– Work and power of the force $\mathrm{F}$ :
\[
A=\int F d r=\int F, d r, P=F v \text {. }
\]
– Iacrement of the kinetie energy of a partiele:
\[
r_{1}-r_{1}=A \text {, }
\]
where $A$ is the work performed by the resultant of all the lorces acting on the particle.
– Work performed by the forces of a feld is ecual to the decrease of the potential esericy of a particle in the given feld:
\[
A=U_{1}-U_{r}
\]
– Relationship between the force of a beld and the potential energy of a particle in the field:
\[
\mathbf{r}=-\boldsymbol{v} \boldsymbol{v}
\]
i.e. the force is equal to the antigradient of the petential energy. tial fild: Increment of the tetal mechanical energy of o particle in a diven potes-
\[
E_{s}-E_{1}=A_{\text {entr }}
\]
that is, by the forces not belonging to those of the cior feld.
– Iscrement of the tetal mechanical energy of a system:
\[
E_{3}-E_{1}=A_{\text {ext }}+A_{\text {int }}^{\text {nemest }} \text {, }
\]
where $E=T+U$, and $U$ is the inherent poteatial energy of the system.
– Law of momentum variation of a system:
\[
\mathbf{i p} / d t=\boldsymbol{F} \text {, }
\]
where $F$ is the resaltant of all eaternal lotes.
– Equatios of motios of the system’s centre of inertia:
\[
n \frac{d \mathbf{v}_{c}}{d t}=\boldsymbol{r} \text {. }
\]
where $F$ is the resultant of all exiternal ferces.
– Kinetie energy of a system
\[
r=\tilde{f}+\frac{m \underline{z}}{2},
\]
where $F$ is its kinetie energy in the system of eentre of inertia.
– Equation of dynamica of a body with variable mass:
\[
=\frac{d v}{d t}=F+\frac{d m}{d t}=
\]
shere $\mathrm{o}$ is the velocity of the mapated (pained) sobatance relative to the body censidered.
$\mathbf{3}$
– Law of ancular mementum variatios of a system:
\[
\frac{d \mathrm{M}}{d i}=\mathrm{N} \text {, }
\]
shere $\mathbf{M}$ is the angular momentum of the system, and $\mathrm{N}$ is the tetal moment of all eriternal forces.
– Angular momentum of a nystem:
\[
\mathbf{M}=\overline{\mathbf{M}}+|\mathbf{r} \mathbf{c} \mathbf{P}| \text {. }
\]
shere $\mathbf{M}$ is its angular mementum in the system of the centre of inertis, $\mathrm{r}_{\mathrm{C}}$ is the radius vecter of the centre of inertia, and $p$ is the momentum of the system.
1.118. A particle has shifted along some trajectory in the plane $x y$ from point $l$ whose radius vector $r_{1}=i+2 j$ to point 2 with the radius vector $r_{n}=2 i-3 j$. During that time the particle experienced the action of certain forces, one of which being $F=3 i+4 j$. Find the work performed by the force F. (Here $r_{1}, r_{4}$, and $F$ are given in $S I$ units).
1.119. A locomotive of mass $m$ starts moving so that its velocity varies according to the law $v=a \sqrt{3}$, where $a$ is a constant, and $s$ is the distance covered. Find the total work performed by all the forces which are acting on the locomotive during the first $t$ seconds after the beginaing of motion.
1.120. The kinetic enercy of a partiele moving along a circle of radius $R$ depends on the distance covered $s$ as $T=a r^{2}$, where $a$ is a constant. Find the force acting on the particle as a function of $s$.
1.121. A body of mass $m$ was slowly hauled up the hill (Fig, 1.29) by a force $F$ which at each point was directed along a tangent to the trajectory. Find the work performed by this force, if the height of the hill is $h$, the length of its base 1 , and the coefficient of friction $k$.
1.122. A dise of mass $m=50 \mathrm{~g}$ slides with the sero initial velocity down an inclined plane set at an angle $\alpha=30^{\circ}$ to the horizontal: having traversed the distance $t=50 \mathrm{~cm}$ along the horizontal plane, the dise stops. Find the work performed by the friction forces over the whole distance, assuming the friction coefficieat $k=0.15$ for both inclined and horizontal planes.
