R-Bench/R-Bench-V · Datasets at Hugging Face

catagory stringclasses 4 values	question stringlengths 39 1.54k	image imagewidth (px) 74 4.95k ⌀	answer stringlengths 1 103
Physics	As shown in the figure, in the coordinate system Oxyz, the x-axis and z-axis lie in the plane of the paper, while the y-axis is perpendicular to the paper, pointing inward. Two infinitely large metal plates P and Q are located at x = -d and x = d, respectively. A uniform magnetic field of magnitude B lies in the Oxz plane and makes an angle α with the z-axis. A positively charged particle with charge q (> 0) and mass m is initially located at the origin O and moves with an initial velocity v in the positive y-direction. Neglecting the effect of gravity, and assuming no electric field is applied between the plates, what should the initial speed v be so that the particle just reaches the plates in space without touching them?		$\frac{qBd}{2m\cos\alpha}$
Physics	As shown in the figure, triangle Oab is a right-angled triangle with its two perpendicular sides formed by insulated dielectric rods, and the hypotenuse ab formed by a thin metallic rod. The length of Ob is equal to ab/2 = l. The triangle is placed in a uniform magnetic field with magnetic induction B, which is perpendicular to the plane of the triangle and directed into the page. The triangle rotates at a constant angular velocity $\omega$ about an axis that passes through point O and is parallel to the magnetic field direction. Determine the potential difference $U_{ab} = U_a - U_b$ between points a and b.		$Bl^2\omega$
Physics	As shown in the figure, a closed wire loop consists of 12 straight segments, each of length l. Except for segments 1-8, 2-9, and 3-4 (represented by dashed lines), which have negligible resistance, the remaining nine segments each have a resistance of r. A uniform magnetic field B is directed perpendicular to the plane of the loop and points into the page. The boundary of the magnetic field, labeled MN, is parallel to the side of the loop formed by segments 5-6-7, as shown. The loop is pulled uniformly to the right with a constant velocity v, gradually moving it out of the magnetic field region. Determine the work done by the pulling force during this process.		$\frac{2B^2 l^3 v}{r}$
Physics	A straight conductor MN and a metal ring with radius r are placed overlapping on a smooth horizontal surface. The straight conductor MN is fixed and located at a distance of r/2 from the center O of the stationary metal ring, and there is good electrical contact at the point where the ring intersects the conductor. A uniform magnetic field with magnetic induction $B_0$, perpendicular to the horizontal surface and in the direction shown in the figure, is present in the space. Determine the velocity of the metal ring immediately after the magnetic field is rapidly removed. Assume the electrical resistance per unit length of both the straight conductor and the ring is $r_0$, and the mass per unit length of the ring is $m_0$.		$\frac{27}{32\pi + 72\sqrt{3}} \cdot \frac{rB_{0}^{2}}{2\pi m_{0}r_{0}}$
Physics	A thin wire forms a loop along the edges of one face of a cube, with its self-inductance denoted as L₁, as shown in diagram (a). Using the same wire, a loop is formed along the edges of the cube, yielding a self-inductance of L₂, as shown in diagram (b). Determine the self-inductance of a loop formed using the same wire along the edges of the cube as shown in diagram (c), where the loops in all diagrams are indicated by bold black lines.		3(L_2 - L_1)
Physics	A uniformly thick and homogeneous conductive wire with a nonzero constant resistivity is bent into an equilateral triangular loop with side length \( a \). The triangle rotates at a constant angular velocity \( \omega \) about one of its sides, AB, which serves as the axis of rotation. A uniform magnetic field with magnetic induction \( B \) is oriented perpendicular to the rotation axis AB. Determine the electric potential difference between points B and A on the wire loop, as shown in the figure.		$\frac{\sqrt{3}}{12}B\omega a^2\sin\omega t$
Physics	As shown in the figure, two identical metal rings each have a radius R, mass m, and resistance r. They are both placed in the same uniform magnetic field, with the field direction perpendicular to the plane of the rings and pointing into the page. The magnetic flux density is $B_0$. The contact points A and C between the two rings are in good electrical contact, with negligible contact resistance, and the angle between the rings is $a = \frac{\pi}{3}$. If the magnetic field is removed, what speed will each ring acquire? Assume that the displacement of the rings during the removal of the magnetic field is negligible and that friction can be ignored.		$\frac{9\sqrt{3}R^3B_0^2}{10mr}$
Physics	As shown in the diagram, the circuit contains two resistors each with resistance \( R_1 = R_2 = R \), and one capacitor with capacitance \( C \); two ideal AC ammeters, \( A_1 \) and \( A_2 \), measure currents of 0.3 A and 0.2 A respectively. The circuit is connected to an AC power supply with an RMS voltage of 36 V and a frequency of 50 Hz. Determine the capacitance \( C \) of the capacitor.		$13.5\mathrm{\mu F}$
Physics	As shown in the figure, the inductances of the two coils are $L_1 = 10,\mathrm{mH}$ and $L_2 = 20,\mathrm{mH}$; the capacitances of the two capacitors are $C_1 = 10,\mathrm{nF}$ and $C_2 = 5,\mathrm{nF}$; the resistor has resistance $R = 100,\mathrm{k\Omega}$. The switch $S$ has been closed for a long time. It is then opened, and after a time interval $t_0$, the currents through $L_1$ and $L_2$ are measured to be $i_{01} = 0.1,\mathrm{A}$ and $i_{02} = 0.2,\mathrm{A}$ respectively, and the voltage is $U_0 = 40,\mathrm{V}$. Calculate the current in the wire segment AB (take the current as positive when flowing from A to B).		-0.1 (A)
Physics	A U-shaped glass capillary tube with uniform thickness and open ends is placed in a vertical plane in an atmosphere with pressure \( p_0 \). The two vertical arms of the tube each have a height of \( h \), and the horizontal section connecting them has a length of \( 2h \). The tube has a radius \( r \) (where \( r << h \)). The horizontal section is completely filled with mercury of density \( \rho \), as shown in the figure. One of the vertical arms is then sealed at the top, so that the gas enclosed within remains at pressure \( p_0 \). If the U-tube begins to rotate uniformly around the axis of the other (open) vertical arm, what must the angular velocity be for the length of mercury in the horizontal section to remain stable at \( \frac{5h}{3} \)? (Assume that, during this motion, the inclination of the mercury surface inside the tube can be neglected.)		$\sqrt{\dfrac{3(3p_0 + 2\rho gh)}{35\rho h^2}}$
Physics	(Multiple Choice) The diagram shows a vertical cross-section of a circular pool. AB is the diameter of the water surface, MN is the diameter of the pool bottom, and O is the center of the circular pool bottom. It is known that ON is 11.4 m, AM and BN are sloped edges, the water depth is 5.0 m, the refractive index of water is 4/3, and the water is extremely clear, with absorption by water being negligible. Points a, b, c, and d in the diagram are four light sources. The sky is blue and the water surface is flat. A diver is lying on his back in a groove at the center of the pool bottom, with his eyes located at point O. From this position, the maximum visible diameter of the water surface is AB. Which of the four light sources a, b, c, and d can have their light reach the diver’s eyes through total internal reflection?		c，d
Physics	The diagram shows a tank filled with a liquid of refractive index \( n \), with a symmetric, ridge-shaped thin transparent cover ADB placed over its central part. The cover has a vertex angle of \( 2\varphi \), encloses air inside, and is fully submerged in the liquid. A bright point C is located at the midpoint of the tank bottom AB. Assume the tank is sufficiently wide, and the thin cover walls have negligible effect on light refraction. Determine the condition under which an eye located above the liquid surface in the plane of the diagram can see the bright point from the side (give the minimum value of \( \tan \varphi \)).		$\sqrt{n^2 -1}-1$
Physics	As shown in the figure, in a vacuum, there is a small sphere with a uniform texture, radius \( r \), and refractive index \( n \) (where \( n > n_0 \), and \( n_0 \) is the refractive index of vacuum). A narrow laser beam with frequency \( \nu \) travels in a straight line BC through the vacuum. The line segment CD passes at a distance \( l \) from the center O of the sphere (with \( l < r \)). The laser beam enters the sphere (which acts as an optical medium) at point C on the surface of the sphere through refraction, and then exits the sphere at point D on the surface through another refraction back into the vacuum. Assume the frequency of the laser beam remains unchanged through both refractions. Find the magnitude of the average force exerted by a photon in the laser beam on the sphere during the two refraction processes.		$\frac{lh\nu}{r^{2}}\left[n-\sqrt{1-\frac{(n^{2}-1)l^{2}}{r^{2}-l^{2}}}\right]$
Physics	As shown in the figure, L₁ and L₂ are two coaxial thin convex lenses with a common principal axis OO’. Lens L₁ has a focal length of f₁ = 10 cm and an aperture (diameter) of d₁ = 4.0 cm; lens L₂ has a focal length of f₂ = 5.0 cm and an aperture of d₂ = 2.0 cm. The distance between the two lenses is a = 30 cm. AB is a uniformly luminous circular disk with a diameter d = 2.0 cm, aligned coaxially with the lenses and placed 20 cm to the left of L₁, forming an image on a screen P located to the right of L₂ and perpendicular to OO’. It is observed that the central part of the image on the screen is brighter than the edges. To make the edges of the image as bright as the center—without changing the position or size of the image—a third coaxial thin lens L₃ is to be inserted along the axis OO’. What is the minimum required aperture (diameter) of lens L₃?		2.0 (cm)
Physics	A slender rod ABC rests against a step in a vertical plane, with end A free to slide along the horizontal ground toward the step, while the rod remains in contact with the edge of the step. When the angle between the rod and the horizontal ground is $\phi$ (as shown in the figure), point B is exactly at the edge of the step, and the magnitude of the velocity of end C is exactly twice that of end A. Determine the ratio of the length of segment BC to that of segment AB.		$\frac{\sqrt{4-\cos^2 \phi}}{\sin\phi}$
