A block of mass 1 kg kept on a rough horizontal surface ( μ = 0.4 ) is attached to a light spring (force constant = 200 N / m ) whose other end is attached to a vertical wall. The block is pushed to c

Question

A block of mass 1 kg kept on a rough horizontal surface ( μ = 0.4 ) is attached to a light spring (force constant = 200 N / m ) whose other end is attached to a vertical wall. The block is pushed to compress the spring by a distance d and released. Find the value(s) of 'd' for which (spring + block) system loses its entire mechanical energy in form of heat.

TestHub · Accepted Answer

All mechanical energy becomes heat only if the block finally stops at the natural length. Each half-oscillation the turning point moves $\frac{2 μ m g}{k} = \frac{2 \times 0.4 \times 1 \times 10}{200} = 4 cm$ closer to the natural length, since

$\frac{1}{2} k (x_{i}^{2} - x_{i + 1}^{2}) = μ m g (x_{i} + x_{i + 1}) \Rightarrow x_{i} - x_{i + 1} = \frac{2 μ m g}{k}$

So d = 4 cm (stops at natural length directly) and d = 8 cm (4 cm extension, then back to natural length) both work; at the intermediate stop the spring force 8 N exceeds friction 4 N, so the return journey happens. d = 6, 10 cm leave stored spring energy.

Ans. (A,C)

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