Question

A long block A is at rest on a smooth horizontal surface. A small block B, whose mass is half of A, is placed on A at one end and projected along A with some velocity u. The coefficient of friction between the blocks is μ.

(a) The blocks will reach a final common velocity μ/3.

(b) The work done against friction is two-thirds of the initial kinetic energy of B.

(c) Before the blocks reach a common velocity, the acceleration of A relative to B is 2/3 μg.

(d) Before the blocks reach a common velocity the acceleration of A relative to B is 3/2 μg.

Answer 1

Correct Answer is: (a, b, & d)

As there are no external forces acting on the ‘A + B’ system, its total momentum is conserved. If the masses of A and B are 2m and m respectively, and v is the final common velocity, mu = (m + 2m)v  or v = u/3.

Work done against friction = loss in KE = 1/2 mu² - 1/2 (3m)v²

= 1/2 mu² - 1/2 (3m) u²/9 = 1/2 mu² [ 1 - 1/3 ] = 2/3 x 1/2 mu² .

The force of friction between the blocks is μmg.

Acceleration of A (to the right) = a₁ = μmg/2m = μg/2.

Acceleration of B (to the left) = a₂ =  μmg/m = μg.

Acceleration of A relative to B = a₁ = (-a₂) = 3/2 μg.