Physics	Two rods, each of length $l$, are connected by a hinge at point $P$, as shown in the figure. One end of the first rod is fixed at point $O$ via a hinge, while the free end $Q$ of the second rod begins to move with a constant velocity $v$ in both magnitude and direction. At the initial moment, the velocity $v$ is directed along the angle bisector of the angle $2\alpha$ between the two rods. Determine the magnitude of the acceleration of the hinge point $P$ connecting the two rods after a very short time has elapsed since the motion begins.		$\dfrac{v^2}{4l\sin^2 \alpha \cos \alpha}$
Physics	Two motorboats are towing a barge, with their velocities given by $v_1$ and $v_2$, respectively. The angle between the two velocity vectors is $\alpha$, and at this moment, the velocity vectors $v_1$ and $v_2$ are aligned with the directions in which the motorboats are moving, as shown in the figure. Determine the magnitude of the barge’s velocity.		$\dfrac{\sqrt{v_1^2+v_2^2-2v_1 v_2 \cos\alpha}}{\sin \alpha}$
Physics	As shown in the figure, a light rope of length l has identical small balls of mass m attached to both ends, and a third ball of mass M attached at the center; all three balls are initially at rest on a smooth horizontal surface, with the rope fully stretched. Now, the central ball M is given an impulse, resulting in an initial velocity v perpendicular to the rope. Determine the tension in the rope at the instant just before the two end balls collide.		$\dfrac{M^2mv^2}{(M+2m)^2l}$
Physics	As shown in the figure, a prism ABC with a triangular cross-section is constrained to move along a smooth track such that its edge AB can only slide along the smooth rail DE. A smooth small ball, having the same mass m as the prism ABC, moves in the same horizontal plane along a direction perpendicular to the rail DE and collides elastically with the initially stationary prism ABC. Find the magnitude of the velocity of the prism ABC after the collision.		$\dfrac{2v_0}{3}$
Physics	As shown in the figure, a uniform solid cylinder of mass m and radius R is wedged between a wooden plank and a vertical wall. The coefficient of kinetic friction between the cylinder and both the wall and the plank is 0.75. The plank is very light and its mass can be neglected. One end of the plank, point O, is connected to the wall via a smooth hinge, while the other end, point A, is suspended with a hanging mass m'. The length of OA is L, and the angle between the plank and the vertical wall is 53°, corresponding to a right triangle with side ratios of 3:4:5. What is the minimum value of m' required to keep the system in equilibrium?		$\dfrac{25Rm}{4L}$
Physics	A uniform thin wooden rod of length $l$ is suspended vertically by a fine thread and positioned above the horizontal surface of water in a bucket, as shown in the figure. As the bucket is slowly raised, the rod gradually becomes submerged in water. When the submerged depth of the rod exceeds a certain value $h$, the rod begins to tilt. Determine the value of $h$, given that the density of the rod is $\rho$ and the density of water is $\rho_0$.		$l\left(1-\sqrt{1-\dfrac{\rho}{\rho_0}}\right)$
Physics	Rod AB is placed inside a cylindrical container. Point A of the rod is hinged at the junction between the cylinder wall and the bottom, while point C of the rod rests against the edge of the cylinder. Both points A and C lie in a vertical plane that passes through the axis of the cylinder, as shown in the figure. The rod forms an angle $\alpha$ with the horizontal. The rod is then moved along the cylinder's edge to a new contact point C', such that the angle $\angle C'OC$ is $\varphi$. What is the minimum coefficient of friction required for the rod to remain in equilibrium at position C'?		$\frac{\tan\frac{\varphi}{2}}{\sqrt{1+\cot^{2}\alpha\cos^{2}\frac{\varphi}{2}}}$
Physics	Three strings are floating on the water surface with their endpoints connected, as shown in the figure. Strings 1 and 2 are each 1.5 cm long, and string 3 is 1 cm long. A certain impurity is dropped at point A inside the circular region, causing the surface tension coefficient of the water at that point to decrease by a factor of 2.5. Determine the tension in each string. The original surface tension coefficient of water is given as 0.07 N/m.		0(N); 1.67e-4(N); 1.67e-4(N)
Physics	Mercury is poured onto a clean, horizontal, flat glass plate, and due to gravity and surface tension, it approximately forms a circular disk shape (with its sides bulging outward). A vertical cross-section through the axis of the disk is shown in the figure. Given the density of mercury $\rho = 13.6 \times 10^3 \mathrm{kg/m^3}$ and the surface tension coefficient $\sigma = 0.49 \mathrm{N/m}$, estimate the thickness $h$ of the disk when its radius is very large. (Provide the answer with one significant figure.)		3e-3 (m)
Physics	In the figure, object a is a fixed uniformly charged sphere with radius R, and O is its center. Given that the electric potential on the sphere's surface is U = 1000 V when the potential at infinity is taken to be zero, an electron b is located near a point O' far from the center O. The electron is moving toward sphere a with a kinetic energy of \( E_k = 2000\ \mathrm{eV} \) in a direction parallel to the line connecting O and O'. Let l be the perpendicular distance from the electron's initial path to the O–O′ line. To allow the electron to reach and collide with the surface of the charged sphere a, determine the maximum possible value of l.		$\dfrac{\sqrt{6}}{2}R$
Physics	A square network made of uniform resistive wires, as shown in the figure, consists of 9 identical small squares, with each side of a small square having a resistance of $r = 8\,\Omega$. A battery with an electromotive force of $E = 5.7\,\mathrm{V}$ and negligible internal resistance is connected to the network. If points C and D are connected by an ideal wire (with negligible resistance), determine the current flowing through this wire.		0.267 (A)
Physics	As shown in the figure, $R_1 = 2\Omega$, $R_2 = R_3 = 6\Omega$, the total resistance of $R_4$ is $6\Omega$, and the internal resistance of the power supply is $r = 1\Omega$. When switch K is moved to the upper position, the voltmeter reads 5V. When the sliding contact is set at a certain position, the power consumed by $R_4$ reaches its maximum; what is this maximum power value?		2 (W)
Physics	In the circuit shown, all power sources have zero internal resistance, and points B and C are connected to the right to an infinite network composed of alternating $1.0\Omega$ and $2.0\Omega$ resistors; determine the amount of charge on the plate of the $10\mathrm{\mu F}$ capacitor that is connected to point D.		1.3e-4 (C)
Physics	As shown in the left diagram, the resistors $R_1$ and $R_2$ each have a resistance of $1\mathrm{k\Omega}$, and the electromotive force is $E = 6\,\mathrm{V}$. Two identical diodes $D$ are connected in series within the circuit, and the $I_D$-$U_D$ characteristic curve of diode $D$ is provided in the right diagram. Calculate the power dissipated by resistor $R_1$.		16 (mW)
Physics	Modern material growth and microfabrication techniques have enabled the creation of microstructured semiconductor devices that confine the motion of electrons within a single plane (two-dimensional), allowing electrons to travel through the device like bullets, unaffected by scattering from impurity atoms. This property holds promising potential for new applications. As shown in the left diagram, a four-terminal, cross-shaped, two-dimensional electron gas semiconductor allows current entering from terminal 1 to exit from terminals 2, 3, or 4, controlled by a magnetic field. Studying the following model structure aids in understanding the current flow in such a four-terminal cross-shaped conductor. In the right diagram, a, b, c, and d represent the cross-sections of four closely spaced cylindrical rods, each with radius R, forming four narrow slits labeled 1, 2, 3, and 4. A uniform magnetic field exists in the vacuum region enclosed by the cylinders and slits, directed perpendicularly into the plane of the page. Let $B$ represent the magnetic induction. A particle of mass m and positive charge q enters the magnetic field through slit 1 with velocity v in a direction tangent to both cylinders a and b. Assume the particle undergoes exactly one perfectly elastic collision with one of the cylinder surfaces, with negligible collision time and no change in charge or influence from friction. Determine the value of B that allows the particle to exit through slit 2, traveling in a direction tangent to both cylinders b and c.		$\frac{mv}{3qR}$
Physics	As shown in the figure, two plane mirrors A and B have their reflecting surfaces perpendicular to the plane of the paper, and the line of intersection between the two mirrors passes through point O in the diagram. The angle between the two mirror surfaces is $\alpha = 15^\circ$. From point C on mirror A, a light ray is emitted in the plane of the paper at an angle of $\beta = 30^\circ$ to mirror A. After undergoing multiple reflections between the two mirrors, the light ray eventually ceases to interact with the mirrors. Assuming both mirrors are sufficiently large and the distance CO is 1 meter, how much time elapses from the moment the light ray leaves point C to its final reflection?		9.1e-9 (s)
Physics	A thick glass tank has a bottom thickness of 5 cm and contains water to a depth of 4 cm, as shown in the figure. Given that the refractive indices of glass and water are 1.8 and 1.33 respectively, what is the apparent distance from the water surface to the bottom surface of the glass tank when viewed vertically from above?		6.34 (cm)
Physics	A horizontal parallel-plane glass plate H, 3.0 cm thick and with a refractive index of n = 1.5, has a small object S located 2.0 cm below its bottom surface. Above the glass plate is a thin convex lens L with a focal length f = 30 cm, and its principal axis is perpendicular to the surface of the glass plate. The object S lies on the principal axis of the lens, as shown in the figure. If an observer above the lens sees the image of S aligned with the actual position of S along the principal axis, what is the distance between the lens and the upper surface of the glass plate?		1.0 (cm)
