\[
=\frac{d v}{d i}=\boldsymbol{V} \text {. }
\]
– The same equation expreased is projections on the tangent and the nermal of the point’s injectory:
\[
=\frac{d r_{1}}{d t}=F_{\mathrm{s}} \quad=\frac{y^{\prime}}{R}=F_{\mathrm{n}} .
\]
$\Sigma^{\prime}$. The equatios of dyamics of a point is the non-isertial relerence frame $K^{\prime}$, which roties with a coastant angular velocity o about as axis translating with an acceleration $w_{i}$ :
where in is the radius vecter of the point relative to the axis of rotation of the $\boldsymbol{R}^{\prime}$ frame.
1.59. An serostat of mass $m$ starts coming down with a constant acceleration $w$. Determine the ballast mass to be dumped for the aerastat to reach the upward acceleration of the same magnitude. The air drag is to he neglected.
1.60. In the arrangement of Fig. 1.9 the masses $m_{6}, m_{1}$, and $m_{2}$ of bodies are equal, the masses of the pulley and the threads are negligible, and there is no friction in the pulley. Find the accel. oration $w$ with which the body $m_{0}$ comes down, and the tension of the thread binding together the bodies $m_{1}$ and $m_{1}$, if the coefficient of friction between these bodies and the horizontal surface is equal to $\boldsymbol{k}$. Consider possible cases.
1.61. Two teuching bars $I$ and 2 are placed on an inclined plane forming an angle $a$ with the horizontal (Fig. 1.10). The masses of the bars are equal to $m_{1}$ and $m_{4}$, and the coefficients of friction be-
Fig. 1.9.
Fie. 1.10.
tween the inclined plane and these bars are equal to $k_{1}$ and $k_{2}$ rospectively, with $k_{1}>k_{1}$. Find:
(a) the force of interaction of the bars in the process of motion:
(b) the minimum value of the angle $a$ at which the bars start sliding down.
1.62. A small body was launched up an inclined plane set at an angle $a=15^{\circ}$ against the herizontal. Find the coefficient of friction, if the time of the ascent of the body is $\eta=2.0$ times leas than the time of its descent.
1.63. The following parameters of the arrangement of Fig. 1.11 are available: the angle $a$ which the inclined plase forms with the horizental, and the coefficient of friction $k$ between the bedy $\mathbf{m}$, and the inclined plane. The masses of the pulley and the threads, as well as the friction in the pulley, are negligible. Assuming both bodies to be motionless at the initial moment, find the mass ratio $m_{2} / m_{1}$ at which the body $m_{1}$
(a) starts coming down;
(b) starts going up:
(c) is at rest.
1.64. The inclined plane of Fig. 1.11 forms an angle $a=30^{\circ}$ with the herizental. The mass ratio $m_{2} / m_{1}=\eta=2 / 3$. The coefficient of friction between the body $m_{1}$ and the inclined plane is equal to $k=$ $=0.10$. The masses of the pulley and the threads are negligible. Find the magnitude and the direction of acceleration of the body $m_{4}$ when the formerly stationary system of masses starts moving.
1.65. A plank of mass $m_{1}$ with a bar of mass $m_{\text {, }}$, placed on it lies on a smooth horizontal plase. A horizontal force growing with time $t$ as $F=$ at (a is constant) is applied to the bar. Find how the accelerations of the plank $w_{1}$ and of the bar $w$, depend on $t$, if the coefficient of friction between the plank and the bar is equal to $k$. Draw the approximate plots of these dependences.
1.66. A small body $\boldsymbol{A}$ starts sliding down from the tep of a wedge (Fig. 1.12) whose base is equal to $l=2.10 \mathrm{~m}$. The coefficient of friction between the body and the wedse surface is $k=0.140$. At what value of the angle $\alpha$ will the time of sliding be the least? What will it be equal to?
1.67. A bar of mass $m$ is pulled by means of a thread up an inclined plane forming an angle $\alpha$ with the horizontal (Fig. 1.13). The coef-
Fig. 1.11.
Fig. 1.12.
ficient of friction is equal to $k$. Find the angle $\beta$ which the thread must form with the inclined plane for the tension of the thread to be minimum. What is it equal to?
1.68. At the moment $t=0$ the force $F=a t$ is applied to a small body of mass $m$ resting on a smooth horizontal plane ( $a$ is a constant).
Fig. 1.13.