1.123. Two bars of masses $m_{1}$ and $m_{1}$ connected by a non-deformed Ilght spring rest on a horizontal plane. The coefficient of friction between the bars and the surface is equal to $\boldsymbol{k}$. What minimum constant force has to be applied in the horizontal direction to the bar of mass $m_{4}$ in order to shift the other bar?
1.124. A chain of mass $m=0.80 \mathrm{~kg}$ and length $t=1.5 \mathrm{~m}$ rests on a rough-surfaced table so that one of its ends hangs over the edge. The chain starts sliding ofl the table all by itself provided the overhanging part equals $\eta=1 / 3$ of the chain length. What will be the total work performed by the friction forces acting on the chain by the moment it slides completely of the table?
1.125. A body of mass $m$ is thrown at an angle $a$ to the horizontal with the initial velocity $v_{0}$. Find the mean power developed by gravity over the whole time of motion of the body, and the instantaneous power of gravity as a function of time.
1.136. A particle of mass $m$ moves along a circle of radius $\boldsymbol{R}$ with a normal acceleration varying with time as $w_{n}=a f^{2}$, where $a$ is a constant. Find the time dependence of the power developed by all the forces acting on the particle, and the mean value of this power averaged over the first $t$ seconds after the beginaing of motion.
1.127. A small body of mases $m$ is located on a horitontal plane at the point $O$. The body aequires a horizontal velocity $v_{0}$. Find:
(a) the mean power developed by the friction force during the whole time of motion, if the friction coefficient $k=0.27, m=1.0 \mathrm{~kg}$. and $v_{0}=1.5 \mathrm{~m} / \mathrm{s}$;
(b) the maximum instantaneous power developed by the friction force, if the friction coefficient varies as $k=\alpha x$, where $\alpha$ is a constant, and $x$ is the distance from the point $O$.
1.128. A small body of mass $m=0.10 \mathrm{~kg}$ moves in the reterence frame rotating about a stationary axis with a constant angular velocity $6=5.0 \mathrm{rad} / \mathrm{s}$. What work does the centrifugal force of inertia perform during the transfer of this body aleng as arbitrary path from point $I$ to point 2 which are located at the distances $n_{1}=\mathbf{3 0} \mathrm{cm}$ and $r_{1}=50 \mathrm{~cm}$ from the rotation axis?
1.129. A system consists of two spriags consected in series and having the stiffness coefficients $k_{1}$ and $k_{2}$. Find the minimum work to be performed in order to stretch this system by $\Delta l$.
1.130. A body of mass $m$ is hauled from the Earth’s surface by applying a force $\boldsymbol{F}$ varying with the beight of ascent $y$ as $\boldsymbol{F}=2$ (ay -1) $\mathrm{mg}$. where $a$ is a positive constant. Find the work performed by this force and the increment of the body’s potential energy in the gravitational field of the Earth over the first half of the ascent.
1.131. The potential energy of a particle in a certais field has the form $U=a / r^{2}-b / r$, where $a$ and $b$ are positive constants, $r$ is the distance from the centre of the field. Find:
(a) the value of $r_{0}$ corresponding to the equilibrium position of the particle; examine whether this position is steady;
(b) the maximum magnitude of the attraction force; draw the plots $U(r)$ and $F,(r)$ (the projections of the force on the radius vector $\mathrm{r}$ ).
1.132. In a certain two-dimensional field of force the potential energy of a particle has the form $U=\alpha z^{2}+\beta y^{2}$, where $a$ and $\beta$ are positive constants whose magaitudes are diflerent. Find out:
(a) whether this field is central;
(b) what is the shape of the equipotential surfaces and also of the surfaces for which the magnitude of the vector of force $F=$ const.
1.133. There are two stationary fields of force $\boldsymbol{F}=$ ayi and $\mathbf{F}=$
– axi + byi, where $i$ and f are the unit vectors of the $x$ and $y$ axes, and $a$ and $b$ are constants. Find out whether these fields are potential.
1.134. A body of mass $m$ is pushed with the initial velocity $v_{0}$ up an inclined plane set at an angle $a$ to the horizontal. The friction coefficient is equal to $k$. What distance will the body cover before it stops and what work do the friction forces perform over this distance?
