LOM-06-SUBJECTIVE UNSOLVED-II

LEVEL - II (BRUSH UP YOUR CONCEPTS) 1. A man of mass m has fallen into a a ditch of width d and two of this friends are slowly pulling him out using a light rope and two fixed pulleys as shown in figure. Show that the force (assumed equal for both the friends) exerted by each friend on the rope increases as the man moves up. Find the force when the man is at a depth h. 2. The elevator shown in figure is descending with an acceleration of 2 m/s2. The mass of the block A is 0.5 kg. What force is exerted by the block A on the block B? 2 m/s2 3. The force of buoyancy exerted by the atmosphere on a balloon is B in the upward direction and remains constant. The force of air resistance on the balloon acts opposite to the direction of velocity and is proportional to it. The balloon carries a mass M and is found to fall down near the earth’s surface with a constant velocity v. How much mass should be removed from the ballon so that it may rise with a constant velocity v? 4. In figure m1 = 5 kg, m2 = 2 kg and F = 10 N. Find the acceleration of either block. Describe the motion of m1 if the string breaks but F continues to act. 5. Two touching bars 1 and 2 are placed on an inclined plane forming an angle α with the horizontalas shown in figure. The masses of the bars are equal to m1 and m2 , and the coefficients of friction between the inclined plane and the bars are equal to k1 and k2 respectively, with k1  k2 . 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. 6. A small body A starts sliding down from the top of fixed wedge (as shown in the figure) 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 ? 7. At the moment t  0 the force F  at is applied to a small body of mass m F resting on a smooth hrozontal plane (a is a constant). The permanent direction of this force forms an angle α with the horizon- tal. 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. 8. Two identical blocks A and B each of mass M are connected through a light inextensible string. Coefficient of friction between blocks and surfaces are  as shown. Initially string is relaxed in its normal   length. Force F is applied on block A as shown. Find the force of friction on blocks and tension in the string. (a) If (b) If F  3 μMg 4 F  3  Mg . 2 9. The friciton coefficient between the board and the floor shown in figure is µ. Find the maximum force that the man can exert on the rope so that the board does not slip on the floor. 10. The inclined plane of forms an angle α  30 with the horizontal. The mass ratio m2 / m1  η  2 / 3 . The coefficient of friction between the body m1 and m2 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 th body m2 when the formerly stationary system of masses starts moving. SUBJECTIVE LEVEL - III (CHECK YOUR SKILLS) M 1. As shown in the figure blocks of masses 2 M , M and 2 are connected through a light string as shown, pulleys are light and smooth. Friction is only between block C and floor. System is released from rest. Find the acceleration of blocks A, B and C and tension in the string. 2. Find the mass M of the hanging block in figure which will prevent the smaller block from slipping over the triangular block. All the surfaces are frictionless and the string and the pulleys are light. ' 3. Figure shows a man of mass 60 kg standing on a light weighing machine kept in a box of mass 30 kg. The box is hanging from a pulley fixed to the ceiling through a light rope, the other end of which is held by the man himself. If the man manages to keep the box at rest, what is the weight shown by the machine ? What should be acceleration of man to get his correct weight on the machine ? 4. Figure shows a small block of mass m kept at the left end of a larger block of m mass M and length l. The system can slide on a horizontal road. Both the blocks are started towards right with an initial velocity v. The friction coeffi- l cient between the road and the bigger block is µ and that between the blocks is  / 2 . Find the time elapsed before the smaller block separates from the big- ger block. 5. Find the accelerations of rod A and wedge B in the arrangement shown in figure. If the ratio of the mass of the wedge of that of the rod equals η , and the friction A between all contact surfaces is negligible. B  6. A heavy chain with a mass ρ per unit length is pulled by the constant force P along a horizontal surface consisting of a smooth section and a rough section. The chain is initially at rest on the rough surface with x = 0. If the coefficient of friction Rough x=0 Smooth between the chain and the rough surface is μk , determine the velocity v of the chain when x = L. The force P is greater than μk ρgL in order to initiate motion. 7. A bar of mass m is pulled by means of a thread up an inclined plane forming an angle α with the horizontal as shown in figure. The coefficient of friction is equal to µ. 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 ? 8. In the arrangement shown in figure, 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 lower end of the rod, began moving with constant accelerations. In t seconds after the begining of motion the ball got opposite the upper end of the rod. The rod length equals l. Find the friction force between the ball and the thread. 9. A plank of mass m1 with a bar of mass m2 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 w1 and of the bar w2 depend on t, if teh coefficient of friction between the plank and teh bar is equal to k. Draw the approximate plots of these dependences. 10. 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 accelera- tion w. How soon will the bodies collide ?

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