Fig. 1.14.

The permanent direction of this force forms an angle $\alpha$ with the horizontal (Fig. 1.14). Find:
(a) the velocity of the body at the moment of its breaking off the plane;
(b) the distance traversed by the body up to this moment.
1.69. A bar of mass $m$ resting on a smooth horizontal plane starts moving due to the force $F=\mathrm{mg} / 3$ of constant magnitude. In the process of its rectilinear motion the angle $\alpha$ between the direction of this force and the horizontal varies as $\alpha=a s$, where $a$ is a constant, and $s$ is the distance traversed by the bar from its initial position. Find the velocity of the bar as a function of the angle $\alpha$.
1.70. A horizontal plane with the coefficient of friction $k$ supports two bodies: a bar and an electric motor with a battery on a block. A thread attached to the bar is wound on the shaft of the electric motor. The distance between the bar and the electric motor is equal to $l$. When the motor is switched on, the bar, whose mass is twice
as great as that of the other body, starts moving with a constant acceleration $w$. How soon will the bodies collide?
1.71. A pulley fixed to the ceiling of an elevator car carries a thread whose ends are attached to the loads of masses $m_{1}$ and $m_{2}$. The car starts going up with an acceleration $\mathbf{w}_{0}$. Assuming the masses of the pulley and the thread, as well as the friction, to be negligible find:
(a) the acceleration of the load $m_{1}$ relative to the elevator shaft and relative to the car;
(b) the force exerted by the pulley on the ceiling of the car.
1.72. Find the acceleration $w$ of body 2 in the arrangement shown in Fig. 1.15, if its mass is $\eta$ times as great as the mass of bar 1 and
Fig. 1.15.
Fig. 1.16.
the angle that the inclined plane forms with the horizontal is equal to $\alpha$. The masses of the pulleys and the threads, as well as the friction, are assumed to be negligible. Look into possible cases.
1.73. In the arrangement shown in Fig. 1.16 the bodies have masses $m_{0}, m_{1}, m_{2}$, the friction is absent, the masses of the pulleys and the threads are negligible. Find the acceleration of the body $m_{1}$. Look into possible cases.
1.74. In the arrangement shown in Fig. 1.17 the mass of the rod $M$ exceeds the mass $m$ of the ball. The ball has an opening permitting
Fig. 1.17.
Fig. 1.18.
Fig. 1.19.
it to slide along the thread with some friction. The mass of the pulley and the friction in its axle are negligible. At the initial moment the ball was located opposite the lower end of the rod. When set free, both bodies began moving with constant accelerations. Find the friction force between the ball and the thread if $t$ seconds after the beginning of motion the ball got epposite the upper end of the rod. The rod length equals $L$.
1.75. In the arrangement shown in Fig. 1.18 the mass of ball $I$ is $\eta=1.8$ times as great as that of rod 2. The length of the latter is $i=100 \mathrm{~cm}$. The masses of the pulleys and the threads, as well as the friction, are negligible. The ball is set on the same level as the lower end of the rod and then released. How soon will the ball be opposite the upper end of the rod?
1.76. In the arrangement shown in Fig. 1.19 the mass of body $I$ is $\eta=4.0$ times as great as that of body 2 . The height $h=20 \mathrm{~cm}$. The masses of the pulleys and the threads, as well as the friction, are negligible. At a certain moment body 2 is released and the arrangement set in motion. What is the maximum height that body 2 will go up to?
1.77. Find the accelerations of rod $A$ and wedge $B$ in the arrangement shown in Fig. 1.20 if the ratio of the mass of the wedge to that of the rod equals $\eta$. and the friction between all contact surfaces is negligible.
1.78. In the arrangement shown in Fig. 1.21 the masses of the wedge $M$ and the body $m$ are known. The appreciable friction exists
Vie. 1.20 ,
Fis. 1.21.
only between the wedge and the body $m$, the friction coefficient being equal to $k$. The masses of the pulley and the thread are negligible. Find the acceleration of the body $m$ relative to the horizontal surface on which the wedge slides.
1.79. What is the minimum acceleration with which bar A (Fig. 1.22) should be shifted horizontally to keep bodies $I$ and 2 stationary relative to the bar? The masses of the bodies are equal, and the coefficient of friction between the bar and the bodies is equal to $k$. The masses of the pulley and the threads are negligible, the friction in the pulley is absent.