1.135. A small dise $A$ slides down with initial velocity equal to sero from the top of a smooth hill of height $\boldsymbol{H}$ having a horizontal portion (Fig. 1.30). What must be the height of the horizontal portion $h$ to ensure the maximum distance $s$ covered by the disc? What is it equal to?
1.136. A small body $A$ starts sliding from the height $h$ down an inclined groove passing into a half-eirele of radius $h / 2$ (Fig. 1.31).
Fig. 1.30.
Fie. 1.31.
Assuming the friction to be negligible, find the velocity of the body at the highest point of its trajectory (after breaking off the groove).
1.137. A ball of mass $m$ is suspended by a thread of length $L$. With what minimum velocity has the point of suspension to be shifted in the horizontal direction for the ball to move along the circle about that point? What will be the tension of the thread at the moment it will be passing the horizental position?
1.138. A horizental plane supports a stationary vertical cylinder of radius $\boldsymbol{R}$ and a dise $\boldsymbol{A}$ attached to the cylinder by a horizontal thread $A B$ of length $l_{6}$ (Fig. 1.32, top view). An initial velocity $v_{0}$
Fie. 1.32 .
คi. 1.38.
is imparted to the disc as shown in the figure. How long will it move along the plane until it strikes against the cylinder? The friction is assumed to be absent.
1.139. A smooth rubber cord of length $l$ whose coefficient of elasticity is $k$ is suspended by one end from the point $O$ (Fig. 1.33). The other end is fitted with a catch $B$. A small sleeve $A$ of mass $m$ starts falling from the point $O$. Neglecting the masses of the thread and the catch, find the maximum elongation of the cord.
1.140. A small bar $A$ resting on a smooth horizontal plane is attached by threads to a point $P$ (Fig. 1.34) and, by means of a weightless pulley, to a weight $B$ possessing the same mass as the bar itself.
Fie. 1.34.
Fig. 1.35.

Besides, the bar is also attached to a point $O$ by means of a light noodeformed spring of length $l_{0}=50 \mathrm{~cm}$ and stifness $x-5 \mathrm{mg} / l_{t}$. where $m$ is the maes of the bar. The thread $P A$ having been burned, the bar starts moving. Find its velocity at the moment when it is breaking off the plane.
1.141. A horizontal plane supports a plank with a bar of mass $m=1.0 \mathrm{~kg}$ placed on it and attached by a light elastic non-deformed cord of length $I_{0}=40 \mathrm{~cm}$ to a point $O$ (Fig 1.35). The coefficient of friction between the bar and the plank equals $k=0.20$. The plank is slowly shifted to the right until the bar starts sliding over it. It occurs at the moment when the cord deviates from the vertical by an angle $\theta=30^{\circ}$. Find the work that has been performed by that moment by the friction force acting on the bar in the reference frame fixed to the plane.
1.142. A smooth light horizontal rod $A B$ can rotate about a vertical axis passing through its end $\boldsymbol{A}$. The rod is fitted with a small sleeve of mass $m$ attached to the end $\boldsymbol{A}$ by a weizhtless spring of length $l_{0}$ and stiffness $x$. What work must be performed to slowly get this system going and reaching the angular velocity $\omega$ ?
1.143. A pulley fixed to the ceiling carries a thread with bodies of masses $m_{1}$ and $m_{2}$ attached to its ends. The masses of the pulley and the thread are aegligible, friction is absent. Find the acceleration $w_{c}$ of the centre of inertia of this system.
1.144. Two interacting particles form a closed system whose centre of inertia is at rest. Fig. 1.36 illustrates the positions of both particles at a certain moment and the trajectory of the particle of mass $m_{1}$. Draw the trajectory of the particle of mass $m_{2}$ if $m_{3}=m_{1} / 2$.
1.145. A closed chais $A$ of mass $m=0.36 \mathrm{~kg}$ is attached to a vertical rotating shaft by means of a thread (Fig. 1.37), and rotates with a constant angular velocity $\omega=35 \mathrm{rad} / \mathrm{s}$. The thread forms an angle $\theta=45^{\circ}$ with the vertical. Find the distance between the chain’s centre of gravity and the rotation axis, and the tension of the thread.
