ring are well established. There is a magnetic field perpendicular to the plane of the ring, with the upper half having a magnetic flux density of $3 B_{0}$ and the lower half having a magnetic flux density of $B_{0}$, both in the same direction. Now the ring (and spokes) rotate uniformly around the center within the plane at an angular velocity of $\omega$. Assume that at $t=0$, spoke 1 is exactly at the boundary between different magnetic fields and is about to enter the $3 B_{0}$ magnetic field area. Calculate the external torque required to maintain the uniform rotation of the ring. | ||
272 | MECHANICS | Three stars shining together means the sun is at its zenith. However, such a state does not have a large enough stability range. If considering external disturbances, during the long history of stellar evolution, there is a considerable probability that they will break apart into a binary star system and a solitary star, separated by cosmic space, like "divided by a mere stream, silently they gaze but cannot speak." The only thing that can transcend time and space might be the radiation that doesn't seem dazzling from afar and the propagation of gravitational fields that dominate their motion and evolution.
Still considering three celestial bodies, each having equal mass $m$. Take the reference frame in translational motion of the binary system’s center of mass, and use its center of mass as the coordinate origin. The orbit of each sub-star in the binary system is an ellipse with semi-major axis $a$ and eccentricity $e$. Orient the major axis direction as the pole direction of the planar polar coordinate system. Since the distance within the binary system is much smaller than the distance between them and the solitary star $a \ll r$, the binary system's period is much smaller than the time scale of the movement of its center of mass revolving with the solitary star under their mutual gravitational influence. Hence, when calculating the gravitational force exerted by the binary system on the solitary star, we can use the concept of period averaging. Now, we study the situation when the solitary star is at polar coordinates $(r,\theta_{0})$.
Next, consider the reaction exerted by the solitary star on the binary system, and the overall dynamic effect will cause a gradual change in the eccentricity of the binary orbit over time (with the semi-major axis unchanged). Calculate the rate of change $\partial e/\partial t$. Only keep the leading term. It is known that the gravitational constant is $G$. | ||
104 | MODERN | The power of the atomic bomb is immense, but triggering the force of nuclear weapons is not easy. One type of atomic bomb uses plutonium nuclear fission, with the core being a high-purity plutonium sphere with a radius of ${r_{0}}$. The first surrounding layer is uranium, whose sole function is to reflect neutrons overflowing from the center, with a reflectivity of ${R}$. Assume the diameter of the neutron reaction cross-section for plutonium is $d$, and the direction of neutrons reflected by the uranium layer is randomly distributed within a solid angle of $2\pi$. The total number of plutonium atoms is $N_{0}$. If a neutron is generated at the center of the plutonium sphere, find the probability $P$ that the neutron will react with plutonium, using expressions involving $ \alpha=\frac{3N d^{2}}{16r_{0}^{3}}, r_{0}, R$. | ||
607 | MECHANICS | This question discusses electric torpedoes. Since thermal engines generate a large amount of bubbles while operating underwater, which can reveal the target's position, modern naval warfare more often utilizes electric torpedoes. During navigation, a battery-capacitor combined power supply is used; currently, only the capacitor is considered: capacitance $C$, its internal resistance $R_{1}$, leakage resistance $R_{2}$, and rotor internal resistance $R_{3}$. The resistance of the ocean is very large, assuming the torpedo's front cross-sectional area is $A$, the rear end is equipped with a four-blade propeller, which has rectangular blades with length $a$ and width $b$, and the blades' normal forms an angle $\theta$ with the direction of propulsion. The resistance of seawater on the front end and blades is given by $\mathrm{d}F= k v\cdot\mathrm{d}S$, acting perpendicular to $\mathrm{d}S$, where $v$ is the velocity component perpendicular to $\mathrm{d}S$ at that location. It is known that the blades are subject to mechanical torque $\tau$ from the motor. Determine the forward speed $u$ of the torpedo.
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599 | MECHANICS | A perfectly flexible rope with a constant linear mass density $\lambda$ is wound into a semicircle of radius $R$ in the ground reference frame $S$, and it is strictly inextensible. An unknown downward impulse $I_0$ is applied tangentially at the left side at $\theta = \pi$. As a result, an upward tangential velocity $v_0$ is produced at the right free end of the rope (at $\theta = 0$). Determine the magnitude of $I_0$, disregarding relativistic effects.
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482 | MECHANICS | A rigid cubic component is composed only of three rigid homogeneous thin plates on the top, bottom, and right side. Each plate has an edge length of $2l$ and a mass of $m$. Establish a Cartesian coordinate system with the center $O$ of the cube as the origin. The vertical direction has a gravitational acceleration of $g$, with the positive $z$ axis pointing upwards. The $x$ and $y$ axes are parallel to the edges of the horizontal plates. Currently, at points $A(0,-l,-l)$, $B(0,l,-l)$, and $C(-l,0,l)$ on the plates, vertical, inextensible light ropes of length $l$ are used to connect to fixed horizontal supports from the wall (located at $z=0, x\leq 0$) at points $D(0,-l,0)$, $E(0,l,0)$, and $F(-l,0,0)$. Miraculously, the component is in a state of tensegral equilibrium. Based on this, calculate:
If the rope $B E$ is suddenly cut, find the tension$T_{CF}$ in the rope $C F$. | ||
353 | MECHANICS | There is now a circular ring with a radius of $R$, formed by a metal wire with a diameter of $d$. Two mutually perpendicular line segments, AB and CD, pass through the center of the circle. Points A, B, C, and D are located on the ring. An identical force $P$ is applied vertically downward on AB, while an identical force $P$ is applied vertically upward on CD. Find the relative displacement $\Delta c$ between points B and C in the plane perpendicular to the original undeformed plane of the ring. Given: Young’s modulus $E$ and shear modulus $G$. | ||
576 | THERMODYNAMICS | In outer space, there is a magical gas planet that has a layer of solid on its surface but is gas internally. The molecular mass is $m$, and there are a total of $\pmb{N}$ molecules. The planet has a radius of $R$. The solid layer on the surface and the internal gas rotate at the same angular velocity $\omega$. The mass of the solid surface can be ignored, as well as any interactions (including gravitational) between the gas molecules. In this problem, assume the gas can instantaneously reach equilibrium, and the external temperature is $\pmb{T}$ (where $\pmb{T}$ is very small) and constant.
Find the moment of inertia $I$ of the planet around its axis of rotation when the rotation speed is $\omega_{0}$, approximated to the first order of small quantities (using $T, \omega_{0}, N, m, R$). | ||
725 | ELECTRICITY | Between two infinitely large parallel grounded conducting plates (with a distance \(l\) between the plates), a point charge \(q\) is placed, which is at a distance \(z_0\) from the lower plate. Find the total induced charge \(Q_u\) on the upper plate, expressing the result in terms of \(q\), \(z_0\), and \(l\). |
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