A shell acquires the initial velocity $v=320 \mathrm{m} {\mathrm{s}}^{-1}$, having made $n=2.0$ turns inside the barrel whose length is equal to $l=2.0 \mathrm{m}$. Assuming that the shell moves inside the barrel with a uniform acceleration, find the angular velocity of its axial rotation at the moment when the shell escapes the barrel.

A solid body rotates about a stationary axis according to the law, $\phi =at-b{t}^3$, where $a=6.0 \mathrm{rad} {\mathrm{s}}^{-1}$ and $b=2.0$ $\mathrm{rad} {\mathrm{s}}^{-3}$. Find:
(a) the mean values of the angular velocity and angular acceleration, averaged over the time interval between $t=0$ and then complete stop;

(b) the angular acceleration at the moment when the body stops.

A rotating disc moves in the positive direction of the $x$-axis. Find the equation $y\left(x\right)$ describing the position of the instantaneous axis of rotation, if at the initial moment the axis $C$ of the disc was located at the point $O$ after which it moved
$\left(a\right)$ with a constant velocity $v$, while the disc started rotating counterclockwise with a constant angular acceleration $\mathrm{\beta }$ (the initial angular velocity is equal to zero),

$\left(b\right)$ with a constant acceleration $w$ (and the zero initial velocity), while the disc rotates counterclockwise with a constant angular velocity $\omega$.

A ball of radius $R=10.0 \mathrm{cm}$ rolls without slipping down an inclined plane so that its centre moves with constant acceleration $w=2.50 \mathrm{cm} {\mathrm{s}}^{-2}\text{, }t=2.00 \mathrm{s}$ after the beginning of motion, its position corresponds to that shown in the figure. Find, 
$\left(a\right)$ the velocities of the points $A\text{,} B$ and $O$, 
$\left(b\right)$ the accelerations of these points.