1.146. A round cone $A$ of mass $m=3.2 \mathrm{~kg}$ and halfangle $a-10^{\circ}$ rolls uniformly and without slipping along a round conical surface $B$ so that its apex $\delta$ remains stationary (Fig. 1.38). The centre of gravity of the cone $\boldsymbol{A}$ is at the same level as the point $\boldsymbol{O}$ and at a distance $l=17 \mathrm{~cm}$ from it. The cone’s axis moves with angular velocity 4 . Find:
(a) the static friction force acting on the cone $A$, if $\mathrm{em}-1.0 \mathrm{rad} / \mathrm{s}$ :
Fie. 1.36.
(b) at what values of $\omega$ the cone $\boldsymbol{A}$ will roll without sliding. if the coefficient of friction betwees the surfaces is equal to $k=0.25$.
1.147. In the reference frame $K$ two particles travel along the $x$ axis, one of mass $m_{1}$ with velocity $v_{1}$, and the other of mass $m_{2}$ with velocity $\mathrm{v}_{\mathbf{z}}$. Find:
(a) the velocity $\mathbf{V}$ of the reference frame $\boldsymbol{K}^{\prime}$ in which the cumulative kinetie energy of these particles is minimum;
(b) the cumulative kinetic enerry of these particles in the $\boldsymbol{K}^{\prime}$ frame.
1.148. The reference frame, in which the centre of inertia of a given system of particles is at rest, translates with a velocity $\mathbf{V}$ relative
ค.g. 1.37.
ทุ. 1.38.
to an inertial reference frame $K$. The mass of the system of particles equals $m$, and the total energy of the system in the frame of the ceatre of inertis is equal to $\mathcal{E}$. Find the total energy $E$ of this system of particles in the reference frame $K$.
1.149. Two small discs of masses $m_{1}$ and $m_{2}$ interconnected by a weightless spring rest on a smeoth herizontal plane. The discs are set in motion with initial velocities $y_{2}$ and $v_{2}$ whese directions are a*

mutually perpendicular and lie in a horizontal plane. Find the total energy $\mathcal{E}$ of this system in the frame of the centre of inertia.
1.150. A system consists of two small spheres of masses $m_{1}$ and $m_{\text {, }}$ interconnected by a weightless spring. At the moment $t=0$ the spheres are aet in motion with the initial velocities $\mathbf{v}_{1}$ and $\mathbf{v}_{\text {, after }}$ which the system starts moving in the Earth’s uniform gravitational field. Neglecting the air drag, find the time dependence of the total momentum of this system in the process of motion and of the radius vector of its centre of inertia relative to the initial position of the centre.
1.151. Two bars of mases $m_{1}$ and $m_{3}$ connected by a weightless spring of stiffness $\%$ (Fig. 1.39) rest on a smooth horizontal plane.
Fie. 1.39.
Fis. 1.40.

Bar $\mathbf{2}$ is shifted a small distance $x$ to the left and then released. Find the velocity of the centre of inertis of the system after bar $I$ breaks of the wall.
1.152. Two bars connected by a weightless spring of stiffness $x$ and length (in the non-deformed state) $\zeta_{0}$ rest on a horizontal plane. A constant horizental force $F$ starts acting on one of the bars as shewn in Fig. 1.40. Find the maximum and minimum distances between the bars during the subsequent motion of the system, if the masses of the bars are:
(a) equal;
(b) equal to $m_{1}$ and $m_{3}$, and the force $F$ is applied to the bar of mass $m_{2}$.
1.153. A system consists of two identical cubes, each of mass $m$, linked together by the compressed weightless spring of stiffness $x$ (Fig. 1.41). The cubes are also consected by a thread which is burned through at a certain moment. Find:
(a) at what values of $\Delta l$, the initial compression of the spring, the lower cube will bounce up after the thread has been burned through:
(b) to what height $h$ the centre of gravity of this system will rise if the initial compressien of the spring $\Delta t=7 \mathrm{mg} / \mathrm{x}$.
1.154. Two identical buggies $I$ and 2 with ene man in each move without friction due to inertia along the parallel rails toward each other. When the buggies get opposite each other, the men exchange their places by jumping in the direction perpendicular to the motion direction. As a consequence, bugzy
$I$ stops and bugry 2 keeps moving in the same direction, with its velocity becoming equal to $v$. Find the initial velocities of the buggies $\mathrm{v}_{1}$ and $\mathrm{v}_{1}$ if the mass of each buggy (without a man) equals $M$ and the mass of each man $m$.
