– Equation of dynamics of a solid body rotating about a stationary axis a:
\[
r_{n}=N_{n} \text {. }
\]
where $N_{\text {, }}$ is the sleleiraic num of the moments of external forces relative to the a axis.
\[
\begin{array}{l}
– \text { Aecerding to Steiner’s theorem: } \\
\qquad t=I_{e}+\text { mal. }
\end{array}
\]
– Kinetie energy of a solid bedy rotating abeot a stationary axis:
\[
t=\frac{1}{2} t \omega^{*} \text {. }
\]
– Work performed by external forces during the rotation of a solid lindy abevt a stationary axis:
\[
A=\int N_{4} d q \text {. }
\]
– Kinetie energy of a solid body in plane motion:
\[
r=\frac{I_{c} \omega^{2}}{2}+\frac{m v t}{2} \text {. }
\]
– Relationslip between the ancular velocity e’ ef cyposcope precession,
\[
\left[e^{\prime} \mathbf{M}\right]=\mathbf{N} \text {. }
\]
1.234. A thin uniform rod $A B$ of mass $m=1.0 \mathrm{~kg}$ moves translationally with acceleration $w=2.0 \mathrm{~m} / \mathrm{s}^{2}$ due to two antiparallel forces $\boldsymbol{F}_{1}$ and $\mathbf{F}_{\text {, }}$ ( Fig. 1.52). The distance between the points at which these forces are applied is equal to $a=20 \mathrm{~cm}$. Besides, it is known that $F_{1}=5.0 \mathrm{~N}$. Find the length of the rod.
1.235. A force $\mathbf{F}=\boldsymbol{A} \mathbf{i}+B \mathbf{j}$ is applied to a point whose radius vector relative to the origin of coordinates $O$ is equal to $r=a i+$ $+b \mathrm{j}$, where $a, b, A, B$ are constants, and $i, j$ are the unit vectors of
n. the $x$ and $y$ axes. Find the moment $\mathbf{N}$ and the arm $l$ of the force $\mathbf{F}$ relative to the point $O$.
1.236. A force $F_{1}=A j$ is applied to a point whose radius vector $r_{1}=a i$, while a force $F_{1}=B i$ is applied to the point whose radius vector $\mathrm{r}_{1}=\mathrm{bj}$. Both radius vectors are determined relative to the origin of coordinates $O, I$ and $j$ are the unit vectors of the $x$ and $y$
Fie. 1.52 .
Fie. 1.53.
axes, a, b, A, B are constants. Find the arm $l$ of the resultant force relative to the point $O$.
1.237. Three forces are applied to a square plate as shown in Fig. 1.53. Find the modulus, direction, and the point of application of the resultant force, if this point is taken on the side BC.
1.238. Find the moment of inertia
(a) of a thin uniform rod relative to the axis which is perpendicular to the rod and passes through its end, if the mass of the rod is $m$ and its length $l_{\text {; }}$
(b) of a thin uniform rectangular plate relative to the axis passing perpendicular to the plane of the plate through one of its vertices, if the sides of the plate are equal to $a$ and $b$, and its mass is $m$.
1.239. Calculate the moment of inertis
(a) of a copper uniform dise relative to the symmetry axis perpendicular to the plane of the dise, if its thickness is equal to $b=2.0 \mathrm{~mm}$ and its radius to $R=100 \mathrm{~mm}$;
(b) of a uniform solid cone relative to its symmetry axis, if the mass of the cone is equal to $m$ and the radius of its base to $\boldsymbol{R}$.
1.240. Demonstrate that in the case of a this plate of arbitrary shape there is the following relationship between the moments of inertia: $I_{1}+I_{3}=I_{3}$. where subindices $1_{1}, 2$, and 3 define three mutually perpendicular axes passing through one point, with axes $\mathbf{1}$ and 2 lying in the plane of the plate. Using this relationship, find the moment of inertia of a thin uniform round dise of radius $\boldsymbol{R}$ and mass $m$ relative to the axis coinciding with one of its dismeters.
