Question

A block of mass 1kg kept on a rough horizontal surface ($\mu=0.4$) 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.

• A. 4cm
• B. 6cm
• C. 8cm
• D. 10 cm

Answer 1

Answer: (A)

4cm

Answer 2

The system loses its entire mechanical energy as heat, meaning the initial spring potential energy is completely converted into work done by friction. The final state is zero kinetic energy and zero spring potential energy (block at rest at spring's natural length). Thus, the work done by friction is equal to the initial spring potential energy: $\frac{1}{2}kd^2 = \mu mg L$. For the block to stop exactly at the natural length, the total distance travelled $L$ must be equal to the initial compression $d$. Substituting $L=d$ gives $\frac{1}{2}kd^2 = \mu mg d$. Solving for $d$, we get $d = \frac{2\mu mg}{k}$. Plugging in the given values ($m=1 \text{ kg}$, $\mu=0.4$, $k=200 \text{ N/m}$, $g=10 \text{ m/s}^2$), we find $d = \frac{2 \times 0.4 \times 1 \times 10}{200} = 0.04 \text{ m} = 4 \text{ cm}$.