Question

A block B is attached to two unstretched springs S₁ and S₂ of spring constants K₁ and K₂ respectively. The other ends of the springs are attached to two masses M₁ and M₂ not attached to the walls 1 and 2. The springs and the supports have negligible mass and there is no friction. The block B is displaced towards wall 1 through a small distance ‘a’ and released. The block returns and moves a maximum distance a/2 towards wall 2. The displacements ‘a and a/2 are measured with respect to the equilibrium position of block B. The ratio K₂/K₁ is

(1) 1/4

(2) 4

(3) 2

(4) 1/2

Answer 1

(2) 4

When mass M is moved a distance ‘a’ towards wall 1; spring S₁ gets compressed by ‘a’. However spring S₂ has no change in its length because mass M₂ is free to move. The energy U₁ of system is due to compression of spring S₁ only.

On being released, block B moves towards equilibrium position, overshoots the position and compresses

S₂ by a/2 before coming to rest. Now mass M1 detatches itself from wall and there is no change in length of S₁. The energy of system, U₂ , is now due to compression of spring S₂ only.

From law of conservation of energy U₁ = U₂ i.e