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(BRUSH UP YOUR CONCEPTS) 1. Two smooth spheres A and B, of equal radius but mass m and M, are free to move on a horizontal table. A is projected with speed u towards B which is at rest. On impact, the line joining their centres is inclined at an angle to the velocity of A before impact. If e is the coefficient of restitution between the spheres, find the speed with which B begins to move. If A’s path after impact is perpen- dicular to its path before impact, show that tan 2 eM m . M m 2. A man of mass m moves on a plank of mass M with a constant velocity urel . with respect to the plank, as shown in figure. (i) If the plank rests on a smooth horizontal surface, then deter- mine its velocity with respect to ground. (ii) If the man travels a distance L with respect to the plank, then L find the distance traveled by the plank with respect to ground. 3. An imperfectly elastic particle is projected from a point in a horizontal place with velocity u at an angle to the horizon. If e be the coefficient of restitution, then calculate, (i) the time after which it ceases to rebound from the plane (ii) its range (iii) tangent to the angle of projection at the nth rebound. 4. The centres of the spheres 1, 2 and 3 lie on a single straight line. Sphere 1 is moving with an (initial) velocity v1 directed along this line and hits sphere 2. Sphere 2, acquiring after collision a velocity v2 , hits sphere 3. Both collisions are absolutely elastic . What must be the mass of sphere 2 for the sphere 3 to acquire maximum V1 1 2 m3 m1 2 velocity (the masses m1 and m3 of spheres 1 and 3 are known)? 5. A small empty bucket of mass M is attached to a long inextensible cord of length l. The bucket is released from rest when the cord is in a horizontal position. In its lowest position the bucket scoops up a mass m of water, what is the height of the swing about the lowest position? 6. Two particles A and B lighter particle has mass m, are released from infinity. They move towards each other under their mutual force of attraction. If their speeds are v and 2v respectively find the K.E. of the system. 7. A steel ball with a mass of m = 20 g falls from a height of h1 = 1 m onto a steel plate and rebounds to a height of h2 = 81 cm. Find: the impulse of the force received by the plate during the impact. 8. Two identical blocks A and B of mass M each are kept on each other on a smooth horizontal plane. There exists friction between A and B. If a bullet of mass m hits the lower block with a horizontal velocity v and gets embedded into it. Find the work done by friction between A and B. 9. A ball is dropped from a height h above the landing and bounces down the flight of stairs. Denoting by e the coefficient of restitu- tion, determine the value of h for which the ball will bounce the same height above each step. 10. A wedge of mass M and angle can move freely on a smooth horizontal plane. A smooth sphere of mass m strikes it in a directio

- 1. SUBJECTIVE LEVEL - II (BRUSH UP YOUR CONCEPTS) 1. Two smooth spheres A and B, of equal radius but mass m and M, are free to move on a horizontal table. A is projected with speed u towards B which is at rest. On impact, the line joining their centres is inclined at an angle to the velocity of A before impact. If e is the coefficient of restitution between the spheres, find the speed with which B begins to move. If A’s path after impact is perpen- dicular to its path before impact, show that 2 eM m tan M m . M urel m L 2. A man of mass m moves on a plank of mass M with a constant velocity rel u . with respect to the plank, as shown in figure. (i) If the plank rests on a smooth horizontal surface, then deter- mine its velocity with respect to ground. (ii) If the man travels a distance L with respect to the plank, then find the distance traveled by the plank with respect to ground. 3. An imperfectly elastic particle is projected from a point in a horizontal place with velocity u at an angle to the horizon. If e be the coefficient of restitution, then calculate, (i) the time after which it ceases to rebound from the plane (ii) its range (iii) tangent to the angle of projection at the nth rebound. m1 1 2 3 V1 m2 m3 4. The centres of the spheres 1, 2 and 3 lie on a single straight line. Sphere 1 is moving with an (initial) velocity 1 v directed along this line and hits sphere 2. Sphere 2, acquiring after collision a velocity 2 v , hits sphere 3. Both collisions are absolutely elastic . What must be the mass of sphere 2 for the sphere 3 to acquire maximum velocity (the masses 1 m and 3 m of spheres 1 and 3 are known)? 5. A small empty bucket of mass M is attached to a long inextensible cord of length l. The bucket is released from rest when the cord is in a horizontal position. In its lowest position the bucket scoops up a mass m of water, what is the height of the swing about the lowest position? 6. Two particles A and B lighter particle has mass m, are released from infinity. They move towards each other under their mutual force of attraction. If their speeds are v and 2v respectively find the K.E. of the system. 7. A steel ball with a mass of m = 20 g falls from a height of h1 = 1 m onto a steel plate and rebounds to a height of h2 = 81 cm. Find: the impulse of the force received by the plate during the impact. 8. Two identical blocks A and B of mass M each are kept on each other on a smooth horizontal plane. There exists friction between A and B. If a bullet of mass m hits the lower block with a horizontal velocity v and gets embedded into it. Find the work done by friction between A and B.
