A block of mass m attached to a single spring of natural length L and spring constant k oscillates on a smooth horizontal surface with a time period T₀ . The spring is then cut into two pieces with their lengths in the ratio 1:3 . The same block is placed on the smooth horizontal surface, and the two new spring pieces are attached to opposite sides of the block. The other ends of the spring pieces are fixed to rigid

Step-by-step solution

The initial time period of the block is T₀ = 2 m k . When the spring is cut into two pieces in the ratio 1:3 , their lengths are L₁ = L 4 and L₂ = 3L 4 . Since the spring constant is inversely proportional to the length of the spring ( k 1 L ), the spring constants of the two pieces are: k₁ = 4k k₂ = 4k 3 When the block is attached to these two springs on opposite sides and the other ends are fixed to walls, the springs act as a parallel combination. The equivalent spring constant is: k_ eq = k₁ + k₂ = 4k + 4k 3 =