(i) O is a fixed peg at a height H above a perfectly inelastic smooth horizontal plane. A light inextensible string of length `L (gt H)` has one end attached to `O` and the other end is attached to a heavy particle. The particle is held at the level of O with string horizontal and just taut and released from rest. Find the height of the particle above the plane when it comes to rest for the first time after the release.

(ii) The bob of a pendulum has mass m and the length of pendulum is `l`. It is initially at rest with the string vertical and the point of suspension at a height `2l` above the floor. A particle P of mass `(m)/(2)` moving horizontally along –ve x-direction with velocity `sqrt(2gl)` collides with the bob and comes to rest. The bob swings and when it comes to rest for the first time, another particle Q of mass m moving horizontally along y direction collides with the bob and sticks to it. It is observed that the bob now moves in a horizontal circle.

(a) Find tension in string just before the second collision.
(b) Find the height of the circular path above the floor.
(c) Find the time period of the circular motion.
(d) The string breaks during the circular motion at time `t = 0`. At what time the bob will hit the floor ?

Answer

D

Explanation

In perfectly inelastic collision with the horizontal surface the component parallel to the surface will remain unchanged. Similarly when the string becomes taut again, the component perpendicular to its length will remain unchanged.

`cos theta=H/Limpliesv_c=sqrt(2gH)`
`v_cos^(2)theta=(sqrt(2gH))H^2/L^2=v` (say)
`:. h=(v^2)/(2g)=((2gH)H^4/L^4)/(2g)=H^5/L^4`