1.241. A uniform dise of radius $R=20 \mathrm{~cm}$ has a round eut as shows in Fig. 1.54. The mass of the remaining (shaded) portion of the s.
dise equals $m=7.3 \mathrm{~kg}$. Find the moment of inertis of such a dise relative to the axis passing through its centre of inertia and perpendicular to the plane of the dise.
1.242. Using the formula for the moment of inertia of a uniform sphere, find the moment of inertia of a thin spherical layer of mass $m$ and radius $\boldsymbol{R}$ relative to the axis passing through its centre.
1.243. A light thread with a body of mass $m$ tied to its end is wound on a uniform solid cylinder of mass $M$ and radius $\boldsymbol{R}$ (Fig. 1.55). At a moment $t=0$ the system is set in motion. Assuming the friction is the axle of the cylisder to be negligible, find the time dependence of
(a) the angular velocity of the cylinder;
(b) the kinetic energy of the whole system.
1.24. The ends of thin threads tightly wound on the axle of radius $r$ of the Maxwell dise are attached to a horizontal bar. When the dise unwinds, the bar is raised to keep the dise at the same height. The mass of the dise with the axle is equal to $m$, the moment of
Fie. 1.54. inertia of the arrangenest relative to its axis is $I$. Find the tension of each thread and the acceleration of the bar.
1.245. A thin horizontal uniform rod $A B$ of mass $m$ and length $t$ can rotate freely about a vertical axis passing through its end $\boldsymbol{A}$. At a certain moment the end $B$ starts experiencing a constant force
Fie. 1.55.
Fig. 1.5e.
$\boldsymbol{F}$ which is always perpendicular to the original position of the stationary rod and directed in a horizontal plane. Find the angular velocity of the rod as a fusction of its rotation angle $\rho$ comited relative to the initial position.
1.246. In the arrangement shown in Fig. 1.56 the mass of the uniform solid cylinder of radius $\boldsymbol{R}$ is equal to $m$ and the mases of two bodies are equal to $m_{1}$ and $m_{3}$. The thread slipping and the friction in the axle of the cylinder are supposed to be absent. Find the angular acceleration of the cylinder and the ratio of tensions $T_{1} / T_{\text {, }}$, of the vertical sections of the thread in the process of motion.
1.247. In the system shown in Fig. 1.57 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 motion.
1.248. A uniform cylinder of radius $R$ is spinned about its axis to the angular velocity $\omega_{0}$ and then placed into a corner (Fig. 1.58).
Fig. 1.57.
Fig. 1.58.

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?
1.249. A uniform disc of radius $R$ is spinned to the angular velocity $\omega$ and then carefully placed on a horizontal surface. How long will the disc be rotating on the surface if the friction coefficient is equal to $k$ ? The pressure exerted by the disc on the surface can be regarded as uniform.
1.250. A flywheel with the initial angular velocity $\omega_{0}$ decelerates due to the forces whose moment relative to the axis is proportional to the square root of its angular velocity. Find the mean angular velocity of the flywheel averaged over the total deceleration time.
1.251. A uniform cylinder of radius $R$ and mass $M$ can rotate freely about a stationary horizontal axis $O$ (Fig. 1.59). 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.
1.252. A uniform sphere of mass $m$ and radius $R$ rolls without slipping down an inclined plane set at an angle $\alpha$ to the horizontal. Find:
(a) the magnitudes of the friction coefficient at which slipping is absent;
(b) the kinetic energy of the sphere $t$ seconds after the beginning of motion.
1.253. A uniform cylinder of mass $m=8.0 \mathrm{~kg}$ and radius $R=$ $=1.3 \mathrm{~cm}$ (Fig. 1.60) starts descending at a moment $t=0$ due to gravity. Neglecting the mass of the thread, find:
50
(a) the tension of each thread and the angular acceleration of the cylinder;
(b) the time dependence of the instantaneous power developed by the gravitational force.
