1.59. An aerostat of mass m starts coming down with a constant acceleration w. Determine the ballast mass to be dumped for the aerostat 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_{0}, 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 acceleration w with which the body m_{0} comes down, and the tension of the thread binding together the bodies m_{1} and m_{2}, if the coefficient of friction between these bodies and the horizontal surface is equal to k. Consider possible cases.

1.61. Two touching bars 1 and 2 are placed on an inclined plane forming an angle α with the horizontal (Fig. 1.10). The masses of the bars are equal to m_{1} and m_{2}, and the coefficients of friction between the inclined plane and these bars are equal to k_{1} and k_{2} respectively, with k_{1} > k_{2}. Find:
(a) the force of interaction of the bars in the process of motion;
(b) the minimum value of the angle α at which the bars start sliding down.

1.62. A small body was launched up an inclined plane set at an angle α = 15° against the horizontal. Find the coefficient of friction, if the time of the ascent of the body is η = 2.0 times less than the time of its descent.

1.63. The following parameters of the arrangement of Fig. 1.11 are available: the angle α which the inclined plane forms with the horizontal, and the coefficient of friction k between the body m_{1} 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_{2} (a) starts coming down;
(b) starts going up;
(c) is at rest.

1.64. The inclined plane of Fig. 1.11 forms an angle α = 30° with the horizontal. The mass ratio m_{2}/m_{1} = η = 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_{2} when the formerly stationary system of masses starts moving.

1.65. A plank of mass m_{1} with a bar of mass m_{2} placed on it lies on a smooth horizontal plane. 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_{2} 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 A starts sliding down from the top of a wedge (Fig. 1.12) whose base is equal to l = 2.10 m. The coefficient of friction between the body and the wedge surface is k = 0.140. At what value of the angle α 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 α with the horizontal (Fig. 1.13). The coefficient of friction is equal to k. Find the angle β 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 = at is applied to a small body of mass m resting on a smooth horizontal plane (a is a constant). The permanent direction of this force forms an angle a 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 = mg/3 of constant magnitude. In the process of its rectilinear motion the angle α between the direction of this force and the horizontal varies as α = as, 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 α.

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 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 η times as great as the mass of bar 1 and the angle that the inclined plane forms with the horizontal is equal to α. 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 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 opposite 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 1 is η = 1.8 times as great as that of rod 2. The length of the latter is l = 100 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 1 is η = 4.0 times as great as that of body 2. The height h = 20 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 η, and the friction between all contact surfaces is negligible.

1.79. What is the minimum acceleration with which bar A (Fig. 1.22) should be shifted horizontally to keep bodies 1 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 1 with bar 2 of mass m placed on it gets a horizontal 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 α?

1.81. Prism 1 of mass m_{1} and with angle α (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 negligible, find the acceleration of the prism.

1.84. An aircraft loops the loop of radius R = 500 m with a constant velocity v = 360 km per hour. Find the weight of the flyer of mass m = 70 kg in the lower, upper, 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 θ, 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 θ 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 body A starts sliding off the top of a smooth sphere of radius R. Find the angle θ (Fig. 1.25) corresponding to the point at which the body breaks off the sphere, as well as the break-off velocity of the body.

1.88. A device (Fig. 1.26) consists of a smooth L-shaped rod located in a horizontal plane and a sleeve A of mass m attached by a weightless spring to a point B. The spring stiffness is equal to χ. The whole system rotates with a constant angular velocity ω 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 circumference of a circular horizontal plane of radius R, the friction coefficient being dependent only on distance r from the centre 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 along which the cyclist can ride with the maximum velocity. What is this velocity?

1.90. A car moves with a constant tangential acceleration w_{τ} = 0.62 m/s^{2} along a horizontal surface circumscribing a circle of radius R = 40 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 moment of time its velocity is equal to zero?

1.91. A car moves uniformly along a horizontal sine curve y = a sin (x/α), where a and α 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 smooth round cone with half-angle θ. Find the tension of the chain if it rotates with a constant angular velocity ω about a vertical axis coinciding with the symmetry axis of the cone.

1.95. Find the magnitude and direction of the force acting on the particle of mass m during its motion in the plane xy according to the law x = a sin ωt, y = b cos ωt, where a, b, and ω are constants.

1.96. A body of mass m is thrown at an angle to the horizontal with the initial velocity v_{0}. Assuming the air drag to be negligible, find:
(a) the momentum increment Δp that the body acquires over the first t seconds of motion;
(b) the modulus of the momentum increment Δp during the total time of motion.

1.97. At the moment t = 0 a stationary particle of mass m experiences a time-dependent force F = at(τ - t), where a is a constant vector, τ 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 F = F_{0} sin ωt, where F_{0} and ω 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 moment t = 0 a particle of mass m starts moving due to a force F = F_{0} cos ωt, where F_{0} and ω 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 motorboat 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 F = -rv, 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 (beginning with the moment t = 0), during which its velocity decreases η 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 α with the horizontal. The friction coefficient depends on the distance x covered as k = ax, 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 horizontal plane with the friction coefficient k. At the moment t = 0 a horizontal force is applied to it, which varies with time as F = at, where a is a constant vector. Find the distance traversed by the body during the first t seconds after the force action began.

1.104. A body of mass m is thrown straight up with velocity v_{0}. Find the velocity v' with which the body comes down if the air drag equals kv^{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 magnitude is constant and whose vector rotates in that plane with a constant angular velocity ω. 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 α 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 φ if the friction coefficient k = tan α and at the initial moment φ_{0} = π/2.

1.107. A chain of length l is placed on a smooth 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 when its upper end is released? It is assumed that the length of the chain l < πR/2.

1.108. A small body is placed on the top of a smooth sphere of radius R. Then the sphere is imparted a constant acceleration w_{0} 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 θ_{0} between the vertical and the radius vector drawn from the centre of the sphere to the break-off point; calculate θ_{0} for w_{0} = g.

1.109. A particle moves in a plane 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^{n}, where n is a constant. At what value of n will the motion of the particle along the circle be steady?

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 off the line, and in what direction, will the bullet hit the target. The shot was fired in the horizontal direction at the latitude φ = 60°, the bullet velocity v = 900 m/s, and the distance from the target equals s = 1.0 km.

1.112. A horizontal disc rotates with a constant angular velocity ω = 6.0 rad/s about a vertical axis passing through its centre. A small body of mass m = 0.50 kg moves along a diameter of the disc with a velocity v' = 50 cm/s which is constant relative to the disc. Find the force that the disc exerts on the body at the moment when it is located at the distance r = 30 cm from the rotation axis.

1.113. A horizontal smooth rod AB rotates with a constant angular velocity ω = 2.00 rad/s about a vertical axis passing through its end A. A freely sliding sleeve of mass m = 0.50 kg moves along the rod from the point A with the initial velocity v_{0} = 1.00 m/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 cm from the rotation axis.

1.114. A horizontal disc of radius R rotates with a constant angular velocity ω about a stationary vertical axis passing through its edge. Along the circumference of the disc a particle of mass 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_{in} acting on the particle in the reference frame fixed to the disc turns into zero. Find:
(a) the acceleration w' of the particle relative to the disc;
(b) the dependence of F_{in} on the distance from the rotation axis.

1.115. A small body of mass m = 0.30 kg starts sliding down from the top of a smooth sphere of radius R = 1.00 m. The sphere rotates with a constant angular velocity ω = 6.0 rad/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.

1.117. At the equator a stationary (relative to the Earth) body falls down from the height h = 500 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.