Physics	A black sphere with radius R is placed inside a cylindrical tube whose inner surface reflects but does not absorb light. A point light source S is located at a distance 2R from the center of the sphere O, with both O and S lying along the axis of the cylinder, as shown in the figure. To ensure that all light emitted by the point source toward the right hemisphere is eventually absorbed by the black sphere, what is the maximum possible inner radius r of the cylinder?		$\dfrac{2\sqrt{3}}{3}R$
Physics	As shown in the figure, a narrow monochromatic light beam is incident on two parallel glass plates. The beam is parallel to the optical axis $OO_1$, which is perpendicular to the glass plates and passes through their center. The beam is at a distance $R = 3\,\mathrm{cm}$ from the axis, and the thickness of the glass plates is $H = 3\,\mathrm{mm}$. The refractive index of the glass varies radially according to the relation $n(r) = n_0\left[1 - \left(\frac{r}{r_0}\right)^2\right]$, where $n_0 = 1.5$ and $r_0 = 9\,\mathrm{cm}$. Determine the angle between the emergent light beam and the optical axis.		1.91°
Physics	As shown in the figure, in a Lloyd's mirror experiment, a point light source S is located 2 mm above the mirror plane; the mirror is positioned midway between the light source and the screen. The length of the mirror is l = 40 cm, the distance between the screen and the light source is D = 1.5 m, and the wavelength of the light is 500 nm. Determine the number of interference fringes observed on the screen.		12.25
Physics	Three identical capacitors are connected as shown in the diagram. Initially, capacitor 1 carries a charge Q with its upper plate positively charged, while capacitors 2 and 3 are uncharged. A wire is used to connect points a and b; then a and b are disconnected, and points a and c are connected; after disconnecting a and c, points a and b are reconnected; finally, points a and d are connected. Determine the charge on the lower plate of capacitor 2 at the end of this process.		Q/12
Physics	In the network shown in the figure, the current values and directions on some branches, certain component parameters, and the electric potentials at junction points are known (all relevant values and parameters are marked in the figure). Using the given data and parameters, determine the magnitude of the current in the branch containing the resistor $R_x$, taking the current as positive when it flows from top to bottom.		2 (A)
Physics	The cross-section of a thermometer is shown in the diagram. It is known that the thin mercury column A is located at a distance of 2R from the vertex O of the cylindrical surface, where R is the radius of the cylindrical surface. Point C represents the position of the central axis of the cylinder. The refractive index of the glass is n = 3/2, and E represents the position of the human eye. Determine the magnification of the image of the mercury column as seen by the eye within the illustrated cross-section.		3
Physics	A smooth horizontal circular table has a radius R = 1.00 m and a vertical post fixed near its center O. The intersection between the post and the tabletop forms a convex, smooth, closed curve C, as shown in the figure. A soft, light, inextensible string has one end fixed to a point on curve C, and the other end is tied to a small block of mass m = 75 g. The block is placed on the table and the string is pulled taut; then the block is given an initial velocity of 4.0 m/s perpendicular to the string. As the block moves on the tabletop, the string wraps around the post. It is known that the string breaks when the tension reaches 2.0 N, and before the string breaks, the block always remains on the tabletop. When the string is just about to break, the line from the table center O to the point where the taut string contacts the closed curve is perpendicular to the taut part of the string. What is the horizontal distance from the point where the block lands to the center O of the table, given that the table height is h = 0.80 m and the block does not collide with the post during its motion on the table? Take the gravitational acceleration as $10\ \mathrm{m/s^2}$.		2.5 (m)
Physics	In certain heavy machinery and lifting equipment, a double-block electromagnetic brake is commonly used. Its simplified schematic diagram is shown in the figure, with $O_1$ and $O_2$ as fixed hinges. When the power is on, the lever A is pressed downward, pulling spring S through hinges $C_1$, $C_2$, and $C_3$, causing brake pads $B_1$ and $B_2$ to disengage from the brake wheel D, allowing the machinery to operate normally. When the power is cut off, the lever A loses its downward force (neglecting the weight of lever A and all other linkages and brake components in the figure), and the spring contracts, pressing the brake pads into contact and generating a braking force. At this moment, $O_1C_1$ and $O_2C_2$ are in a vertical position. Given that a minimum braking torque of M = 1100 N•m is required to decelerate the uniformly rotating brake wheel D, with a coefficient of friction of 0.40 between the brake pads and the brake wheel, an unstretched spring length of L = 0.300 m, a brake wheel diameter d = 0.400 m, and diagram dimensions a = 0.065 m, $h_1$ = 0.245 m, and $h_2$ = 0.340 m, determine the minimum required stiffness coefficient k of the spring to achieve the necessary braking effect.		1.49e4 (N/m)
Physics	A cylinder A with radius R rests stationary on a horizontal surface and is in contact with a vertical wall. Another thinner cylinder B, having the same mass as A and radius r, is placed on top of cylinder A such that it also touches the wall, as shown in the figure. Cylinder A is held in place by hand while placing B, and then released. The coefficient of static friction between cylinder A and the ground is 0.20, and the coefficient of static friction between the two cylinders is 0.30. If the system remains in equilibrium after release, what is the minimum value of the ratio r/R?		0.289940828402367
Physics	A pendulum with a string of length l (with the bob considered a point mass and the string's mass negligible) is set up such that a fixed peg is placed at point C on the vertical line passing through the pivot point O, at a distance x below O (where x < l), as shown in the figure. As the pendulum swings, the string may be obstructed by the peg. When l is fixed and x varies, the motion of the pendulum bob after being obstructed differs accordingly. Initially, the pendulum is pulled to a position to the left of the vertical line (with the bob's height not exceeding that of point O) and then released to swing freely. If, after being blocked by the peg, the bob just manages to strike the peg, determine the minimum value of x.		0.464l
Physics	A rigid, uniform, thin circular ring with radius R and mass $m_0$ is initially at rest on a smooth horizontal table. There is a small hole $P_0$ on the ring, and a particle of mass m located on the table can freely pass through this hole. Initially, the particle enters the ring through hole $P_0$ with velocity v and undergoes N elastic collisions with the smooth inner wall of the ring before exiting through the same hole $P_0$ (as shown in the figure). During the entire process from entering to exiting the hole, the line connecting the ring's center O to the particle rotates exactly 360° relative to the ring. Find the magnitude of the velocity of the ring's center relative to the table after the particle exits.		$\dfrac{2m}{m+m_0}v\sin\dfrac{\pi}{N+1}$
Physics	Three identical, smooth-surfaced small balls labeled A, B, and C are involved in the setup. Balls B and C are suspended from the ceiling by two massless, inextensible strings, each with a length of L = 2.00 m, such that the two balls just touch each other. A right-handed coordinate system Oxyz is established with the point of contact O as the origin, the z-axis pointing vertically upward, and the x-axis aligned with the line connecting the centers of balls B and C, as shown in the figure. Ball A is projected towards balls B and C and collides simultaneously with them. Before the collision, ball A moves in the positive y-direction with a speed of 4.00 m/s. After the collision, ball A rebounds in the negative y-direction with a speed of 0.40 m/s. Determine the maximum displacement of balls B and C from point O after being struck.		1.13 (m)
Physics	Three uniform rigid rods, each 2.00 meters in length, form an equilateral triangular frame ABC. Point C is suspended from a frictionless horizontal axis, allowing the entire frame to rotate about this axis. Rod AB serves as a track on which an electric toy squirrel can move, as shown in the figure. It is observed that the squirrel is moving along the track while the frame remains stationary. Determine the period of the squirrel's motion.		2.64 (s)
Physics	A massless inextensible string passes through a bead of mass m (considered as a particle). The lower end of the string is fixed at point A, while the upper end is attached to a light ring that can slide along a fixed horizontal rod (the mass of the ring and the friction between the ring and the rod are negligible). The rod lies in the same vertical plane as point A. Initially, the bead is held close to the ring with the string taut, as shown in the figure. Given: the length of the string is L, the vertical distance from point A to the rod is h, and the maximum tension the string can withstand is T. If the string breaks before the bead reaches its lowest point during descent (assuming no friction between the bead and the string), find the speed of the bead at the moment the string breaks.		$\sqrt{2gL\left(1-\dfrac{mg}{2T}\right)}$
Physics	Three steel balls A, B, and C are connected by two light, rigid rods of length l, standing vertically on a horizontal surface as shown in the figure. The mass of ball A is 2m, and the masses of balls B and C are each m. There is a vertical wall located at a horizontal distance $a = \dfrac{5\sqrt{2}}{8}l$ from the rods. Due to a slight disturbance, the two rods slide outward in opposite directions, causing ball B to move downward, and as a result, ball C collides with the wall. Assume that the speed of ball C remains unchanged before and after the collision, all friction is negligible, and the diameters of the balls are much smaller than l. Find the magnitude of the velocity of ball B at the instant it touches the ground.		$\dfrac{1}{8}\sqrt{(38+45\sqrt{2})gl}$
