The figure shows a block of mass m resting on a horizontal smooth floor at a distance l from a rigid wall. Block is pushed toward the right by a distance 3 l 2 and released from rest. When block passes from its mean position, another block of mass m 1 is gently placed on it which sticks to it due to friction. The value of m 1 so that the combined block just touches the left wall is η 4 m . Find the value of η .

Step-by-step solution

When block P is released from rest from a distance 3 l 2 toward right from mean position, this will be the amplitude of oscillation so velocity of block when passing from its mean position is given as v = A ω = 3 l 2 k m If mass m 1 is added to it and just after if velocity of combined block becomes v 1 , from momentum conservation, we have mv = (m + m 1 )v 1 ∴ v 1 = m m + m 1 ⋅ 3 l 2 k m If this is the velocity of combined block at mean position, it is given as v 1 = A 1 ω 1 where ω 1 = k m + m 1 where A 1 and ω 1