A weight is hung freely from the end of a spring. A boy then slowly pushes the weight upwards until the spring become slack. The gain in gravitational energy of the weight during this process is equal to:

 

A block attached to spring, pulled by a constant horizontal force $F$, is kept on a smooth surface as shown in the figure. Initially, the spring is in its natural state. Then the maximum positive work that the applied force $F$ can do is :[ Given that spring does not break]

In a simple pendulum, the breaking strength of the string is double the weight of the bob. The bob is released from rest when the string is horizontal. The string breaks when it makes an angle $\mathrm{\theta }$ with the vertical:

The elastic collision between two bodies, $A$ and $B$, can be considered using the following model. $A$ and $B$ are free to move along a common line without friction. When their distance is greater than $d=1 \mathrm{m}$, the interacting force is zero; when their distance is less than $d$, a constant repulsive force $F=6 \mathrm{N}$ is present. The mass of body $A$ is $m_A=1\mathrm{kg}$ and it is initially at rest; the mass of body $B$ is $m_B=3\mathrm{kg}$ and it is approaching body $A$ head-on with a speed $v_0=2 \mathrm{m} {\mathrm{s}}^{-1}$. Find the minimum distance between $A$ and $B$:-