1.254. 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
Fig. 1.59 .
Fig. 1.60.
the ceiling of an elevator car. The carstarts going up with an acceleration $w_{0}$. Find the acceleration $w^{\prime}$ of the cylinder relative to the car and the force $\mathbf{F}$ exerted by the cylinder on the ceiling (through the threads).
1.255. A spool with a thread wound on it is placed on an inclined smooth plane set at an angle $\alpha=30^{\circ}$ to the horizontal. The free end of the thread is attached to the wall as shown in Fig. 1.61. The mass of the spool is $m=200 \mathrm{~g}$, its moment of inertia relative to its own axis $I=0.45 \mathrm{~g} \cdot \mathrm{m}^{2}$, the radius of the wound thread layer $r=3.0 \mathrm{~cm}$. Find the acceleration of the spool axis.
1.256. A uniform solid cylinder of mass $m$ rests on two horizontal planks. A thread is wound on the cylinder. The hanging end of the thread is pulled vertically down with a constant force $F$ (Fig. 1.62).
Fig. 1.61.
Fig. 1.62.

Find the maximum magnitude of the force $F$ which still does not bring about any sliding of the cylinder, if the coefficient of friction between the cylinder and the planks is equal to $k$. What is the ac4*

celeration $w_{\max }$ of the axis of the cylinder rolling down the inclined plane?
1.257. A spool with thread wound on it, of mass $m$, rests on a rough horizontal surface. Its moment of inertia relative to its own axis is equal to $I=\gamma m R^{2}$, where $\gamma$ is a numerical factor, and $R$ is the outside radius of the spool. The radius of the wound thread layer is equal
Fig. 1.63.
to $r$. The spool is pulled without sliding by the thread with a constant force $\mathbf{F}$ directed at an angle $\alpha$ to the horizontal (Fig. 1.63). Find:
(a) the projection of the acceleration vector of the spool axis on the $x$-axis;
(b) the work performed by the force $\mathbf{F}$ during the first $t$ seconds after the beginning of motion.
1.258. The arrangement shown in Fig. 1.64 consists of two identical uniform solid cylinders, each of mass $m$, on which two light threads
Fig. 1.64.
Fig. 1.65.
are wound symmetrically. Find the tension of each thread in the process of motion. The friction in the axle of the upper cylinder is assumed to be absent.
1.259. In the arrangement shown in Fig. 1.65 a weight $A$ possesses mass $m$. a pulley $B$ possesses mass $M$. Also known are the moment of inertia $I$ of the pulley relative to its axis and the radii of the pulley
$R$ and $2 R$. The mass of the threads is negligible. Find the acceleration of the weight $A$ after the system is set free.
1.260. A uniform solid cylinder $A$ of mass $m_{1}$ can freely rotate about a horizontal axis fixed to a mount $B$ of mass $m_{2}$ (Fig. 1.66). A constant horizontal force $F$ is applied to the end $K$ of a light thread tightly wound on the cylinder. The friction between the mount and the supporting horizontal plane is assumed to be absent. Find:
(a) the acceleration of the point $K$;
(b) the kinetic energy of this system $t$ seconds after the beginning of motion.
1.261. A plank of mass $m_{1}$ with a uniform sphere of mass $m_{2}$ placed on it rests on a smooth horizontal plane.
Fig. 1.66.
A constant horizontal force $F$ is applied to the plank. With what accelerations will the plank and the centre of the sphere move provided there is no sliding between the plank and the sphere?
1.262. A uniform solid cylinder of mass $m$ and radius $R$ is set in rotation about its axis with an angular velocity $\omega_{0}$, then lowered with its lateral surface onto a horizontal plane and released. The coefficient of friction between the cylinder and the plane is equal to $k$. Find:
(a) how long the cylinder will move with sliding;
(b) the total work performed by the sliding friction force acting on the cylinder.
