Rotational Motion

A disk and a sphere roll off two inclined planes of the same altitude and length. Which of the two objects will get to the bottom of the respective plane first? How does the result depend on the masses and diameters of the disk and the sphere?

A shaft whose diameter is $d$ and length is $l$ is rotating without friction in bearings with an angular velocity ${\omega }_0$. A sleeve of height $h$and outer diameter $D$ is fitted on the shaft (the materials of the sleeve and the shaft are the same). At first the sleeve is not connected with the shaft and is at rest. Then at some moment the sleeve is clamped to the shaft. What will be the common angular velocity of the shaft plus the sleeve?

A top is spinning in the direction designated by the arrow in the figure. In what direction does the precession of the top occur?

The axis $AB$ of a gyroscope is mounted in a frame that can rotate about the axis $CD$. This frame is mounted, via vertical supports $C{C}^{\text{'}}$ and $D{D}^{\text{'}}$, on a horizontal platform which, in turn, can rotate about the axis $EF$. At first the platform is at rest and the gyroscope is rotating in the direction designated by arrow $1$. Then the platform begins to rotate in the direction designated by arrow $2$. How will the gyroscope's axis change its position in space?

A homogeneous rod $AB$ is lying on a perfectly smooth floor. A bullet hits the rod and gets stuck in it. The direction of the bullet's initial velocity $v_0$ is perpendicular to the rod, and the point where the bullet hits the rod lies at a distance $x$ from the middle of the rod. The mass of the bullet is $m$ and the mass of the rod is $M$. Will a velocity directed in opposition to $v_0$ be imparted to end $A$ at the first moment after the collision?

A thin rod of length $2R$ and mass $m$ is standing (vertically) on a perfectly smooth floor. The state of equilibrium in which the rod is at rest is unstable, and the rod falls. Find the trajectories that the various points of the rod describe and the velocity with which the upper end of the rod hits the floor.

A rod whose lower end is sliding along the horizontal plane starts to topple from the vertical position. What will be the velocity of the upper end when this end hits the ground?

A number of types of helicopters, among which are the Soviet-made "Mi" helicopters and the Westland Whirlwinds designed for use by Queen Elizabeth $\mathrm{II}$, utilize one main rotor and a small vertical tail rotor. What is the function of this second rotor?

A vacuum cleaner standing on the floor turns through a small angle when switched on and then stops. Why does this happen?

Strings are wound around a shaft and a sheave of equal mass, and a load is attached to the end of each string (the loads have equal mass). Which of the two loads will descend with a greater acceleration and which of the rotating objects, the shaft or the sheave, has a greater angular acceleration?

A device often used to illustrate the laws of uniformly accelerated motion is the Atwood machine. The machine consists of two loads of mass $m_1$ and $m_2$ attached to the ends of a limp but inextensible string. The string runs over a pulley. The acceleration with which the loads move is $w=\frac{m_1-m_2}{m_1+m_2}g$, whereas the angular acceleration of the pulley is ignored. Is the last assumption true for exact calculations?

liquid has been poured into a cylindrical vessel of mass $M$ (the mass of the vessel bottom can be ighored) and height $H$. The linear density of the liquid, that is, the ratio of the mass of the liquid column to its height, is $\delta$. What is the height $x$ of the column of liquid at which the common center of gravity of the liquid plus the vessel is in the lowest position?

A stunt rider on a unicycle is riding around the arena of a circus in a circle of radius $R$. The radius of the wheel of the unicycle is $r$ and the angular velocity with which the wheel rotates is $\omega$. What is the angular acceleration of the wheel? (Ignore the fact that the wheel axis is inclined.)