Question

A block of mass $m$ is attached to one end of a massless spring of spring constant $k$. The other end of spring is fixed to a wall. The block can move on a horizontal rough surface. The coefficient of friction between the block and the surface is $\mu$. Then the compression of the spring for which maximum extension of the spring becomes half of maximum compression is:

• A. $\frac{2mg\mu }{k}$
• B. $\frac{mg\mu }{k}$
• C. $\frac{4mg\mu }{k}$
• D. None of these

Answer 1

Answer: (C)

$\frac{4mg\mu }{k}$

Energy stored in a spring is given by $E=\frac{1}{2}{kx}^2$ where $x$ is the compression/extension in the spring.

Mass is at rest when its compression is $x$.

When it extends, maximum extension is $x/2$.

The difference in potential energies of the two positions equals the work done against the friction.

$E_i-E_f=$ Work done against friction

$\therefore \frac{1}{2}k{x}^2-\frac{1}{2}k{\left(\frac{x}{2}\right)}^2=\mu mg(x+\frac{x}{2})$
$\therefore x=\frac{4\mu mg}{k}$