Question

A spring of spring constant k placed horizontally on a rough horizontal surface is compressed against a block of mass m placed on the surface so as to store maximum energy in the spring. If the coefficient of friction between the block and the surface is $\mu$, the potential energy stored in the spring is:

• A. $\frac{{\mu }^2{m}^2{g}^2}{k}$
• B. $\frac{2{\mu }^2{g}^2}{k}$
• C. $\frac{{\mu }^2{m}^2{g}^2}{2k}$
• D. $\frac{3{\mu }^{2}m{g}^2}{k}$

Answer 1

The correct option is C $\frac{{\mu }^2{m}^2{g}^2}{2k}$

Restoring force of spring $F=-kx$
coefficient of kinetic friction $F=\mu mg$
at equilibrium,the spring along with mass will return to its mean position.
so,$-kx=\mu mg;x=\frac{-\mu mg}{k}$
the potential energy stored the spring is $\frac{1}{2}k{x}^2=\frac{1}{2}k(\frac{-\mu mg}{k}{)}^2=\frac{1}{2}k(\frac{{\mu }^2{m}^2{g}^2}{k})$