1.263. A uniform ball of radius $r$ rolls without slipping down from the top of a sphere of radius $R$. Find the angular velocity of the ball at the moment it breaks off the sphere. The initial velocity of the ball is negligible.
1.264. A uniform solid cylinder of radius $R=15 \mathrm{~cm}$ rolls over a horizontal plane passing into an inclined plane forming an angle
Fig. 1.67.
Fig. 1.68.
$\alpha=30^{\circ}$ with the horizontal (Fig. 1.67). Find the maximum value of the velocity $v_{0}$ which still permits the cylinder to roll onto the inclined plane section without a jump. The sliding is assumed to be absent.
1.255. A small body $A$ is fixed to the inside of a thin rigid hoop of radius $\boldsymbol{H}$ and mass equal to that of the body $A$. The hoop rolls without slipping over a herisontal plane; at the moments when the body $A$ gets into the lower position, the centre of the hoop moves with velocity
Fie. 1.69 .
ก.g. 1.20,
$v_{0}$ (Fig. 1.68). At what values of $v_{0}$ will the hoop move without bouneing?
1.266. Determine the kinetic energy of a tractor crawler belt of mass $m$ if the tractor moves with velocity $v$ (Fig. 1.69).
1.267. A uniform sphere of mass $m$ and radius $r$ rolls without sliding over a horizontal plane, rotating about a horizontal axle $O A$ (Fig. 1.70). In the process, the centre of the sphere moves with velocity $v$ along a circle of radias $\boldsymbol{R}$. Find the kinetic energy of the sphere.
1.268. Demonstrate that in the reference frame rotating with a constant angular velocity $\omega$ about a stationary axis a body of mass $m$ experiences the resultant
(a) centrifugal force of inertia $\mathbf{F}_{e f}=$ $=m \omega^{2} \mathbf{R}_{C}$, where $\mathbf{R}_{C}$ is the radius vector of the body’s centre of inertia relative to the rotation axis;
(b) Coriolis force $\boldsymbol{F}_{\mathrm{ser}}=2 \mathrm{~m}$ |viel, where $\mathbf{v} \dot{c}$ is the velocity of the body’s centre of inertia in the rotating reference frame.
1.259. A midpoint of a thin uniform rod $A B$ of mass $m$ and length $l$ is rigidly fixed to a rotation axle $O O^{\prime}$ as shown in Fig. 1.71. The rod is set into rotation with a constant angular velocity $\omega$. Find the resultant moment of the centrifugal forces of inertia relative to the point $C$ in the reference frame fixed to the axle $C O^{\prime}$ and to the rod.
1.270. A conical pendulum, a thin uniform rod of length $l$ and imass $m$, rotates uniformly about a vertical axis with angular velocity – (the upper end of the rod is hinged). Find the angle $\theta$ between the rod and the vertical.
1.271. A uniform cube with edge a rests on a horizontal plane whose friction coeffeient equals $k$. The cube is set in motion with an initial velocity, travels some distance over the plane and comes to a stand-
54
still. Explain the disappearance of the angular momentum of the cube relative to the axis lying in the plane at right angles to the cube’s motion direction. Find the distance between the resultants of gravitational forces and the reaction forees exerted by the supporting plane.
1.272. A smooth uniform $\operatorname{rod} A B$ 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 $\boldsymbol{A}$. Find the veloeity $v^{\prime}$ of the sleeve relative to the rod at the moment it reaches its other end $B$.
1.273. 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 the impulse $J=3.0 \mathrm{~N}$-s in a horizontal direction perpendicular to the rod. As a result, the rod obtains the momentum $p=3.0 \mathrm{~N}$-s. Find the force with which one half of the rod will act on the other in the process of motion.