1.155. Two identical buggies move one after the other due to inertia (without friction) with the same velocity $\mathbf{v}_{0}$. A man of mass $m$ rides the rear bugry. At a certain moment the man jumps into the front bugry with a velocity u relative to his buggy. Knowing that the mass of each bugzy is equal to $M$, find the velocities with which the bugzies will move after that.
1.156. Two men, each of mass $m$, stand on the edge of a stationary bugry of mass $M$. Assuming the friction to be neglizible, find the velocity of the bugzy after both men jump off with the same horizontal velocity a relative to the bugzy: (i) simultaneously; (2) one after the other. In what case will the velocity of the buzgy be greater and how many times?
1.157. A chain hangs on a thread and touches the surface of a table by its lower end. Show that after the thread has been burned through, the force exerted on the table by the falling part of the chain at any moment is twice as great as the force of pressure exerted by the part already resting on the table.
1.158. A steel ball of mass $m=50 \mathrm{~g}$ falls from the height $h-$ $-1.0 \mathrm{~m}$ on the horizental surface of a massive slab. Find the cumulative momentum that the ball imparts to the slab after numerous bounces, if every impact decreases the velocity of the ball $\eta=1.25$ times.
1.159. A raft of mass $M$ with a man of mass $m$ aboard stays motionless on the surface of a lake. The man moves a distance $\mathrm{I}^{\prime}$ relative to the raft with velocity $v^{\prime}(t)$ and then stops. Assuming the water resistance to be negligible, find:
(a) the displacement of the raft 1 relative to the shore;
(b) the herizontal component of the force with which the man acted on the raft during the motion.
1.160. A stationary pulley carries a rope whose one end supports a ladder with a man and the other end the counterweight of mass $M$. The man of mass $m$ elimbs up a distance $r^{\prime}$ with respect to the ladder and then stops. Neglecting the mass of the rope and the friction in the pulley axle, find the displacement I of the centre of inertia of this system.
1.161. A cannon of mass $M$ starts sliding freely down a smooth inclined plane at as angle $a$ to the horizontal. After the cannon covered the distance $l$, a shot was fired, the shell leaving the cannon in the horizontal direction with a momentum p. As a consequence, the cannon stopped. Assuming the mass of the shell to be aegligible, as compared to that of the cannon, determine the duration of the shot.
1.162. A horinontally flying bullet of mass $m$ gets stuck in a body of mass $M$ suspended by two identical threads of length $l$ (Fig. 1.42).
As a result, the threads swerve through an angle $\theta$. Assuming $m \subset M$, find:
(a) the velocity of the bullet before striking the body;
(b) the fraction of the bullet’s initial kinetic energy that turned into heat.
1.163. A body of mass $M$ (Fig. 1.43) with a small dise of mass $m$ placed on it rests on a smooth horizontal plane. The dise is set in
Fie. 1.42.
Fig. 1.43.
motion in the horizontal direction with velocity $v$. To what height (relative to the initial level) will the disc rise after breaking of the body M? The friction is assumed to be absent.
1.164. A small dise of mass $m$ slides down a smooth hill of height $h$ without initial velocity and gets onto a plank of mass $M$ lying on
Fie. 1.44.
the horizontal plane at the base of the hill (Fig. 1.44). Due to friction between the dise and the plank the dise slows down and, beginning with a certais moment, moves in ene piece with the plank.
(1) Find the total work performed by the friction forces in this process.
(2) Can it be stated that the result obtained does not depend on the chetee of the reference frame?
1.165. A stone falls down without initial velocity from a height $h$ onto the Earth’s surface. The air drag assumed to be negligible, the stone hits the ground with velocity $v_{0}=\sqrt{2 g h}$ relative to the Earth. Obtain the same formula in terms of the reference frame \”Falling\” to the Earth with a constant velocity $v_{v}$.