Physics	As shown in the figure, an n-layer bracket is composed of identical triangular plates connected by hinges, with two forces of magnitude P applied at the top; neglecting the weight of the triangular plates, determine the magnitude of the horizontal component of the reaction force at the fixed hinge support A.		0.5P
Physics	Twenty identical resistors, each with resistance R, are connected as shown in the diagram; determine the equivalent resistance between points A and B.		2R
Physics	A tetrahedral framework is composed of uniform wires made of the same material and having identical cross-sectional areas, with each edge having a resistance of a, b, or c as shown in the diagram. Determine the equivalent resistance between points A and B.		$\dfrac{1}{2}\left(\dfrac{ab}{a+b}+\dfrac{ac}{a+c}\right)$
Physics	As shown in the figure, a smooth hemispherical container with diameter a has its edge just touching a smooth vertical wall. A uniform straight rod AB has its end A resting against the wall and end B in contact with the bottom of the container. When the rod is inclined at an angle of 60° to the horizontal, it remains in equilibrium. Find the length of the rod.		$a\left(1+\dfrac{\sqrt{13}}{13}\right)$
Physics	A uniform rod AB is placed on two inclined planes that are perpendicular to each other, as shown in the figure. Assuming the angle of friction at each contact surface is $\varphi$, determine the possible range of the angle $\theta$ between the rod AB and the inclined plane AO when the system is in equilibrium. It is given that the inclined plane BO makes an angle $\alpha$ with the horizontal.		$\alpha - 2\varphi \leq \theta \leq \alpha + 2\varphi$
Physics	As shown in the left diagram, it is a common-emitter single-stage amplifier circuit, where \( E = 6\,\mathrm{V} \), \( R_b = 270\,\mathrm{k\Omega} \), and \( R_c = 2\,\mathrm{k\Omega} \); given that the transistor's output characteristic curves are shown in the right diagram, determine the collector voltage \( U_c \).		Approximately 3.2 (V) (3.1 to 3.3 are all correct)
Physics	The correct way to use a pressure cooker (as shown in the left diagram) is as follows: after sealing the cooker with its lid, heat it until the water inside begins to boil, then attach the pressure regulator weight. At this point, it can be assumed that all the air inside has been expelled and only saturated steam remains; continued heating will cause the water temperature to rise until the steam pressure lifts the weight, indicating the cooker has reached its intended temperature. For a certain pressure cooker with a target temperature of 120°C, suppose a user deviates from the process and attaches the pressure regulator when the water temperature is only 90°C (it can be assumed that the steam inside is saturated at this point). When continued heating causes the regulator to begin releasing steam, what is the temperature inside the cooker at that moment? Given: atmospheric pressure $p_0 = 1.013\times 10^5\mathrm{Pa}$; saturated vapor pressure at 90°C $p_{s0} = 7.010\times 10^4\mathrm{Pa}$; saturated vapor pressure at 120°C $p_0 = 1.985\times 10^5\mathrm{Pa}$. The function relationship between saturated vapor pressure $p_s$ and temperature $t$ (°C) between 90°C and 120°C is shown in the graph on the right.		about 114.5°C (114°C to 115°C are all correct)
Physics	A toy train is composed of many connected cars and moves at a constant speed along a horizontal track before entering a vertical loop-the-loop (as shown in the figure). The total length of the train is \( L \), and the radius of the loop is \( R \), where \( R \) is much larger than the length of an individual car, but \( L > 2\pi R \). What minimum initial speed must the train have to ensure that all cars remain on the track and none lose contact while traversing the loop?		$\sqrt{gR\left(3+\dfrac{4\pi R}{L}\right)}$
Physics	Three steam trains are moving at constant speeds along a straight railway track. Three streams of steam rings blown from the trains are photographed from above, as shown in the image. The first train is traveling at 50 km/h, and the second at 70 km/h, with their directions of motion indicated by arrows in the image. Determine the speed of the third train.		40 (km/h)
Counting	How many triangles are there in the given figure?		24
Counting	In the figure, there are six circles whose outlines divide the space into multiple regions, with each region lying inside a certain number of circles. What is the maximum number of circles that cover a single region in the diagram?		5
Counting	Which of the four colors covers the largest area?		Yellow
Counting	How many unit squares does the line segment pass through in the given grid diagram?		16
Counting	The figure shows a polygon on a grid diagram; please calculate how many grid cells are completely contained within the polygon.		7
Counting	How many bricks are missing from the wall?		6
Counting	How many 2×2 squares are there in this figure?		8
Counting	How many lines in the figure will each of the segments AB, AC, AD, and AE intersect respectively?		3, 4, 3, 2
Counting	First, draw a line segment connecting points A and C in the figure (the segment passes through point B). How many triangles are formed in the new figure? (Note: A large triangle can consist of several smaller triangles.)		10
Counting	How many quadrilaterals are there in the picture in total?		150
Counting	In the figure, there are a total of ( ) triangles.		10
Counting	How many rectangles are there in the figure?		8
Counting	How many circles are there in the image?		5
Counting	In the picture, there are ( ) rectangles and ( ) circles.		10；4
Counting	There are ( ) rectangles, ( ) triangles, and ( ) circles in the picture.		7；9；5
Counting	In the picture, there are ( ) triangles and ( ) parallelograms.		10；4
Counting	In the picture, there are ( ) rectangles and ( ) triangles.		9；12
Counting	How many different routes are there from Xiaoqiang's home to the sports stadium?		5
Counting	The diagram shows a floor made up of three different types of tiles labeled Type 1, Type 2, and Type 3. How many tiles of each type are used in total?		12,16,8
Counting	If the area of each square is 1 cm², what is the total area of the shaded region?		35
Counting	How many bricks are missing in the diagram?		9
Counting	How many "X" symbols are there in the image in total?		120
Counting	Given triangle ABC, two points D and E are chosen on line segment BC, and lines AD and AE are drawn; two points F and G are chosen on line segment AC, and lines BF and BG are drawn. How many quadrilaterals are there in the resulting figure?	Not supported with pagination yet	9
Counting	Given triangle ABC, two points D and E are selected on line segment BC (with D being closer to point B), and lines AD and AE are drawn. Two points F and G are selected on line segment AC (with F being closer to point C), and lines BF and BG are drawn. Line BF intersects AD at point H and AE at point J; line BG intersects AD at point I and AE at point K. Question: How many quadrilaterals in the figure have H, I, J, or K as one of their vertices, respectively?	Not supported with pagination yet	4, 4, 4, 4
Counting	Given triangle AHK, point B lies on segment AH, points I and J lie on segment HK, and point E lies on segment AK. Lines AI and BE intersect at point C, and lines HE and AI intersect at point G. Similarly, lines AJ and BE intersect at point D, and lines AJ and HE intersect at point F. How many triangles are there in the figure?	Not supported with pagination yet	24
Counting	How many triangles can be found in the figure?		29
Counting	How many triangles are there in the figure?		35
Counting	How many triangles are there in the diagram?		15
Counting	As shown in the figure, there are 20 game pieces placed on graph paper (each red dot represents one piece). Using these pieces as vertices, how many distinct squares can be formed?		21
Counting	How many triangles are there in the diagram?		15
Counting	In triangle ABC, points G and H are selected on side AB, and point D is selected on side AC. Lines CG and CH intersect BD at points E and F, respectively. How many triangles are there in the figure?	Not supported with pagination yet	15
Counting	The square in the diagram is divided into 9 equal smaller squares, forming a total of 16 vertices. By selecting any 3 non-collinear points among these vertices, triangles can be formed. Among all such triangles, how many have the same area as the shaded triangle?		48
Counting	How many triangles are there in the diagram?		45
Counting	There are 16 points arranged in a 4×4 grid forming a square on a plane, with equal spacing between adjacent points along each row and column, and a nail is placed at each point. Using these points as vertices and connecting them with lines to form squares, how many distinct squares can be formed in total?	Not supported with pagination yet	20
Counting	Xiao Wang places several identical equilateral triangle paper pieces on a table. In the first round, he places 1 piece; in the second round, he places 3 more pieces around the first triangle; in the third round, he places additional pieces around the shape formed in the second round, and so on. The placement rule is: each new triangle must share at least one full edge with a triangle placed in the previous round, and there must be no overlapping except along the edges. After the 20th round of placement, how many equilateral triangle paper pieces are used in total?	Not supported with pagination yet	571
Counting	How many trapezoids are there in the diagram in total?		28
Counting	What is the maximum number of regions into which a rectangle can be divided by drawing three circles entirely inside it (the circles may be tangent to the rectangle)?	Not supported with pagination yet	15
Counting	How many triangles are there in the diagram?		15
Counting	How many triangles are there in the figure?		76
Counting	Given a regular pentagon with points A, B, C, D, and E as the midpoints of its five sides, if every pair of these points is connected by a line segment, how many trapezoids are formed in the resulting figure?	Not supported with pagination yet	35
Counting	Given a regular pentagon with points A, B, C, D, and E being the midpoints of its sides, how many trapezoids are formed by connecting the following pairs of points: AC, AD, BD, BE, and CE?	Not supported with pagination yet	15
Counting	As shown in the figure, the three large triangles are all equilateral triangles; how many triangles are there in total in the figure?		30
Counting	How many triangles are there in the diagram?		17

