An object is released from rest from the inner edge of a hemispherical bowl, and it falls under gravity. The coefficient of kinetic friction between the object and the bowl is $\mu$. If the object covers an angular displacement $\theta$ with respect to the center of the hemisphere when it stops for the first time, which of the following expressions is correct?

As shown in the figure a block is resting on a frictionless floor and is attached to the free end of a spring. The right edge of the block in equilibrium is at a distance $d$ from the wall. When the spring is compressed by a distance $d / 2$ and released, the time-period of the motion is $T$. Similarly, when compressed by a distance $2 d$ and released, the time-period is $T_0$. Considering elastic collision between the block and the wall, what is the value of the quantity $T_0 / T$ ?

A liquid of density $\rho$ in a container weighs $W$. A cubic block of side $L$ and density $\rho_b<\rho$ is pushed by a stick to completely submerge the block in the liquid without touching the bottom. If $g$ is the acceleration due to the gravity and liquid displaced by the stick is negligible, what is the new weight of the container as registered by the weighing machine below?

A monoatomic ideal gas at pressure $P$ and volume $V$ is first adiabatically compressed to volume $V / 8$ and then is allowed to expand isothermally back to its original volume. What is the final pressure of the gas?