№ 
Condition 
free/or 0.5$ 
101  Having gone through a plank of thickness h, a bullet changed its velocity from v0 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. 

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. 

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. 

104  A body of mass m is thrown straight up with velocity v0. Find the velocity v* with which the body comes down if the air drag equals kv2, where k is a constant and v is the velocity of the body. 

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. 

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 v0. Find how the velocity of the disc depends on the angle φ if the friction coefficient k = tan α and at the initial moment φ0 = π/2. 

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. 

108  A small body is placed on the top of a smooth sphere of radius R. Then the sphere is imparted a constant acceleration w0 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 breakoff; (b) the angle θ0 between the vertical and the radius vector drawn from the centre of the sphere to the breakoff point; calculate θ0 for w0 = g. 

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/rn, where n is a constant. At what value of n will the motion of the particle along the circle be steady? 

110  A sleeve A can slide freely along a smooth rod bent in the shape of a halfcircle of radius R (Fig. 1.28). The system is set in rotation with a constant angular velocity w about a vertical axis OO**. Find the angle T corresponding to the steady position of the sleeve. 

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. 

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. 

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 v0 = 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. 

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 Fin 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 Fin on the distance from the rotation axis. 

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. 

116  A train of mass m = 2000 tons moves in the latitude f= 60° 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 km per hour; (b) in what direction and with what velocity the train should move for the resultant of the inertial forces acting on the train in the reference frame fixed to the Earth to be equal to zero. 

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. 

118  A particle has shifted along some trajectory in the plane xy from point 1 whose radius vector r1 = i + 2j to point 2 with the radius vector r2 = 2i  3j. During that time the particle experienced the action of certain forces, one of which being F = 3i + 4j. Find the work performed by the force F. (Here r1, r2, and F are given in SI units). 

119  A locomotive of mass m starts moving so that its velocity varies according to the law v = a sqrt(s), 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 beginning of motion. 

120  The kinetic energy of a particle moving along a circle of radius R depends on the distance covered s as T = as2, where a is a constant. Find the force acting on the particle as a function of s. 
