$A$ 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.

The potential energy of a certain spring when stretched through a distance $S$ is $10 \, J$. The amount of work (in $J$) that must be done on this spring to stretch it through an additional distance $S$ will be

$A$ mass of $1\, kg$ is hanging from a spring of spring constant $1\, N/m$. If Saroj pulls the mass down by $2\, m$,the work done by Saroj is......$J$.

$A$ block of mass $m$ moving at a speed $v$ compresses a spring through a distance $x$ before its speed becomes one fourth. Find the spring constant of the spring.

$A$ spring has a natural length $l$ with one end fixed to the ceiling. The other end is fitted with a smooth ring which can slide on a horizontal rod fixed at distance $l$ below the ceiling. Initially, the spring makes an angle of $60^{\circ}$ with the vertical, when the system is released from rest. Find the angle of the spring with the vertical, when the velocity of the ring reaches half of the maximum velocity, which the ring can attain during the motion.

When equal forces are applied to two springs with spring constants $1500 \, N/m$ and $3000 \, N/m$,what is the ratio of their stored potential energies?