Physics

As shown in the figure, in the coordinate system Oxyz, the x-axis and z-axis lie in the plane of the paper, while the y-axis is perpendicular to the paper, pointing inward. Two infinitely large metal plates P and Q are located at x = -d and x = d, respectively. A uniform magnetic field of magnitude B lies in the Oxz plane and makes an angle α with the z-axis. A positively charged particle with charge q (> 0) and mass m is initially located at the origin O and moves with an initial velocity v in the positive y-direction. Neglecting the effect of gravity, and assuming no electric field is applied between the plates, what should the initial speed v be so that the particle just reaches the plates in space without touching them?

$\frac{qBd}{2m\cos\alpha}$

Physics

As shown in the figure, triangle Oab is a right-angled triangle with its two perpendicular sides formed by insulated dielectric rods, and the hypotenuse ab formed by a thin metallic rod. The length of Ob is equal to ab/2 = l. The triangle is placed in a uniform magnetic field with magnetic induction B, which is perpendicular to the plane of the triangle and directed into the page. The triangle rotates at a constant angular velocity $\omega$ about an axis that passes through point O and is parallel to the magnetic field direction. Determine the potential difference $U_{ab} = U_a - U_b$ between points a and b.

$Bl^2\omega$

Physics

As shown in the figure, a closed wire loop consists of 12 straight segments, each of length l. Except for segments 1-8, 2-9, and 3-4 (represented by dashed lines), which have negligible resistance, the remaining nine segments each have a resistance of r. A uniform magnetic field B is directed perpendicular to the plane of the loop and points into the page. The boundary of the magnetic field, labeled MN, is parallel to the side of the loop formed by segments 5-6-7, as shown. The loop is pulled uniformly to the right with a constant velocity v, gradually moving it out of the magnetic field region. Determine the work done by the pulling force during this process.

$\frac{2B^2 l^3 v}{r}$

Physics

A straight conductor MN and a metal ring with radius r are placed overlapping on a smooth horizontal surface. The straight conductor MN is fixed and located at a distance of r/2 from the center O of the stationary metal ring, and there is good electrical contact at the point where the ring intersects the conductor. A uniform magnetic field with magnetic induction $B_0$, perpendicular to the horizontal surface and in the direction shown in the figure, is present in the space. Determine the velocity of the metal ring immediately after the magnetic field is rapidly removed. Assume the electrical resistance per unit length of both the straight conductor and the ring is $r_0$, and the mass per unit length of the ring is $m_0$.

$\frac{27}{32\pi + 72\sqrt{3}} \cdot \frac{rB_{0}^{2}}{2\pi m_{0}r_{0}}$

Physics

A thin wire forms a loop along the edges of one face of a cube, with its self-inductance denoted as L₁, as shown in diagram (a). Using the same wire, a loop is formed along the edges of the cube, yielding a self-inductance of L₂, as shown in diagram (b). Determine the self-inductance of a loop formed using the same wire along the edges of the cube as shown in diagram (c), where the loops in all diagrams are indicated by bold black lines.

3(L_2 - L_1)

Physics

A uniformly thick and homogeneous conductive wire with a nonzero constant resistivity is bent into an equilateral triangular loop with side length $ a $. The triangle rotates at a constant angular velocity $ \omega $ about one of its sides, AB, which serves as the axis of rotation. A uniform magnetic field with magnetic induction $ B $ is oriented perpendicular to the rotation axis AB. Determine the electric potential difference between points B and A on the wire loop, as shown in the figure.

$\frac{\sqrt{3}}{12}B\omega a^2\sin\omega t$

Physics

As shown in the figure, two identical metal rings each have a radius R, mass m, and resistance r. They are both placed in the same uniform magnetic field, with the field direction perpendicular to the plane of the rings and pointing into the page. The magnetic flux density is $B_0$. The contact points A and C between the two rings are in good electrical contact, with negligible contact resistance, and the angle between the rings is $a = \frac{\pi}{3}$. If the magnetic field is removed, what speed will each ring acquire? Assume that the displacement of the rings during the removal of the magnetic field is negligible and that friction can be ignored.

$\frac{9\sqrt{3}R^3B_0^2}{10mr}$

Physics

As shown in the diagram, the circuit contains two resistors each with resistance $ R_1 = R_2 = R $, and one capacitor with capacitance $ C $; two ideal AC ammeters, $ A_1 $ and $ A_2 $, measure currents of 0.3 A and 0.2 A respectively. The circuit is connected to an AC power supply with an RMS voltage of 36 V and a frequency of 50 Hz. Determine the capacitance $ C $ of the capacitor.

$13.5\mathrm{\mu F}$

Physics

As shown in the figure, the inductances of the two coils are $L_1 = 10,\mathrm{mH}$ and $L_2 = 20,\mathrm{mH}$; the capacitances of the two capacitors are $C_1 = 10,\mathrm{nF}$ and $C_2 = 5,\mathrm{nF}$; the resistor has resistance $R = 100,\mathrm{k\Omega}$. The switch $S$ has been closed for a long time. It is then opened, and after a time interval $t_0$, the currents through $L_1$ and $L_2$ are measured to be $i_{01} = 0.1,\mathrm{A}$ and $i_{02} = 0.2,\mathrm{A}$ respectively, and the voltage is $U_0 = 40,\mathrm{V}$. Calculate the current in the wire segment AB (take the current as positive when flowing from A to B).

-0.1 (A)

Physics

A U-shaped glass capillary tube with uniform thickness and open ends is placed in a vertical plane in an atmosphere with pressure $ p_0 $. The two vertical arms of the tube each have a height of $ h $, and the horizontal section connecting them has a length of $ 2h $. The tube has a radius $ r $ (where $ r << h $). The horizontal section is completely filled with mercury of density $ \rho $, as shown in the figure. One of the vertical arms is then sealed at the top, so that the gas enclosed within remains at pressure $ p_0 $. If the U-tube begins to rotate uniformly around the axis of the other (open) vertical arm, what must the angular velocity be for the length of mercury in the horizontal section to remain stable at $ \frac{5h}{3} $? (Assume that, during this motion, the inclination of the mercury surface inside the tube can be neglected.)

$\sqrt{\dfrac{3(3p_0 + 2\rho gh)}{35\rho h^2}}$

Physics

(Multiple Choice) The diagram shows a vertical cross-section of a circular pool. AB is the diameter of the water surface, MN is the diameter of the pool bottom, and O is the center of the circular pool bottom. It is known that ON is 11.4 m, AM and BN are sloped edges, the water depth is 5.0 m, the refractive index of water is 4/3, and the water is extremely clear, with absorption by water being negligible. Points a, b, c, and d in the diagram are four light sources. The sky is blue and the water surface is flat. A diver is lying on his back in a groove at the center of the pool bottom, with his eyes located at point O. From this position, the maximum visible diameter of the water surface is AB. Which of the four light sources a, b, c, and d can have their light reach the diver’s eyes through total internal reflection?

c，d

Physics

The diagram shows a tank filled with a liquid of refractive index $ n $, with a symmetric, ridge-shaped thin transparent cover ADB placed over its central part. The cover has a vertex angle of $ 2\varphi $, encloses air inside, and is fully submerged in the liquid. A bright point C is located at the midpoint of the tank bottom AB. Assume the tank is sufficiently wide, and the thin cover walls have negligible effect on light refraction. Determine the condition under which an eye located above the liquid surface in the plane of the diagram can see the bright point from the side (give the minimum value of $ \tan \varphi $).

$\sqrt{n^2 -1}-1$

Physics

As shown in the figure, in a vacuum, there is a small sphere with a uniform texture, radius $ r $, and refractive index $ n $ (where $ n > n_0 $, and $ n_0 $ is the refractive index of vacuum). A narrow laser beam with frequency $ \nu $ travels in a straight line BC through the vacuum. The line segment CD passes at a distance $ l $ from the center O of the sphere (with $ l < r $). The laser beam enters the sphere (which acts as an optical medium) at point C on the surface of the sphere through refraction, and then exits the sphere at point D on the surface through another refraction back into the vacuum. Assume the frequency of the laser beam remains unchanged through both refractions. Find the magnitude of the average force exerted by a photon in the laser beam on the sphere during the two refraction processes.

$\frac{lh\nu}{r^{2}}\left[n-\sqrt{1-\frac{(n^{2}-1)l^{2}}{r^{2}-l^{2}}}\right]$

Physics

As shown in the figure, L₁ and L₂ are two coaxial thin convex lenses with a common principal axis OO’. Lens L₁ has a focal length of f₁ = 10 cm and an aperture (diameter) of d₁ = 4.0 cm; lens L₂ has a focal length of f₂ = 5.0 cm and an aperture of d₂ = 2.0 cm. The distance between the two lenses is a = 30 cm. AB is a uniformly luminous circular disk with a diameter d = 2.0 cm, aligned coaxially with the lenses and placed 20 cm to the left of L₁, forming an image on a screen P located to the right of L₂ and perpendicular to OO’. It is observed that the central part of the image on the screen is brighter than the edges. To make the edges of the image as bright as the center—without changing the position or size of the image—a third coaxial thin lens L₃ is to be inserted along the axis OO’. What is the minimum required aperture (diameter) of lens L₃?