1.166. A particle of mass $1.0 \mathrm{~g}$ moving with velocity $\mathrm{v}_{1}=3.04-$ $-2.0 \mathrm{j}$ experiences a perfectly inelastic collision with another particle of mass $2.0 \mathrm{~g}$ and velocity $\mathrm{v}_{\mathrm{a}}=4.0 \mathrm{j}-6.0 \mathrm{k}$. Find the velocity of the formed particle (both the vector $\mathbf{v}$ and its modulus), if the components of the vectors $\mathbf{v}_{1}$ and $\mathrm{v}_{2}$ are given in the 81 units.
1.167. Find the increment of the kinetic energy of the closed system comprising two spheres of masses $m_{1}$ and $m_{2}$ due to their perfectly inelastic collision, if the initial velocities of the spheres were equal to $\mathbf{v}_{1}$ and $\mathbf{v}_{8}$ :
1.168. A particle of mass $m_{1}$ experienced a perfectly elastic collision with a stationary particle of mass $m$, What fraction of the kinetic energy does the striking particle lose, if
(a) it recoils at right angles to its original motion direction;
(b) the collision is a head-on one?
1.169. Particle $t$ experiences a perfectly elastic collision with a stationary particle 2. Determine their mass ratio, if
(a) after a head-on collision the particles fly apart in the opposite directions with equal velocities;
(b) the particles fly apart symmetrically relative to the initial motion direction of particle $I$ with the angle of divergence $\theta=60^{\circ}$.
1.170. A ball moving translationally collides elastically with another, stationary, ball of the same mass. At the moment of impact the angle between the straight line passing through the centres of the balls and the direction of the initial motion of the striking ball is equal to $a=45^{\circ}$. Assuming the balls to be smooth, find the fraction $\eta$ of the kinetic energy of the striking ball that turned inte potential energy at the moment of the maximum deformation.
1.171. A shell flying with velocity $v=500 \mathrm{~m} / \mathrm{s}$ bursts into three identical fragments so that the kinetic energy of the system increases $\eta=1.5$ times. What maximum velocity can one of the fragments obtain?
1.172. Particle $I$ meving with velocity $v=10 \mathrm{~m} / \mathrm{s}$ experienced a head-on collision with a stationary particle 2 of the same mass. As a result of the collision, the kinetic energy of the system decreased by $\eta=1.0 \%$. Find the magnitude and direction of the velocity of particle $I$ after the collision.
1.173. A particle of mass $m$ having collided with a stationary particle of mass $M$ deviated by an angle $a / 2$ whereas the particle $M$ recoiled at an angle $\theta=30^{\circ}$ to the direction of the initial motion of the particle $m$. How much (is per cent) and in what way has the kinetic energy of this system changed after the collision, if $M / m=$ $=5.0$ ?
1.174. A closed system consists of two particles of masses $m_{1}$ and $m_{2}$ which move at right angles to each other with velocities $n_{2}$ and $v_{\mathrm{r}}$. Find:
(a) the momentum of each particle and
(b) the total kinetic energy of the two particles in the reference frame fixed to their centre of inertia.
1.175. A particle of mass $m_{1}$ collides elastically with a stationary particle of mass $m_{1}\left(m_{1}>m_{1}\right)$. Find the maximum angle through which the striking particle may deviate as a result of the collision.
1.176. Three identical discs $A, B$, and $C$ (Fig. 1.45) rest on a smooth horizontal plane. The dise $A$ is set in motion with velocity $v$ after which it experiences an elastic collision simultaneously with the discs $B$ and $C$. The distance between the centres of the latter discs prior to the collision is $\eta$ times greater than the diameter of each disc. Find the velocity of the dise $A$ after the collision. At what value of $\eta$ will the dise A recoil after the collision; stop; move on?
1.177. A molecule collides with another, stationary, molecule of the same mass. Demonstrate that the angle of diverrence
(a) equals $90^{\circ}$ when the collision is ideally elastic:
(b) differs from $90^{\circ}$ when the collision
Fis. 1.45. is inelastic.
1.178. A rocket ejects a steady jet whose velocity is equal to u relative to the rocket. The gas discharge rate equals $\mu \mathrm{kg} / \mathrm{s}$. Demonstrate that the rocket motion equation in this case takes the form
\[
m \mathrm{w}=\mathbf{F}-\mu \mathrm{u},
\]
where $m$ is the mass of the rocket at a given moment, $=$ is its acceleration, and $\boldsymbol{F}$ is the external force.
