$A$ sphere of mass $m$ is attached to a spring of spring constant $k$ and is held in an unstretched position over an inclined plane as shown in the figure. After letting the sphere go,find the maximum length by which the spring extends,given the sphere only rolls.

$A$ spring of force constant $10\, N/m$ has an initial stretch $0.20\, m$. In changing the stretch to $0.25\, m$,the increase in potential energy is about.....$joule$.

An engine is attached to a wagon through a shock absorber of length $1.5 \, m$. The system with a total mass of $50,000 \, kg$ is moving with a speed of $36 \, km \, h^{-1}$ when the brakes are applied to bring it to rest. In the process of the system being brought to rest,the spring of the shock absorber gets compressed by $1.0 \, m$. If $90 \%$ of the energy of the wagon is lost due to friction,calculate the spring constant.

If a long spring is stretched by $0.02\, m$,its potential energy is $U$. If the spring is stretched by $0.1\, m$,then its potential energy will be

$A$ block of mass $m$ is hung vertically from an elastic thread of force constant $\frac{2mg}{a}$. Initially,the thread is at its natural length and the block is allowed to fall freely. The work done on the block by the Earth when it passes through the equilibrium position will be:

$A$ spring of force constant $K$ is first stretched by distance $a$ from its natural length and then further by distance $b$. The work done in stretching the part $b$ is .............