2.0 (cm)

Physics

A slender rod ABC rests against a step in a vertical plane, with end A free to slide along the horizontal ground toward the step, while the rod remains in contact with the edge of the step. When the angle between the rod and the horizontal ground is $\phi$ (as shown in the figure), point B is exactly at the edge of the step, and the magnitude of the velocity of end C is exactly twice that of end A. Determine the ratio of the length of segment BC to that of segment AB.

$\frac{\sqrt{4-\cos^2 \phi}}{\sin\phi}$

Physics

Two rods, each of length $l$, are connected by a hinge at point $P$, as shown in the figure. One end of the first rod is fixed at point $O$ via a hinge, while the free end $Q$ of the second rod begins to move with a constant velocity $v$ in both magnitude and direction. At the initial moment, the velocity $v$ is directed along the angle bisector of the angle $2\alpha$ between the two rods. Determine the magnitude of the acceleration of the hinge point $P$ connecting the two rods after a very short time has elapsed since the motion begins.

$\dfrac{v^2}{4l\sin^2 \alpha \cos \alpha}$

Physics

Two motorboats are towing a barge, with their velocities given by $v_1$ and $v_2$, respectively. The angle between the two velocity vectors is $\alpha$, and at this moment, the velocity vectors $v_1$ and $v_2$ are aligned with the directions in which the motorboats are moving, as shown in the figure. Determine the magnitude of the barge’s velocity.

$\dfrac{\sqrt{v_1^2+v_2^2-2v_1 v_2 \cos\alpha}}{\sin \alpha}$

Physics

As shown in the figure, a light rope of length l has identical small balls of mass m attached to both ends, and a third ball of mass M attached at the center; all three balls are initially at rest on a smooth horizontal surface, with the rope fully stretched. Now, the central ball M is given an impulse, resulting in an initial velocity v perpendicular to the rope. Determine the tension in the rope at the instant just before the two end balls collide.

$\dfrac{M^2mv^2}{(M+2m)^2l}$

Physics

As shown in the figure, a prism ABC with a triangular cross-section is constrained to move along a smooth track such that its edge AB can only slide along the smooth rail DE. A smooth small ball, having the same mass m as the prism ABC, moves in the same horizontal plane along a direction perpendicular to the rail DE and collides elastically with the initially stationary prism ABC. Find the magnitude of the velocity of the prism ABC after the collision.

$\dfrac{2v_0}{3}$

Physics

As shown in the figure, a uniform solid cylinder of mass m and radius R is wedged between a wooden plank and a vertical wall. The coefficient of kinetic friction between the cylinder and both the wall and the plank is 0.75. The plank is very light and its mass can be neglected. One end of the plank, point O, is connected to the wall via a smooth hinge, while the other end, point A, is suspended with a hanging mass m'. The length of OA is L, and the angle between the plank and the vertical wall is 53°, corresponding to a right triangle with side ratios of 3:4:5. What is the minimum value of m' required to keep the system in equilibrium?

$\dfrac{25Rm}{4L}$

Physics

A uniform thin wooden rod of length $l$ is suspended vertically by a fine thread and positioned above the horizontal surface of water in a bucket, as shown in the figure. As the bucket is slowly raised, the rod gradually becomes submerged in water. When the submerged depth of the rod exceeds a certain value $h$, the rod begins to tilt. Determine the value of $h$, given that the density of the rod is $\rho$ and the density of water is $\rho_0$.

$l\left(1-\sqrt{1-\dfrac{\rho}{\rho_0}}\right)$

Physics

Rod AB is placed inside a cylindrical container. Point A of the rod is hinged at the junction between the cylinder wall and the bottom, while point C of the rod rests against the edge of the cylinder. Both points A and C lie in a vertical plane that passes through the axis of the cylinder, as shown in the figure. The rod forms an angle $\alpha$ with the horizontal. The rod is then moved along the cylinder's edge to a new contact point C', such that the angle $\angle C'OC$ is $\varphi$. What is the minimum coefficient of friction required for the rod to remain in equilibrium at position C'?

$\frac{\tan\frac{\varphi}{2}}{\sqrt{1+\cot^{2}\alpha\cos^{2}\frac{\varphi}{2}}}$

Physics

Three strings are floating on the water surface with their endpoints connected, as shown in the figure. Strings 1 and 2 are each 1.5 cm long, and string 3 is 1 cm long. A certain impurity is dropped at point A inside the circular region, causing the surface tension coefficient of the water at that point to decrease by a factor of 2.5. Determine the tension in each string. The original surface tension coefficient of water is given as 0.07 N/m.

0(N); 1.67e-4(N); 1.67e-4(N)

Physics

Mercury is poured onto a clean, horizontal, flat glass plate, and due to gravity and surface tension, it approximately forms a circular disk shape (with its sides bulging outward). A vertical cross-section through the axis of the disk is shown in the figure. Given the density of mercury $\rho = 13.6 \times 10^3 \mathrm{kg/m^3}$ and the surface tension coefficient $\sigma = 0.49 \mathrm{N/m}$, estimate the thickness $h$ of the disk when its radius is very large. (Provide the answer with one significant figure.)

3e-3 (m)

Physics

In the figure, object a is a fixed uniformly charged sphere with radius R, and O is its center. Given that the electric potential on the sphere's surface is U = 1000 V when the potential at infinity is taken to be zero, an electron b is located near a point O' far from the center O. The electron is moving toward sphere a with a kinetic energy of $ E_k = 2000\ \mathrm{eV} $ in a direction parallel to the line connecting O and O'. Let l be the perpendicular distance from the electron's initial path to the O–O′ line. To allow the electron to reach and collide with the surface of the charged sphere a, determine the maximum possible value of l.

$\dfrac{\sqrt{6}}{2}R$

Physics

A square network made of uniform resistive wires, as shown in the figure, consists of 9 identical small squares, with each side of a small square having a resistance of $r = 8\,\Omega$. A battery with an electromotive force of $E = 5.7\,\mathrm{V}$ and negligible internal resistance is connected to the network. If points C and D are connected by an ideal wire (with negligible resistance), determine the current flowing through this wire.

0.267 (A)

Physics

As shown in the figure, $R_1 = 2\Omega$, $R_2 = R_3 = 6\Omega$, the total resistance of $R_4$ is $6\Omega$, and the internal resistance of the power supply is $r = 1\Omega$. When switch K is moved to the upper position, the voltmeter reads 5V. When the sliding contact is set at a certain position, the power consumed by $R_4$ reaches its maximum; what is this maximum power value?

2 (W)

Physics

In the circuit shown, all power sources have zero internal resistance, and points B and C are connected to the right to an infinite network composed of alternating $1.0\Omega$ and $2.0\Omega$ resistors; determine the amount of charge on the plate of the $10\mathrm{\mu F}$ capacitor that is connected to point D.

1.3e-4 (C)

Physics

As shown in the left diagram, the resistors $R_1$ and $R_2$ each have a resistance of $1\mathrm{k\Omega}$, and the electromotive force is $E = 6\,\mathrm{V}$. Two identical diodes $D$ are connected in series within the circuit, and the $I_D$-$U_D$ characteristic curve of diode $D$ is provided in the right diagram. Calculate the power dissipated by resistor $R_1$.

16 (mW)

Physics

Modern material growth and microfabrication techniques have enabled the creation of microstructured semiconductor devices that confine the motion of electrons within a single plane (two-dimensional), allowing electrons to travel through the device like bullets, unaffected by scattering from impurity atoms. This property holds promising potential for new applications. As shown in the left diagram, a four-terminal, cross-shaped, two-dimensional electron gas semiconductor allows current entering from terminal 1 to exit from terminals 2, 3, or 4, controlled by a magnetic field. Studying the following model structure aids in understanding the current flow in such a four-terminal cross-shaped conductor. In the right diagram, a, b, c, and d represent the cross-sections of four closely spaced cylindrical rods, each with radius R, forming four narrow slits labeled 1, 2, 3, and 4. A uniform magnetic field exists in the vacuum region enclosed by the cylinders and slits, directed perpendicularly into the plane of the page. Let $B$ represent the magnetic induction. A particle of mass m and positive charge q enters the magnetic field through slit 1 with velocity v in a direction tangent to both cylinders a and b. Assume the particle undergoes exactly one perfectly elastic collision with one of the cylinder surfaces, with negligible collision time and no change in charge or influence from friction. Determine the value of B that allows the particle to exit through slit 2, traveling in a direction tangent to both cylinders b and c.

$\frac{mv}{3qR}$

Physics

As shown in the figure, two plane mirrors A and B have their reflecting surfaces perpendicular to the plane of the paper, and the line of intersection between the two mirrors passes through point O in the diagram. The angle between the two mirror surfaces is $\alpha = 15^\circ$. From point C on mirror A, a light ray is emitted in the plane of the paper at an angle of $\beta = 30^\circ$ to mirror A. After undergoing multiple reflections between the two mirrors, the light ray eventually ceases to interact with the mirrors. Assuming both mirrors are sufficiently large and the distance CO is 1 meter, how much time elapses from the moment the light ray leaves point C to its final reflection?

9.1e-9 (s)

Physics

A thick glass tank has a bottom thickness of 5 cm and contains water to a depth of 4 cm, as shown in the figure. Given that the refractive indices of glass and water are 1.8 and 1.33 respectively, what is the apparent distance from the water surface to the bottom surface of the glass tank when viewed vertically from above?

