$A$ particle is moving in a circle of radius $r$ under the action of a force $F = \alpha r^2$ which is directed towards the centre of the circle. The total mechanical energy (kinetic energy + potential energy) of the particle is (take potential energy $= 0$ for $r = 0$).

$A$ person of mass $60 \, kg$ wants to lose $5 \, kg$ by going up and down a $10 \, m$ high stairs. Assume he burns twice as much fat while going up than coming down. If $1 \, kg$ of fat is burnt on expending $7000 \, kcal$,how many times must he go up and down to reduce his weight by $5 \, kg$?

An engine is attached to a wagon through a shock absorber of length $1.5\, \text{m}$. The system with a total mass of $40,000\, \text{kg}$ is moving with a speed of $72\, \text{km/h}$ 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 $1.0\, \text{m}$. If $90\, \%$ of the energy of the wagon is lost due to friction,the spring constant is $....\, \times 10^{5}\, \text{N/m}$.

If a skater of mass $3 \,kg$ has an initial speed of $32 \,m/s$ and a second skater of mass $4 \,kg$ has a speed of $5 \,m/s$. After a perfectly inelastic collision,they move together with a speed of $5 \,m/s$. Calculate the loss in kinetic energy.

$A$ simple pendulum of length $1 \,m$ has a wooden bob of mass $M = 1 \,kg$. It is struck by a bullet of mass $m = 10^{-2} \,kg$ moving with a speed of $u = 2 \times 10^2 \,m/s$. The bullet gets embedded into the bob. The height to which the bob rises before swinging back is (use $g = 10 \,m/s^2$): (in $\,m$)

$A$ body of mass $2.9 \, kg$ is suspended from a string of length $2.5 \, m$ and is at rest. $A$ bullet of mass $100 \, g$ strikes the block horizontally with velocity $150 \, m/s$ and sticks to it. What is the maximum angle made by the string with the vertical after the impact? (Given $g = 10 \, m/s^2$)