The spring is initially in unstretched condition. It is first stretched by a length '$x$' and the work done is $W_1$. Then again it is stretched by further length '$x$' when the work done is $W_2$. The value of $W_2$ is given by

A thin uniform rod AB of mass '$m$' and length '$l$' is hinged at one end A to the ground level. Initially the rod stands vertically and is allowed to fall freely to the ground in the vertical plane. The angular velocity of the rod when its B end strikes the ground is
($g$ = acceleration due to gravity)

The radius of gyration of a solid sphere of radius 'R' and mass 'M' about its diameter is $K_d$ and that about a tangent of a solid sphere is $K_t$. The ratio of $K_d$ to $K_t$ is

A geostationary satellite is orbiting the earth at a height of $4R$ above the surface of the earth, where $R$ is the radius of the earth. Another satellite is orbiting the earth at a height $1.5R$ from the surface of the earth with periodic time 'T' in hour. The value of 'T' in hour is