6.34 (cm)

Physics

A horizontal parallel-plane glass plate H, 3.0 cm thick and with a refractive index of n = 1.5, has a small object S located 2.0 cm below its bottom surface. Above the glass plate is a thin convex lens L with a focal length f = 30 cm, and its principal axis is perpendicular to the surface of the glass plate. The object S lies on the principal axis of the lens, as shown in the figure. If an observer above the lens sees the image of S aligned with the actual position of S along the principal axis, what is the distance between the lens and the upper surface of the glass plate?

1.0 (cm)

Physics

A black sphere with radius R is placed inside a cylindrical tube whose inner surface reflects but does not absorb light. A point light source S is located at a distance 2R from the center of the sphere O, with both O and S lying along the axis of the cylinder, as shown in the figure. To ensure that all light emitted by the point source toward the right hemisphere is eventually absorbed by the black sphere, what is the maximum possible inner radius r of the cylinder?

$\dfrac{2\sqrt{3}}{3}R$

Physics

As shown in the figure, a narrow monochromatic light beam is incident on two parallel glass plates. The beam is parallel to the optical axis $OO_1$, which is perpendicular to the glass plates and passes through their center. The beam is at a distance $R = 3\,\mathrm{cm}$ from the axis, and the thickness of the glass plates is $H = 3\,\mathrm{mm}$. The refractive index of the glass varies radially according to the relation $n(r) = n_0\left[1 - \left(\frac{r}{r_0}\right)^2\right]$, where $n_0 = 1.5$ and $r_0 = 9\,\mathrm{cm}$. Determine the angle between the emergent light beam and the optical axis.

1.91°

Physics

As shown in the figure, in a Lloyd's mirror experiment, a point light source S is located 2 mm above the mirror plane; the mirror is positioned midway between the light source and the screen. The length of the mirror is l = 40 cm, the distance between the screen and the light source is D = 1.5 m, and the wavelength of the light is 500 nm. Determine the number of interference fringes observed on the screen.

12.25

Physics

Three identical capacitors are connected as shown in the diagram. Initially, capacitor 1 carries a charge Q with its upper plate positively charged, while capacitors 2 and 3 are uncharged. A wire is used to connect points a and b; then a and b are disconnected, and points a and c are connected; after disconnecting a and c, points a and b are reconnected; finally, points a and d are connected. Determine the charge on the lower plate of capacitor 2 at the end of this process.

Q/12

Physics

In the network shown in the figure, the current values and directions on some branches, certain component parameters, and the electric potentials at junction points are known (all relevant values and parameters are marked in the figure). Using the given data and parameters, determine the magnitude of the current in the branch containing the resistor $R_x$, taking the current as positive when it flows from top to bottom.

2 (A)

Physics

The cross-section of a thermometer is shown in the diagram. It is known that the thin mercury column A is located at a distance of 2R from the vertex O of the cylindrical surface, where R is the radius of the cylindrical surface. Point C represents the position of the central axis of the cylinder. The refractive index of the glass is n = 3/2, and E represents the position of the human eye. Determine the magnification of the image of the mercury column as seen by the eye within the illustrated cross-section.

3

Physics

A smooth horizontal circular table has a radius R = 1.00 m and a vertical post fixed near its center O. The intersection between the post and the tabletop forms a convex, smooth, closed curve C, as shown in the figure. A soft, light, inextensible string has one end fixed to a point on curve C, and the other end is tied to a small block of mass m = 75 g. The block is placed on the table and the string is pulled taut; then the block is given an initial velocity of 4.0 m/s perpendicular to the string. As the block moves on the tabletop, the string wraps around the post. It is known that the string breaks when the tension reaches 2.0 N, and before the string breaks, the block always remains on the tabletop. When the string is just about to break, the line from the table center O to the point where the taut string contacts the closed curve is perpendicular to the taut part of the string. What is the horizontal distance from the point where the block lands to the center O of the table, given that the table height is h = 0.80 m and the block does not collide with the post during its motion on the table? Take the gravitational acceleration as $10\ \mathrm{m/s^2}$.

2.5 (m)

Physics

In certain heavy machinery and lifting equipment, a double-block electromagnetic brake is commonly used. Its simplified schematic diagram is shown in the figure, with $O_1$ and $O_2$ as fixed hinges. When the power is on, the lever A is pressed downward, pulling spring S through hinges $C_1$, $C_2$, and $C_3$, causing brake pads $B_1$ and $B_2$ to disengage from the brake wheel D, allowing the machinery to operate normally. When the power is cut off, the lever A loses its downward force (neglecting the weight of lever A and all other linkages and brake components in the figure), and the spring contracts, pressing the brake pads into contact and generating a braking force. At this moment, $O_1C_1$ and $O_2C_2$ are in a vertical position. Given that a minimum braking torque of M = 1100 N•m is required to decelerate the uniformly rotating brake wheel D, with a coefficient of friction of 0.40 between the brake pads and the brake wheel, an unstretched spring length of L = 0.300 m, a brake wheel diameter d = 0.400 m, and diagram dimensions a = 0.065 m, $h_1$ = 0.245 m, and $h_2$ = 0.340 m, determine the minimum required stiffness coefficient k of the spring to achieve the necessary braking effect.

1.49e4 (N/m)

Physics

A cylinder A with radius R rests stationary on a horizontal surface and is in contact with a vertical wall. Another thinner cylinder B, having the same mass as A and radius r, is placed on top of cylinder A such that it also touches the wall, as shown in the figure. Cylinder A is held in place by hand while placing B, and then released. The coefficient of static friction between cylinder A and the ground is 0.20, and the coefficient of static friction between the two cylinders is 0.30. If the system remains in equilibrium after release, what is the minimum value of the ratio r/R?

0.289940828402367

Physics

A pendulum with a string of length l (with the bob considered a point mass and the string's mass negligible) is set up such that a fixed peg is placed at point C on the vertical line passing through the pivot point O, at a distance x below O (where x < l), as shown in the figure. As the pendulum swings, the string may be obstructed by the peg. When l is fixed and x varies, the motion of the pendulum bob after being obstructed differs accordingly. Initially, the pendulum is pulled to a position to the left of the vertical line (with the bob's height not exceeding that of point O) and then released to swing freely. If, after being blocked by the peg, the bob just manages to strike the peg, determine the minimum value of x.

0.464l

Physics

A rigid, uniform, thin circular ring with radius R and mass $m_0$ is initially at rest on a smooth horizontal table. There is a small hole $P_0$ on the ring, and a particle of mass m located on the table can freely pass through this hole. Initially, the particle enters the ring through hole $P_0$ with velocity v and undergoes N elastic collisions with the smooth inner wall of the ring before exiting through the same hole $P_0$ (as shown in the figure). During the entire process from entering to exiting the hole, the line connecting the ring's center O to the particle rotates exactly 360° relative to the ring. Find the magnitude of the velocity of the ring's center relative to the table after the particle exits.

$\dfrac{2m}{m+m_0}v\sin\dfrac{\pi}{N+1}$

Physics

Three identical, smooth-surfaced small balls labeled A, B, and C are involved in the setup. Balls B and C are suspended from the ceiling by two massless, inextensible strings, each with a length of L = 2.00 m, such that the two balls just touch each other. A right-handed coordinate system Oxyz is established with the point of contact O as the origin, the z-axis pointing vertically upward, and the x-axis aligned with the line connecting the centers of balls B and C, as shown in the figure. Ball A is projected towards balls B and C and collides simultaneously with them. Before the collision, ball A moves in the positive y-direction with a speed of 4.00 m/s. After the collision, ball A rebounds in the negative y-direction with a speed of 0.40 m/s. Determine the maximum displacement of balls B and C from point O after being struck.

1.13 (m)

Physics

Three uniform rigid rods, each 2.00 meters in length, form an equilateral triangular frame ABC. Point C is suspended from a frictionless horizontal axis, allowing the entire frame to rotate about this axis. Rod AB serves as a track on which an electric toy squirrel can move, as shown in the figure. It is observed that the squirrel is moving along the track while the frame remains stationary. Determine the period of the squirrel's motion.

2.64 (s)

Physics

A massless inextensible string passes through a bead of mass m (considered as a particle). The lower end of the string is fixed at point A, while the upper end is attached to a light ring that can slide along a fixed horizontal rod (the mass of the ring and the friction between the ring and the rod are negligible). The rod lies in the same vertical plane as point A. Initially, the bead is held close to the ring with the string taut, as shown in the figure. Given: the length of the string is L, the vertical distance from point A to the rod is h, and the maximum tension the string can withstand is T. If the string breaks before the bead reaches its lowest point during descent (assuming no friction between the bead and the string), find the speed of the bead at the moment the string breaks.

$\sqrt{2gL\left(1-\dfrac{mg}{2T}\right)}$

Physics

Three steel balls A, B, and C are connected by two light, rigid rods of length l, standing vertically on a horizontal surface as shown in the figure. The mass of ball A is 2m, and the masses of balls B and C are each m. There is a vertical wall located at a horizontal distance $a = \dfrac{5\sqrt{2}}{8}l$ from the rods. Due to a slight disturbance, the two rods slide outward in opposite directions, causing ball B to move downward, and as a result, ball C collides with the wall. Assume that the speed of ball C remains unchanged before and after the collision, all friction is negligible, and the diameters of the balls are much smaller than l. Find the magnitude of the velocity of ball B at the instant it touches the ground.