1.179. A recket moves in the absence of external forces by ejecting a steady jet with velocity o constant relative to the rocket. Find the velocity $\mathbf{v}$ of the rocket at the moment when its mass is equal to $m$, if at the initial moment it possessed the mass $m$, and its velocity was equal to zero. Make use of the formula given in the foregoing problem.
1.180. Find the law according to which the mass of the rocket varies with time, when the rocket moves with a constant acceleration $w$, the external forces are absent, the gas escapes with a constant velocity $u$ relative to the rocket, and its mass at the initial moment equals $m_{\text {r }}$.
1.181. A spaceship of mass $m_{4}$ moves in the absence of external forces with a constant velocity $\mathbf{v}_{0}$. To change the motion direction, a jet engine is switched on. It starts ejecting a gas jet with velocity u which is constant relative to the spaceship and directed at right angles to the spaceship motion. The engine is shut down when the mass of the spaceship decreases to $m$. Through what angle $a$ did the motion direction of the spaceship deviate due to the jet engine operation?
1.182. A cart loaded with sand moves along a herinontal plane due to a constant force $\boldsymbol{F}$ coinciding in direction with the cart’s velocity vecter. In the process, sand spills through a hele in the bottom with a constant velocity $\mu \mathrm{kg} / \mathrm{s}$. Find the acceleration and the velocity of the cart at the moment $t$, if at the initial moment $t=0$ the cart with loaded sand had the mass $m_{0}$ and its velocity was equal to tero. The friction is to be neglected.
1.183. A datcar of mass $m_{0}$ starts moving to the right due to a constant horizontal force F (Fig.’ 1.46). Sasd spills on the flatear
from a stationary hopper. The velocity of loading is constant and equal to $\mu \mathrm{kg} / \mathrm{s}$. Find the time dependence of the velocity and the acceleration of the flatcar in the process of loading. The friction is aezligibly small.
1.184. A chain $A B$ of length $l$ is located in a smooth horizontal tube so that its fraction of length $h$ hangs freely and touches the surface of the table with its end $B$ (Fig. 1.47). At a certain moment
Fis. 1.48.
the ead $A$ of the chain is set free. With what velocity will this end of the chain slip out of the tube?
1.185. The angular momentum of a particle relative to a certain point $O$ varies with time as $\mathbf{M}=\mathbf{a}+\mathrm{bc}^{4}$, where $\mathrm{a}$ and $\mathrm{b}$ are constant vectors, with $a \perp$ b. Find the force moment $\mathbf{N}$ relative to the point $O$ acting on the particle when the angle between the vectors $\mathbf{N}$ and $\mathbf{M}$ equals $45^{\circ}$.
1.186. A ball of mass $m$ is thrown at an angle $a$ to the horizontal with the initial velocity $v_{0}$. Find the time dependence of the magsitude of the ball’s angular momentum vecter relative to the point from which the ball is thrown. Find the angular momentum $M$ at the highest point of the trajectory if $m=130 \mathrm{~g}, \alpha=45^{\circ}$, and $v_{4}=$ $=25 \mathrm{~m} / \mathrm{s}$. The air drag is to be aeglected.
1.187. A dise $\boldsymbol{A}$ of mass $m$ sliding over a smooth horizontal surface with velocity $v$ experiences a perfectly elastic collision with a smooth stationary wall at a point $\delta$ (Fig. 1.48). The angle between the motion direction of the dise and the normal of the wall is equal to a. Find:
(a) the points relative to which the angular momentum $\mathbf{M}$ of the dise remains constant in this process;
(b) the magnitude of the increment of the vector of the disc’s angular momentum relative to the point $O$ ‘ which is located in the plane of the disc’s motion at the distance I from the point $O$.
1.188. A small ball of mass $m$ suspended from the ceiling at a point $O$ by a thread
Fig. 1.48. of length $l$ moves along a horizontal circle with a constant angular velocity 6 . Relative to which points does the angular momentum M of the ball remais constant? Find the magaitude of the increment of the vector of the ball’s angular momentum relative to the point $O$ picked up during half a revolution.