1.274. A thin uniform square plate with side $l$ and mass $M$ can rotate freely about o stationary vertical axis coinciding with one of its sides. A small ball of mass $m$ Aying with velocity $v$ at right angles to the plate strikes elastically the centre of it. Find:
(a) the velocity of the ball $v^{\prime}$ after the impact:
(b) the horizontal component of the resultant force which the axis will exert on the plate after the impact.
1.275. A vertically oriented unilorm rod of mass $M$ and length $t$ cas rotate about its upper end. A horizontally flying bullet of mass $m$ strikes the lower end of the rod and gots stuck in it; as a result, the rod swing through an angle $\alpha$. Assuming that $m<M$, find:
(a) the velocity of the fying bullet:
(b) the momentum increment in the system \”bullet-rod\” during the impact; what causes the change of that momentum;
(c) at what distance $x$ from the upper end of the rod the bullet mast strike for the momentum of the system \”bullet-rod\” to remain constant during the impact.
1.276. A horizontally oriented uniform dise of mass $M$ and radius $\boldsymbol{A}$ rotates freely about a stationary vertical axis passing through its centre. The dise has a radial guide along which can slide without friction a small body of mass $m$. A light thread ranning dows through the hollow axle of the dise is tied to the body. Initially the body was located at the edge of the dise and the whole system rotated with an angular velocity $\omega_{6}$. 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:
(a) the angular velocity of the system in its final state;
(b) the work perlormed by the force $F$.
1.277. A man of mass $m_{1}$ stands on the edge of a horizontal uniform disc of mass $m_{2}$ and radius $\boldsymbol{R}$ which is capable of rotating freely about a stationary vertical axis passing through its centre. At a cer-
ss

tain moment the man starts moving along the edge of the dise; he shifts over an angle $\varphi^{\prime}$ relative to the dise and then stops. In the process of motion the velocity of the man varies with time as $v^{\prime}(\mathrm{f})$. Assuming the dimensions of the man to be negligible, find:
(a) the angle through which the dise had turned by the moment the man stopped;
(b) the force moment (relative to the rotation axis) with which the man acted on the dise in the process of motion.
1.278. 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 $I_{5}$, and the angular velocities to $\omega_{1}$ and $\epsilon_{1}$. When the upper dise fell on the lower one, both disca began rotating, after some time, as a single whole (doe to friction). Find:
(a) the steady-state angular rotation velocity of the discs;
(b) the work performed by the friction forces in this process.
1.279. A small dise and a thin uniform rod of length $l$, whose mass is $\eta$ times greater than the mass of the dise, lie on a smooth horizontal piane. The dise is set in motion, in horizontal direction and perpendicular to the rod, with velocity v, after which it elastically collides with the end of the rod. Find the velocity of the dise and the angular velocity of the rod after the collision. At what value of $\eta$ will the velocity of the dise after the collision be equal to sero? reverse its direction?
1.280. A stationary platform $P$ which can rotate freely about a vertical axis (Fig. 1.72) supports a motor M and a balance weight $\boldsymbol{N}$. The moment of inertia of the platform with the motor and the balance weight relative to this axis is equal to $I$. A light frame is fixed to the motor’s shaft with a uniform sphere $A$ rotating freely with an angular velocity $\omega_{0}$ about $a$ shaft $B B^{\prime}$ coineiding with the axis $O O^{\prime}$. The moment of inertis of the sphere relative to the rotation axis is equal to $I_{\text {. }}$. Find:
(a) the work performed by the motor in turning the shaft $B B^{\prime}$ through $90^{\circ}$; through $180^{\circ}$;
(b) the moment of external forces which maintains the axis of the arrangement in the vertical position after the motor turns the shaft $B B^{\prime}$ through $90^{\circ}$.
1.281. A horizontally oriented uniform $\operatorname{rod} A B$ of mass $m=$ $-1.40 \mathrm{~kg}$ and length $L_{0}=100 \mathrm{~cm}$ rotates freely about a stationary vertical axis $O O^{\prime}$ passing through its ead $A$. The point $A$ is located at the middle of the axis $O O^{\prime}$ whose length is equal to $l=55 \mathrm{~cm}$. At what angular velocity of the rod the horizental component of the force acting on the lower end of the axis $O O^{\prime}$ is equal to nero? What
is in this case the horizontal component of the force acting on the upper end of the axis?