$\dfrac{1}{8}\sqrt{(38+45\sqrt{2})gl}$

Physics

As shown in the figure, an n-layer bracket is composed of identical triangular plates connected by hinges, with two forces of magnitude P applied at the top; neglecting the weight of the triangular plates, determine the magnitude of the horizontal component of the reaction force at the fixed hinge support A.

0.5P

Physics

Twenty identical resistors, each with resistance R, are connected as shown in the diagram; determine the equivalent resistance between points A and B.

2R

Physics

A tetrahedral framework is composed of uniform wires made of the same material and having identical cross-sectional areas, with each edge having a resistance of a, b, or c as shown in the diagram. Determine the equivalent resistance between points A and B.

$\dfrac{1}{2}\left(\dfrac{ab}{a+b}+\dfrac{ac}{a+c}\right)$

Physics

As shown in the figure, a smooth hemispherical container with diameter a has its edge just touching a smooth vertical wall. A uniform straight rod AB has its end A resting against the wall and end B in contact with the bottom of the container. When the rod is inclined at an angle of 60° to the horizontal, it remains in equilibrium. Find the length of the rod.

$a\left(1+\dfrac{\sqrt{13}}{13}\right)$

Physics

A uniform rod AB is placed on two inclined planes that are perpendicular to each other, as shown in the figure. Assuming the angle of friction at each contact surface is $\varphi$, determine the possible range of the angle $\theta$ between the rod AB and the inclined plane AO when the system is in equilibrium. It is given that the inclined plane BO makes an angle $\alpha$ with the horizontal.

$\alpha - 2\varphi \leq \theta \leq \alpha + 2\varphi$

Physics

As shown in the left diagram, it is a common-emitter single-stage amplifier circuit, where $ E = 6\,\mathrm{V} $, $ R_b = 270\,\mathrm{k\Omega} $, and $ R_c = 2\,\mathrm{k\Omega} $; given that the transistor's output characteristic curves are shown in the right diagram, determine the collector voltage $ U_c $.

Approximately 3.2 (V) (3.1 to 3.3 are all correct)

Physics

The correct way to use a pressure cooker (as shown in the left diagram) is as follows: after sealing the cooker with its lid, heat it until the water inside begins to boil, then attach the pressure regulator weight. At this point, it can be assumed that all the air inside has been expelled and only saturated steam remains; continued heating will cause the water temperature to rise until the steam pressure lifts the weight, indicating the cooker has reached its intended temperature. For a certain pressure cooker with a target temperature of 120°C, suppose a user deviates from the process and attaches the pressure regulator when the water temperature is only 90°C (it can be assumed that the steam inside is saturated at this point). When continued heating causes the regulator to begin releasing steam, what is the temperature inside the cooker at that moment? Given: atmospheric pressure $p_0 = 1.013\times 10^5\mathrm{Pa}$; saturated vapor pressure at 90°C $p_{s0} = 7.010\times 10^4\mathrm{Pa}$; saturated vapor pressure at 120°C $p_0 = 1.985\times 10^5\mathrm{Pa}$. The function relationship between saturated vapor pressure $p_s$ and temperature $t$ (°C) between 90°C and 120°C is shown in the graph on the right.

about 114.5°C (114°C to 115°C are all correct)

Physics

A toy train is composed of many connected cars and moves at a constant speed along a horizontal track before entering a vertical loop-the-loop (as shown in the figure). The total length of the train is $ L $, and the radius of the loop is $ R $, where $ R $ is much larger than the length of an individual car, but $ L > 2\pi R $. What minimum initial speed must the train have to ensure that all cars remain on the track and none lose contact while traversing the loop?

$\sqrt{gR\left(3+\dfrac{4\pi R}{L}\right)}$

Physics

Three steam trains are moving at constant speeds along a straight railway track. Three streams of steam rings blown from the trains are photographed from above, as shown in the image. The first train is traveling at 50 km/h, and the second at 70 km/h, with their directions of motion indicated by arrows in the image. Determine the speed of the third train.

40 (km/h)

Counting

How many triangles are there in the given figure?

24

Counting

In the figure, there are six circles whose outlines divide the space into multiple regions, with each region lying inside a certain number of circles. What is the maximum number of circles that cover a single region in the diagram?

5

Counting

Which of the four colors covers the largest area?

Yellow

Counting

How many unit squares does the line segment pass through in the given grid diagram?

16

Counting

The figure shows a polygon on a grid diagram; please calculate how many grid cells are completely contained within the polygon.

7

Counting

How many bricks are missing from the wall?

6

Counting

How many 2×2 squares are there in this figure?

8

Counting

How many lines in the figure will each of the segments AB, AC, AD, and AE intersect respectively?

3, 4, 3, 2

Counting

First, draw a line segment connecting points A and C in the figure (the segment passes through point B). How many triangles are formed in the new figure? (Note: A large triangle can consist of several smaller triangles.)

10

Counting

How many quadrilaterals are there in the picture in total?

150

Counting

In the figure, there are a total of ( ) triangles.

10

Counting

How many rectangles are there in the figure?

8

Counting

How many circles are there in the image?

5

Counting

In the picture, there are ( ) rectangles and ( ) circles.

10；4

Counting

There are ( ) rectangles, ( ) triangles, and ( ) circles in the picture.

7；9；5

Counting

In the picture, there are ( ) triangles and ( ) parallelograms.

10；4

Counting

In the picture, there are ( ) rectangles and ( ) triangles.

9；12

Counting

How many different routes are there from Xiaoqiang's home to the sports stadium?

5

Counting

The diagram shows a floor made up of three different types of tiles labeled Type 1, Type 2, and Type 3. How many tiles of each type are used in total?

12,16,8

Counting

If the area of each square is 1 cm², what is the total area of the shaded region?

35

Counting

How many bricks are missing in the diagram?

9

Counting

How many "X" symbols are there in the image in total?

120

Counting

Given triangle ABC, two points D and E are chosen on line segment BC, and lines AD and AE are drawn; two points F and G are chosen on line segment AC, and lines BF and BG are drawn. How many quadrilaterals are there in the resulting figure?

Not supported with pagination yet

9

Counting

Given triangle ABC, two points D and E are selected on line segment BC (with D being closer to point B), and lines AD and AE are drawn. Two points F and G are selected on line segment AC (with F being closer to point C), and lines BF and BG are drawn. Line BF intersects AD at point H and AE at point J; line BG intersects AD at point I and AE at point K. Question: How many quadrilaterals in the figure have H, I, J, or K as one of their vertices, respectively?

Not supported with pagination yet

4, 4, 4, 4

Counting

Given triangle AHK, point B lies on segment AH, points I and J lie on segment HK, and point E lies on segment AK. Lines AI and BE intersect at point C, and lines HE and AI intersect at point G. Similarly, lines AJ and BE intersect at point D, and lines AJ and HE intersect at point F. How many triangles are there in the figure?

Not supported with pagination yet

24

Counting

How many triangles can be found in the figure?

29

Counting

How many triangles are there in the figure?

35

Counting

How many triangles are there in the diagram?

15

Counting

As shown in the figure, there are 20 game pieces placed on graph paper (each red dot represents one piece). Using these pieces as vertices, how many distinct squares can be formed?

21

Counting

How many triangles are there in the diagram?

15

Counting

In triangle ABC, points G and H are selected on side AB, and point D is selected on side AC. Lines CG and CH intersect BD at points E and F, respectively. How many triangles are there in the figure?

Not supported with pagination yet

15

Counting

The square in the diagram is divided into 9 equal smaller squares, forming a total of 16 vertices. By selecting any 3 non-collinear points among these vertices, triangles can be formed. Among all such triangles, how many have the same area as the shaded triangle?

48

Counting

How many triangles are there in the diagram?

45

Counting

There are 16 points arranged in a 4×4 grid forming a square on a plane, with equal spacing between adjacent points along each row and column, and a nail is placed at each point. Using these points as vertices and connecting them with lines to form squares, how many distinct squares can be formed in total?

Not supported with pagination yet

20

Counting

Xiao Wang places several identical equilateral triangle paper pieces on a table. In the first round, he places 1 piece; in the second round, he places 3 more pieces around the first triangle; in the third round, he places additional pieces around the shape formed in the second round, and so on. The placement rule is: each new triangle must share at least one full edge with a triangle placed in the previous round, and there must be no overlapping except along the edges. After the 20th round of placement, how many equilateral triangle paper pieces are used in total?

Not supported with pagination yet

571

Counting

How many trapezoids are there in the diagram in total?

28

Counting

What is the maximum number of regions into which a rectangle can be divided by drawing three circles entirely inside it (the circles may be tangent to the rectangle)?

Not supported with pagination yet

15

Counting

How many triangles are there in the diagram?

15

Counting

How many triangles are there in the figure?

76

Counting

Given a regular pentagon with points A, B, C, D, and E as the midpoints of its five sides, if every pair of these points is connected by a line segment, how many trapezoids are formed in the resulting figure?

Not supported with pagination yet

35

Counting

Given a regular pentagon with points A, B, C, D, and E being the midpoints of its sides, how many trapezoids are formed by connecting the following pairs of points: AC, AD, BD, BE, and CE?

Not supported with pagination yet

15

Counting

As shown in the figure, the three large triangles are all equilateral triangles; how many triangles are there in total in the figure?

30

Counting

How many triangles are there in the diagram?

17