LEVEL – II
Q:1. A block of mass m moving with a velocity v hits a light spring of stiffness K attached rigidly to a stationary sledge of mass M. Neglecting friction between all contacting surface, find the maximum compression of the spring.
Ans:$\displaystyle = (\sqrt{\frac{M m}{(M+m)k}})v_0 $
Q:2. 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 above the lowest position?
Ans: $ \displaystyle (\frac{M}{M+m})^2 l $
Q:3. A uniform thin rod of mass M and length L is standing vertically along the y-axis on a smooth horizontal surface, with its lower end at the origin (0,0). A slight disturbance at t = 0 causes the lower end to slip on the smooth surface along the positive x-axis, and the rod starts falling.
(a) What is the path followed by the centre of mass of the rod during its fall.
(b) Find the equation of trajectory of a point on the rod located at a distance r from the lower end. What is the shape of the path of this point?
Ans: (a)centre of mass will fall vertically
(b)$ \displaystyle \frac{x^2}{(l/2 – r)^2} + \frac{y^2}{r^2} = 1 $ ; which is an ellipse
Q:4. A block of mass M with a semicircular track of radius R rest on a horizontal frictionless surface. A uniform cylinder of radius r and mass m is released from rest at the top point A. The cylinder slips in the semicircular frictionless track. How far the block moved when the cylinder reaches the bottom of the track?
Ans: $ \displaystyle \frac{m}{M + m}(R-r) $
Q:5. A block of mass M is hanging from a rigid support by an in-extensible light string. A ball of mass m hits it with a vertical velocity v at its bottom. Find the change in momentum of the ball assuming inelastic collision.
Ans: $ \displaystyle -\frac{M m}{M + m}v $
Q:6. 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.
Ans: $ \displaystyle \frac{M m^2 v^2}{2(m + 2M((m+M)} $
Q:7. A cannon and a supply of cannon balls are inside a sealed railroad car. The cannon fires to the right, the car recoils to the left. The cannon balls remain in the car after hitting the far wall. Show that no matter how the cannon balls are fired, the railroad car cannot travel more than l, assuming it starts from rest .
Ans: $ \displaystyle x_c = \frac{m l}{M + m} < l $
Q:8. Two wooden blocks of mass M1 = 1 kg, M2 = 2.98 kg lie separately side by side smooth surface. A bullet of mass m = 20 gm strikes the block M1 and pierces through it, then strikes the second block and sticks to it. Consequently both the blocks move with equal velocities. Find the percentage change in speed of the bullet when it escapes from the first block.
Ans: $ \displaystyle \frac{M_2 – M_1 + m}{M_2 + m}\times 100 $
Q:9. A steel ball is suspended by a light inextensible string of length l from a fixed point O. When the ball is in equilibrium it just touches a vertical wall as shown in the figure. The ball is first taken aside such that string becomes horizontal and then released from rest. If co-efficient of restitution is e, then find the maximum deflection of the string after nth collision.
Ans: $ \displaystyle \theta = cos^{-1}(1-e^{2n}) $
Q:10. A body of mass M with a small disc of mass m placed on it rests on a smooth horizontal plane. The disc is set in motion in the horizontal direction with velocity v. To what height (relative to the initial level) will the disc rise after breaking off the body M? All surfaces are frictionless.
Ans: $ \displaystyle \frac{M+m}{M }\frac{v^2}{2 g} $