1.80. Prism $I$ with bar 2 of mass $m$ placed on it gets a borizontal acceleration $w$ directed to the left (Fig. 1.23). At what maximum value of this acceleration will the bar be still stationary relative to the prism, if the coefficient of friction between them $k<\cot$ a?
24
1.81. Prism $I$ of mass $m_{1}$ and with angle $a$ (see Fig. 1.23) rests on a horizontal surface. Bar 2 of mass $m_{2}$ is placed on the prism. Assuming the friction to be aegligible, find the acceleration of the prism.
1.82. In the arrangement shown in Fig. 1.24 the masses $m$ of the bar and $M$ of the wedge, as well as the wedge angle $\alpha$, are known.
ก.s. 1.22.
Fie. 1.23.

The masses of the pulley and the thread are aegligible. The friction is absent. Find the acceleration of the wedge $M$.
1.83. A particle of mass $m$ moves along a circle of radius $\boldsymbol{R}$. Find the modulus of the average vecter of the force acting on the particle over the distance equal to a quarter of the circle, if the particle moves
(a) uaiformly with velecity is,
(b) with censtant tangential acceleration un, the initial velocity being equal to zero.
1.84. An aircraft loops the loop of radius $\boldsymbol{R}=500 \mathrm{~m}$ with a constant velocity $v=$ – $360 \mathrm{~km}$ per hour. Find the weight of the fyer of mass $m=70 \mathrm{~kg}$ in the lower, upper,

Fie. 1.24. and middle points of the loop.
1.85. A small sphere of mass $m$ suspended by a thread is first taken aside so that the thread forms the right angle with the vertical and then released. Find:
(a) the total acceleration of the sphere and the thread tension as a function of $\theta$, the angle of deflection of the thread from the vertical;
(b) the thread tension at the moment when the vertical component of the sphere’s velocity is maximum;
(c) the angle $\theta$ between the thread and the vertical at the moment when the total acceleration vector of the sphere is directed horizontally.
1.86. A ball suspended by a thread swings in a vertical plane so that its acceleration values in the extreme and the lowest position are equal. Find the thread deflection angle in the extreme position.
1.87. A small bedy $A$ starts sliding off the top of a smooth sphere of radius $\boldsymbol{A}$. Find the angle $\theta$ (Fig. 1.25) corresponding to the point at which the body breaks off the sphere, as well as the break-of velocity of the body.
1.88. A device (Fig. 1.26) consists of a smooth L-shaped rod locatod in a horizontal plane and a sleeve $A$ of mass $m$ attached by a weight-
less spring to a point $B$. The spring stiffness is equal to $x$. The whole system rotates with a constant angular velocity $\omega$ about a vertical axis passing through the point $O$. Find the elongation of the spring. How is the result affected by the rotation direction?
1.89. A cyclist rides along the circumferesce of a circular horizontal plane of radius $\boldsymbol{R}$, the friction coefficient being dependent only on
Fie. 1.25.
Fie. 1.26.
distance $r$ from the ceatre $O$ of the plane as $k=k_{0}(1-r / R)$, where $k_{0}$ is a constant. Find the radius of the circle with the centre at the point aleng which the eyelist can ride with the maximum velocity. What is this velocity?
1.90. A car moves with a constant tangential acceleration we, $=$ $-0.62 \mathrm{~m} / \mathrm{s}^{2}$ along a herinontal surface circumscribing a circle of radius $\boldsymbol{R}=40 \mathrm{~m}$. The coefficient of sliding friction between the wheels of the car and the surface is $k=0.20$. What distance will the car ride without sliding if at the initial mement of time its velocity is equal to zero?
1.91. A car moves uniformly aleng a horizontal sine curve $y=$ $=a \sin (x / a)$, where $a$ and $a$ are certain constants. The coefficient of friction between the wheels and the road is equal to $k$. At what velocity will the car ride without sliding?
1.92. A chain of mass $m$ forming a circle of radius $R$ is slipped on a smeoth round cone with half-angle 9 . Find the tension of the chain if it rotates with a constant angular velocity a about a vertical axis coinciding with the symmetry axis of the cone.
1.93. A fixed pulley carries a weightless thread with masses $m_{1}$ and $m_{2}$ at its ends. There is friction between the thread and the pulley. It is such that the thread starts slipping when the ratio $m_{2} / m_{1}=$ -n. Find:
(a) the friction coefficient:
(b) the acceleration of the masses when $m_{v} / m_{1}=\eta>\eta_{c}$.