- 2. h d d 9. A ball is dropped from a height h above the landing and bounces down the flight of stairs. Denoting by e the coefficient of restitu- tion, determine the value of h for which the ball will bounce the same height above each step. 10. A wedge of mass M and angle can move freely on a smooth horizontal plane. A smooth sphere of mass m strikes it in a direction perpendicular to its inclined face with a speed v and rebounds. If the coefficient of restitution is e, then find the speed of the sphere and that of the wedge after impact.
- 3. SUBJECTIVE LEVEL - III (CHECK YOUR SKILLS) M m v 1. A body of mass M with a small disc m placed on it rests on a smooth horizontal surface. The disc is set in motion in the horizontal direction with a velocity v. To what height relative to the initial level will the disc rise after breaking off from the body M. Friction can be assumed to be absent. m M M h 2. The inclined surfaces of two movable wedges of the same mass M are smoothly conjugated with the horizontal plane (as shown in figure). A washer of mass m slides down the left wedge from a height h. To what maximum height will the washer rise along the right wedge? Friction should be neglected. m1 m2 r 3. A symmetric block of mass m1 with a notch of hemispherical shape of radius r rests on a smooth horizontal surface near the wall as shown in figure. A small washer of mass m2 slides without friction from the initial position shown in figure. Find the maximum velocity of the block. v2 = 40 m/s B m m A 600 300 v1 = 30 m/s 4. The magnitude and direction of the velocities of two iden- tical frictionless balls before they strike each other are as shown. Assuming e = 0.90, determine the magnitude and direction of the velocity of the each ball after the impact. h H x 5. A girl of mass M holding a bag of mass m slips from the roof of a tall building of height H and starts falling vertically as shown in figure. When at a height h from the ground, she notices that the ground below her is pretty hard, but there is a pond at a horizontal distance x from the line of fall. In order to save herself she throws the bag horizontally (with respect to herself) in the direction oppo- site to the pond. Calculate the minimum horizontal velocity im- parted to the bag so that the girl lands in the water. If the girl just succeeds to avoid the hard ground, where will the bag land ? 6 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 l relative to the raft with velocity v (t) and then stops. Assuming the water resis- tance to be negligible, find: (a) the displacement of the raft l relative to the shore; (b) the horizontal component of the force with which the man acted on the raft during the motion.
- 4. 7. A particle A of mass m moving on a smooth horizontal surface collides with a stationary particle B of mass 2m directly, situated at a distance d from a wall. The coefficeint of restitution between A and B and between B and the wall is e = 1/4. Calculate the distance from the wall where they collide again. Assume that the entire motion takes place along a straight line perpendicular to the wall. 8. A wedge of mass M rests on an absolutely smooth, horizontal surface. A block of mass m is placed on the wedge, inclined at an angle to the horizontal. All the surfaces are frictionless. Assuming that the system was at rest at the initial moment, find the velocity of the wedge when the block slides down the plane through a vertical height h. A B u l 9. Two particles A and B of equal mass m each are attached by a string of length 2l and initially placed over a smooth horizontal table in the position shown in figure. Particle B is projected across the table with speed u perpendicular to AB as shown in figure. Find the velocities of each particle after the string becomes taut and the magnitude of the impulsive tension. 10. A sphere A is of mass m and another sphere B of identical size but of mass 2m, move towards each other with velocity ĵ 2 î and ĵ 3 î respectively. They collide when their line of centres is parallel to . ĵ î If e = 1/2, find the velocities of A and B after impact.