1.189. A ball of mass $m$ falls down without initiat velocity from a height $\boldsymbol{h}$ over the Earth’s surface. Find the increment of the ball’s angular momentum vector picked up during the time of falling (relative to the point $O$ of the relerence frame moving translationally in a horizontal direction with a velocity $\boldsymbol{V}$. The ball starts falling from the point $O$. The air drag is to be neglected.
1.190. A smooth horizontal dise rotates with a constant angular velocity $\omega$ about a stationary vertical axis passing through its centre, the point $O$. At a moment $t=0$ a dise is set is motion from that
Fis. 1.49.
Fig. 1.50.
point with velocity $v_{0}$. Find the angalar momentum $M(t)$ of the dise relative to the point $O$ in the relerence frame fixed to the disc. Make sure that this angular momentum is caused by the Coriolis force.
1.191. A particle moves along a closed trajectory in a central field of force where the particle’s potential energy $U=k r^{2}(k$ is a positive constant, $r$ is the distance of the particle from the centre $O$ of the field). Find the mase of the particle if its minimum distance from the point $O$ equals $r_{1}$ and its velocity at the point farthest from $O$ equals $v_{2}$ :
1.192. A small ball is suspended from a point $O$ by a light thread of length $l$. Then the ball is drawn aside so that the thread deviates through an angle $\theta$ from the vertical and set in motion in a horizontal direction at right angles to the vertical plase in which the thread is located. What is the initial velocity that has to be imparted to the ball so that it could deviate through the maximum angle $\pi / 2$ in the process of motion?
1.193. A small body of mass $m$ tied to a non-stretchable thread moves over a smooth horizontal plase. The other end of the thread is being drawn into a hole $O$ (Fig. 1.49) with a constant velocity. Find the thread tension as a function of the distance $r$ between the body and the hole if at $r=r_{0}$ the angular velocity of the thread is equal to $\omega_{0}$ :
1.194. A light non-stretchable thread is wound on a massive fixed pulley of radius $\boldsymbol{R}$. A small body of mass $m$ is tied to the free end of the thread. At a moment $t=0$ the system is released and starts moving. Find its angular momentum relative to the pulley axle as a function of time $t$.
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1.195. A uniform sphere of mass $m$ and radius $\boldsymbol{H}$ starts rolling without slipping down an inclined plane at an angle $\alpha$ to the horizontal. Find the time dependence of the angular momentum of the sphere relative to the point of contact at the initial moment. How will the obtained result change in the case of a perfectly smooth inclined plane?
1.196. A certain system of particles possesses a total momentum $p$ and an angular momentum $M$ relative to a point $O$. Find its angular momentum $\mathbf{M}^{\prime}$ relative to a point $O^{\prime}$ whose position with respect to the point $O$ is determined by the radius vector $r_{6}$. Find out when the angular momentum of the system of particles does not depend on the choice of the point $O$.
1.197. Demonstrate that the angular momentum $M$ of the system of particles relative to a point $O$ of the relerence frame $K$ can be represented as
\[
\mathbf{M}=\mathbf{M}+\left|\mathbf{r}_{\mathrm{C}} \mathbf{p}\right| \text {. }
\]
where $\mathbf{M}$ is its proper angular momentum (in the reference frame moving translationally and fixed to the centre of inertia), $r_{C}$ is the radius vector of the centre of inertia relative to the point $O_{;} P$ is the total momentum of the system of particles in the relerence frame $\boldsymbol{K}$.
1.198. A ball of mass $m$ moving with velocity $v_{0}$ experiences a head-on elastic collision with one of the spheres of a stationary rigid dumbbell as whown in Fig. 1.50. The mass of each sphere equals $m / 2$, and the distance between them is $L$. Disregarding the size of the spheres, find the proper angular momentum $\vec{M}$ of the dumbell after the collision, i.e. the angular momentum in the reference frame moving translationally and fixed to the dumbbell’s centre of inertia.
1.199. Two small identical discs, each of mass $m$, lie on a smooth borizontal plane. The dises are interconnected by a light non-deformed spring of length $L_{0}$ and stiffness $x$. At a certain moment one of the dises is set in motion in a horizontal direction perpendieular to the spring with velocity $v_{0}$. Find the maximum elongation of the spring in the process of motion, if it is known to be considerably less than uaity.