1.94. A particle of mass $m$ moves along the internal smooth surface of a vertical cylinder of radius $\boldsymbol{R}$. Find the force with which the particle acts on the cylinder wall if at the initial moment of time its velocity equals $v_{0}$ and forms an angle $a$ with the horizontal.
1.95. Find the magaitude and direction of the force acting on the particle of mass $m$ during its motion in the plane $x y$ according to the faw $z=a$ sin ot, $y=b$ cos ot, where $a$, b, and $a$ are constants.
1.96. A body of mass $m$ is thrown at an angle to the horizontal with the initial velocity $\mathbf{v}_{8}$. Assuming the air drag to be negligible, find:
(a) the momentum increment Ap that the body acquires over the first $t$ seconds of motion;
(b) the modulus of the momentum increment Ap during the total time of motion.
1.97. At the moment $t=0$ a stationary particle of mass $m$ experiences a time-dependent force $\mathrm{F}=\mathrm{at}(\mathrm{r}-t)$, where $\mathrm{a}$ is a constant vector, $\mathrm{t}$ is the time during which the given force acts. Find:
(a) the momentum of the particle when the action of the force discontinued:
(b) the distance covered by the particle while the force acted.
1.98. At the moment $t=0$ a particle of mass $m$ starts moving due to a force $\mathbf{F}=\mathbf{F}_{6}$ sin et, where $\mathbf{F}_{\text {, and }}$, are constants. Find the distance covered by the particle as a function of $t$. Draw the approximate plot of this function.
1.99. At the momeat $t=0$ a particle of mass $m$ starts moving due to a force $\boldsymbol{F}=\boldsymbol{F}_{0} \cos \boldsymbol{\epsilon}$, where $\boldsymbol{F}$, and $\oplus$ are constants. How long will it be moving until it stops for the first time? What distance will it traverse during that time? What is the maximum velocity of the particle over this distance?
1.100. A moterboat of mass $m$ moves along a lake with velocity $v_{0}$. At the moment $t=0$ the engine of the boat is shut down. Assuming the resistance of water to be proportional to the velocity of the boat $\boldsymbol{F}=-r \mathbf{v}$, find:
(a) how long the motorboat moved with the shutdown engine;
(b) the velocity of the motorboat as a function of the distance covered with the shutdown engine, as well as the total distance covered till the complete stop:
(c) the mean velocity of the motorboat over the time interval (beginaing with the mement $t=0$ ), during which its velocity decreases on times.
1.101. Having gone through a plank of thickness $h$, a bullet changed its velocity from $v_{0}$ to $v$. Find the time of motion of the bullet in the plank, assuming the resistance force to be proportional to the square of the velocity.
1.102. A small bar starts sliding down an inclined plane forming an angle $a$ with the horizontal. The friction coefficient depends on the distance $x$ covered as $k=a x$, where $a$ is a constant. Find the distance covered by the bar till it stops, and its maximum velocity over this distance.
1.103. A body of mass $m$ rests on a herizontal plane with the friction coefficient $k$. At the moment $t=0$ a horizontal force is applied to it, which varies with time as $\mathbf{F}=$ at, where a is a constant vector.
Find the distance traversed by the bedy during the first $t$ seconds after the force action began.
1.104. A body of mass $m$ is thrown straight ep with velocity $v_{0}$. Find the velocity of with which the bedy comes down if the air draz equals $k v^{2}$, where $k$ is a constant and $v$ is the velocity of the body.
1.105. A particle of mass $m$ moves in a certain plane $P$ due to a force $F$ whose marnitude is constant and whose vector rotates in that plane with a constant angular velocity 6 . Assuming the particle to be stationary at the moment $t=0$, find:
(a) its velocity as a function of time;
(b) the distance covered by the particle between two successive stops. and the mean velocity over this time.
1.106. A small disc $A$ is placed on an inclined plane forming an angle $a$ with the horizontal (Fig. 1.27) and is imparted an initial velocity $v_{0}$. Find how the velocity of the disc depends on the angle $\varphi$ if the friction coefficient $k=\tan \alpha$ and at the initial moment $\varphi_{0}-$ $=\pi / 2$.
1.197. A chain of length $t$ is placed on s smoeth spherical surface of radius $R$ with one of its ends fixed at the top of the sphere. What will be the acceleration $w$ of each element of the chain whea its upper end is released? It is assumed that the length of the chain $l<\frac{1}{2} \pi R$.
