In the system shown in the figure, the masses of the bodies are known to be $m_1$ and $m_2$, the coefficient of friction between the body $m_1$ and the horizontal plane is equal to $k$, and a pulley of mass $m$, is assumed to be a uniform disc. The thread does not slip over the pulley. At the moment $t=0$, the body $m_2$, starts descending. Assuming the mass of the thread and the friction in the axle of the pulley to be negligible, find the work performed by the friction forces acting on the body $m_1$, over the first $t$ seconds, after the beginning of the motion.

A uniform cylinder of radius $R$ is spinned about its axis to the angular velocity ${\omega }_0$ and then placed in a corner. The coefficient of friction between the corner walls and the cylinder is equal to $k$. How many turns will the cylinder accomplish before it stops?

A uniform cylinder, of radius $R$ and mass $M$, can rotate freely about a stationary horizontal axis $O$. A thin cord, of length $l$ and mass $m$, is wound on the cylinder in a single layer. Find the angular acceleration of the cylinder as a function of the length $x$ of the hanging part of the cord. The wound part of the cord is supposed to have its centre of gravity on the cylinder axis.

A uniform cylinder of mass $m=8.0 \mathrm{kg}$, and radius $R=1.3 \mathrm{cm}$, starts descending, at a moment $t=0$, due to gravity. Neglecting the mass of the thread, find:

$\left(\mathrm{a}\right)$ the tension of each thread and the angular acceleration of the cylinder;
$\left(\mathrm{b}\right)$ the time dependence of the instantaneous power developed by the gravitational force.

$\left(\text{Gravitational acceleration} g=9.8 \mathrm{m} {\mathrm{s}}^{-2}\right)$

Thin threads are tightly wound on the ends of a uniform solid cylinder, of mass $m$. The free ends of the threads are attached to the ceiling of an elevator car. The car starts going up with an acceleration ${\omega }_0$. Find the acceleration $\omega \text{'}$, of the cylinder, relative to the car, and the force $F$, exerted by the cylinder on the ceiling (through the threads).

A spool, with a thread wound on it, is placed on an inclined smooth plane, set at an angle $\alpha =30°$, to the horizontal. The free end of the thread is attached to the wall, as shown in the figure. The mass of the spool is $m=200 \mathrm{g}$, its moment of inertia, relative to its own axis, is $I=0.45 \mathrm{g} {\mathrm{m}}^2$ and the radius of the wound thread layer is $r=3.0 \mathrm{cm}$. Find the acceleration of the spool axis.

$\left(\text{Gravitational acceleration }g=9.8 \mathrm{m} {\mathrm{s}}^{-2}\right)$