1.282. The middle of a uniform rod of mass $m$ and length $/$ is rigidly fixed to a vertical axis $O O^{\prime}$ so that the angle between the rod and the axis is equal to $\theta$ (see Fig. 1.71). The ends of the axis $O O^{\prime}$ are provided with bearings. The system rotates without friction with an angular velocity $\omega$. Find:
(a) the magnitude and direction of the rod’s angular momentum $M$ relative to the point $C$, as well as its angular momentum relative to the rotation axis:
(b) how much the modulas of the vector M relative to the point $C$ increases during a half-turn:
(c) the moment of external forces $N$ acting on the axle $O O^{\prime}$ in the process of retation.
1.283. A top of mass $m=0.50 \mathrm{~kg}$, whose axis is tilted by an angle $\theta=30^{\circ}$ to the vertical, precesses due to gravity. The moment of inertis of the top relative to its symmetry axis is equal to $I=$ $=2.0 \mathrm{~g} \cdot \mathrm{m}^{2}$, the angular velocity of rotation about that axis is equal to $\omega=350 \mathrm{rad} / \mathrm{s}$, the distance from the point of rest to the centre of inertie of the top is $l=10 \mathrm{~cm}$. Find:
(a) the angular velocity of the top’s precession;
(b) the magnitude and direction of the horizontal component of the reaction force acting on the top at the point of rest.
1.284. A gyroscope, a uniform dise of radius $R=5.0 \mathrm{~cm}$ at the end of a rod of length $l=10 \mathrm{~cm}$ (Fig. 1.73), is mounted on the floor of an elevator car going up with a constant acceleration $w=2.0 \mathrm{~m} / \mathrm{s}^{2}$. The other end of the rod is hinged at the point $O$. The gyroscope precesses with an angular velocity $n=0.5$ rps. Neglecting the friction and the mass of the rod, find the proper angular velocity of the disc.
1.285. A top of mass $m=1.0 \mathrm{~kg}$ and moment of inertis relative to its ewn axis $I=4.0 \mathrm{~g} \cdot \mathrm{m}^{2}$ spins with an angular velocity
Fig. 1.72.
Fis. 1.72
The distance between the bearings in which the axle of the dise is mounted is equal to $l=15 \mathrm{~cm}$. The axle is forced to oscillate about a horizontal axis with a period $T=1.0$ s and amplitude $\varphi_{\mathrm{m}}=20^{\circ}$. Find the maximum value of the gyroscopic forces exerted by the axle on the bearings.
1.288. A ship moves with velocity $v=36 \mathrm{~km}$ per hour along an are of a cirele of radius $A=200 \mathrm{~m}$. Find the moment of the gyroscopic forces exerted on the bearings by the shaft with a dywheel whose moment of inertia relative to the rotation axis equals $I=$ $=3.8 \cdot 10^{\circ} \mathrm{kg} \cdot \mathrm{m}^{2}$ and whose rotation velocity $n=300 \mathrm{rpm}$. The rotation axis is oriented along the length of the ship.
1.289. A locomotive is propelled by a turbine whose axle is parallel to the axes of wheels. The turbine’s rotation direction coincides with that of wheels. The moment of inertia of the turbine rotor relative to its own axis is equal to $I=240 \mathrm{~kg} \cdot \mathrm{m}^{2}$. Find the additional force exerted by the gyroscopic forees on the rails when the locomotive moves along a circle of radius $R=250 \mathrm{~m}$ with velocity $v=$ $-50 \mathrm{~km}$ per hour. The gauge is equal to $l=1.5 \mathrm{~m}$. The angular velocity of the turbine equals $n=1500 \mathrm{rpm}$.