1.108. A small body is placed on the top of a smooth sphere of radius $A$. Then the sphere is imparted a constant acceleration $w_{c}$ in the horizontal direction and the body begins sliding down. Find:
(a) the velocity of the body relative to the sphere at the moment of break-off;
(b) the angle $\theta_{0}$ between the vertical and the radius vector drawn from the centre of the sphere to the break-off point; calculate $\theta_{4}$ for $\omega_{0}=g$.
1.109. A particle moves in a plase under the action of a force which is always perpendicular to the particle’s velocity and depends on a distance to a certain point on the plane as $1 / r^{m}$, where $n$ is a constant. At what value of $n$ will the motion of the particle along the circle be ateady?
1.110. A sleeve $A$ can slide freely along a smeoth rod bent in the shape of a half-eirele of radius $\boldsymbol{R}$ (Fig, 1.28). The system is set in rotation with a constant angular velocity $\omega^{\circ}$ about a vertical axis $O O^{\prime}$. Find the angle $\theta$ corresponding to the steady position of the sleeve.
1.111. A rifle was aimed at the vertical line on the target located precisely in the northern direction, and then fired. Assuming the air drag to be negligible, find how much ofl the line, and in what direction, will the bullet hit the target. The shot was fired in the horizontal
direction at the latitude $\varphi=60^{\circ}$, the bullet velocity $v=900 \mathrm{~m} / \mathrm{s}$, and the distance from the target equals $s=1.0 \mathrm{~km}$.
1.112. A horizontal dise rotates with a constant angular velocity $0=6.0 \mathrm{rad} / \mathrm{s}$ about a vertical axis passing through its centre. A small body of mase $m=0.50 \mathrm{~kg}$ moves along a dismeter of the dise with a velocity $v^{\prime}=50 \mathrm{~cm} / \mathrm{s}$ which is constant relative to the disc. Find the force that the dise exerts on the body at the moment when it is located at the distance $r=30 \mathrm{~cm}$ from the rotation axis.
1.113. A horizontal smooth rod $A B$ rotates with a constant angular velocity $\omega=2.00 \mathrm{rad} / \mathrm{s}$ about a vertical axis passing through its end $A$. A freely sliding sleeve of mass $m=0.50 \mathrm{~kg}$ moves along the rod from the point $A$ with the initial velocity $v_{2}=1.00 \mathrm{~m} / \mathrm{s}$. Find the Coriolis force acting on the sleeve (in the reference frame fixed to the rotating rod) at the moment when the sleeve is located at the distance $r=50 \mathrm{~cm}$ from the rotation axis.
1.114. A borizontal disc of radius $R$ rotates with a constant angular velocity $\omega$ about a stationary vertical axis passing through its edge. Along the circumference of the dise a particle of nass $m$ moves with a velocity that is constant relative to the disc. At the moment when the particle is at the maximum distance from the rotation axis, the resultant of the inertial forces $F_{\text {in }}$ acting on the particle in the reference frame fixed to the dise turns inte zero. Find:
(a) the acceleration $t^{\prime}$ of the particle relative to the disc;
(b) the dependence of $F_{i n}$ on the distance from the rotation axis.
1.115. A small body of mass $m=0.30 \mathrm{~kg}$ starts sliding down from the top of a smooth sphere of radius $R=1.00 \mathrm{~m}$. The sphere rotates with a constant angular velocity o $=6.0 \mathrm{rad} / \mathrm{s}$ about a vertical axis passing through its centre. Find the centrifugal force of inertia and the Coriolis force at the moment when the body breaks off the surface of the sphere in the reference frame fixed to the sphere.
1.116. A train of mass $m=2000$ tons moves in the latitude $9-$ $=60^{\circ}$ North. Find:
(a) the magnitude and direction of the lateral force that the train exerts on the rails if it moves along a meridian with a velocity $v=$ $=54 \mathrm{~km}$ per hour;
(b) in what direction and with what velocity the trais ahould move for the resultant of the inertial forces acting on the train in the relerence frame fixed to the Earth to be equal to zero.
1.117. At the equator a stationary (relative to the Earth) body falls down from the height $h-500 \mathrm{~m}$. Assuming the air drag to be negligible, find how much off the vertical, and in what direction, the body will deviate when it hits the ground.