I E Irodov Solutions for Chapter: PHYSICAL FUNDAMENTALS OF MECHANICS, Exercise 5: DYNAMIC OF A SOLID BODY

A smooth uniform rod $AB$, of mass $M$ and length $l$, rotates freely with an angular velocity ${\omega }_0$, in a horizontal plane, about a stationary vertical axis passing through its end $A$. A small sleeve of mass $m$ starts sliding along the rod from the point $A$. Find the velocity $v\text{'}$ of the sleeve, relative to the rod, at the moment it reaches its other end $B$.

A uniform rod of mass $m=5.0 \mathrm{kg}$ and length $l=90 \mathrm{cm}$, rests on a smooth horizontal surface. One of the ends of the rod is struck with an impulse $J=3.0 \mathrm{N} \mathrm{s}$, in the horizontal direction, perpendicular to the rod. As a result, the rod obtains the momentum $p=3.0 \mathrm{N} \mathrm{s}$. Find the force that one half of the rod will apply on the other, in the process of motion.

A thin uniform square plate, with side $l$ and mass $M$, can rotate freely about a stationary vertical axis, coinciding with one of its sides. A small ball of mass $m$, flying with velocity $\overrightarrow{v}$, at right angles to the plate, strikes elastically at the centre of it. Find:
$\left(\mathrm{a}\right)$ the velocity of the ball $\overrightarrow{v^{\text{'}}}$ after the impact;
$\left(\mathrm{b}\right)$ the horizontal component of the resultant force, which the axis will exert on the plate after the impact.

A vertically oriented uniform rod, of mass $M$ and length $l$, can rotate about its upper end. A horizontally flying bullet, of mass $m$, strikes the lower end of the rod and gets stuck on it; as a result, the rod swings through an angle $\alpha$. Assuming that $m\ll M,$ find
$\left(\mathrm{a}\right)$ the velocity of the flying bullet
$\left(\mathrm{b}\right)$ the momentum increment in the system "bullet-rod" during the impact; what causes the change of that momentum
$\left(\mathrm{c}\right)$ at what distance $x$, from the upper end of the rod, the bullet must strike, for the momentum of the system "bullet-rod" to remain constant during the impact.

A horizontally oriented uniform disc, of mass $M$ and radius $R$, rotates freely about a stationary vertical axis, passing through its centre. The disc has a radial guide, along which can slide, without friction, a small body of mass $m$. A light thread, running down through the hollow axle of the disc, is tied to the body. Initially, the body was located at the edge of the disc and the whole system was rotated with an angular velocity ${\omega }_0$. Then, by means of a force $F$ applied to the lower end of the thread, the body was slowly pulled to the rotation axis. Find:
$\left(\mathrm{a}\right)$ the angular velocity of the system in its final state;
$\left(\mathrm{b}\right)$ the work performed by the force $F$.

A man of mass $m_1$ stands on the edge of a horizontal uniform disc of mass $m_2$ and radius $R$, which is capable of rotating freely about a stationary vertical axis passing through its centre. At a certain moment, the man starts moving along the edge of the disc; he shifts over an angle $\phi \text{'}$, relative to the disc, and then stops. In the process of motion, the velocity of the man varies with time as $v\text{'}\left(t\right)$. Assuming the dimensions of the man to be negligible, find:
$\left(\mathrm{a}\right)$ the angle through which the disc had turned by the moment the man stopped;
$\left(\mathrm{b}\right)$ the force moment (relative to the rotation axis) which the man applied on the disc, in the process of motion.

Two horizontal discs rotate freely about a vertical axis passing through their centres. The moments of inertia of the discs, relative to this axis, are equal to $I_1$ and $I_2,$ and their angular velocities are $\overrightarrow{{\omega }_1}$. and $\overrightarrow{{\omega }_2}$. When the upper disc fell on the lower one, both discs began rotating, after some time, as a single whole (due to friction). Find:
$\left(\mathrm{a}\right)$ the steady-state angular rotation velocity of the discs;
$\left(\mathrm{b}\right)$ the work performed by the friction forces in this process.

A small disc and a thin uniform rod, of length $l,$ whose mass is $\eta$ times greater than the mass of the disc, lie on a smooth horizontal plane. The disc is set in motion, in horizontal direction and perpendicular to the rod, with velocity $\overrightarrow{v}$, after which it elastically collides with the end of the rod. Find the velocity of the disc and the angular velocity of the rod, after the collision. At what value of $\eta$, will the velocity of the disc, after the collision, be equal to zero? Reverse its direction?