Problem database

A bus is driving along a straight highway at constant speed $v$. You have noticed the bus when it was at some point $A$. From what area near the highway can you catch up with this bus if your running speed is $u < v$? Draw this area for $u = v/2$.

A ball is launched along a pool table with sides $a$ and $b$ from the middle of side $b$. At what angle to the side of the table must the ball begin to move to return to the same point from which it began its movement?

A coin is placed on a plane whose slope tangent is equal to the coefficient of friction. In the horizontal direction along the plane, the coin was given a velocity $v$. Find the steady speed of the coin.

A beam of charged particles of different masses, having the same velocity $v$, was directed along the normal to two grid electrodes, between which the same force $F$ acts on each particle. At what minimum mass of particles in the beam will all of them reach the second grid, if the width of the gap between the electrodes is equal to $l$?

Between a stationary ball of mass $m_1$ and an impinging ball of mass $m_2$, there is a stationary ball. What is the mass of the intermediate ball at which the ball of mass $m_2$ acquires the greatest velocity after impact? All strokes are central and elastic.

A spool of thread lies on a horizontal plane. The reel is pulled by a thread. At what angles $\alpha$ between the force and the horizontal will the coil accelerate towards the taut thread?

A ball is fixed in the middle of a string of length $2$l. What is the total force acting on the ball from the side of the string if its transverse displacement from the equilibrium position is $x \ll$, and the string tension force $F$ does not depend on the displacement? Why, for small displacements, can we consider the dependence of the force acting on the ball on $x$ to be linear? What is the direction of this force with respect to the displacement? Find how the potential energy of the ball depends on the small displacement $x$. What is the velocity of the ball when it passes its equilibrium position if its maximum displacement is $x_0$? The mass of the ball is $m$.

The surface of bodies vibrating with ultrasonic frequency seems slippery to the touch, and objects placed on this surface "float" on it from the slightest force applied to them. Explain this.

Two balls of mass $m_1$ and $m_2$, attached to identical springs, can oscillate by sliding on a bar of mass $M$ without friction. The bar lies on a horizontal plane. The balls are connected by a thread, the tension of which is $F$. The thread is burned out. At what smallest coefficient of friction between the plane and the bar the bar will not move?

The tensile strength of ceramics and glasses is much less than that of compression. Because of the impact on the left end of the glass rod a compression wave - a "half-wave" sine wave with stress amplitude $\sigma_0$ and length $L$ ran. Which section of the rod will break away if the tensile strength $\sigma < \sigma_0$? Consider the cases $\sigma_0 \gg \sigma$ and $\sigma_0 \approx \sigma$.

An end-to-end collision of two elastic rods of the same cross-section takes place. Considering the compression waves generated by the impact and their reflection from the free ends, prove that the result of the collision is the same as in a frontal absolutely elastic collision of bodies if the ratio of lengths of rods equals the ratio of speeds of sound in these rods.

An elastic rod of length $l_1$, flying with speed $v$, collides with the end of a fixed rod of length $l_2$, $\frac{l_1}{c_1} > \frac{l_2}{c_2}$, where $c_1$ and $c_2$ are sound speeds in one and the other rod respectively. The cross-section of the rods and the density of their material are the same. Determine the velocity of the centers of mass of the rods after the collision.

Ultrasonic wave propagates through the air in a narrow corridor without noticeable attenuation for a long distance. The corridor was partitioned with a soundproof screen of some thickness. The power of the passing wave decreased many times. Then a double-thick screen was installed instead of the former one. It was found that through this screen the ultrasound passes almost without attenuation. What's the matter? The wave frequency is $1$ MHz, the sound velocity in the screen material is $5$ km/s. Find the thickness of the sound shields.

How will the natural frequencies of a steel ball change when its radius is doubled?

Between rigid parallel walls there is air. One of the walls starts transverse harmonic motion with amplitude $A_0$ and frequency $\omega$. The distance between the walls is $L \gg A_0$. To what amplitude of displacement in the beam will this wall "rock" the air? Estimate the swing time if the speed of sound in the air is $c$.

A string is excited by passing an alternating current through it, so that the magnetic force on the side of a small magnet $M$ changes harmonically. The frequency of the current corresponds to the third harmonic of the string. The length of the string is $l$. Where should the magnet be placed to make the amplitude of oscillations greatest?

. Determine the mass of a body connected through an elastic support of stiffness k and mass m to a rigid floor if the first resonant frequency of longitudinal vibrations of this system is $\omega$.

". . . In 1941 the Germans invented the cumulative anti-tank shell. On the cone of the shell is a fuse. On impact it causes detonation and ignites the entire charge. The projectile penetrates the entire armor. In 1944 such German shells came into our hands and into the hands of the Allies. A wide experiment was started. Many additional effects and paradoxes were discovered. They began to find out what was flying, what was penetrating? At first they thought it was an armor-busting projectile, that the armor was pierced by a jet of hot gas. No, it turned out that it is metal, and in the most inexplicable way: in front of the plate at a speed of $8$ km/s, inside the plate at $4$ km/s, behind the plate again at $8$ km/s" (from the opening speech of Academician M.A. Lavrentiev, Presidium Chairman of SB AS USSR, to the students of the Summer Physics and Mathematics School in 1971). Explain this phenomenon. Determine with what speed the wall of a metal conical cavity overlapping the charge moved, if the angle at the apex of the cavity is $30^{\circ}$ .

On the surface of a liquid of density $\rho$ floats a cylindrical thin-walled glass, half submerged in the liquid.

a. How much will the glass sink into the liquid if it is placed upside down on the surface of the liquid? The height of the glass is $h$, the air pressure is $P_0$.

b. To what depth do you need to submerge the upside-down glass so that it, together with the air contained in it, goes to the bottom?

Solve problems 5.8.5 if the initial tangent of the angle of incidence is $\frac{1}{m}$, where $m$ is an integer.