A block of mass $M_1$ resting on a frictionless horizontal surface connected to a spring of spring constant $k$ that is anchored to a wall. A block of mass $M_2=\alpha M_1$ is placed on the top of the first block. The two bodies move as a unit with S.H.M. What is the maximum amplitude of oscillation so that the two bodies move a unit? The coefficient of friction between two bodies is $\mu$.where $\alpha$ is a constant.

A load of mass $m$ falls from a height $h$ on the scale pan hung from a spring as shown. If the spring constant is $k$ and the mass of the scale pan is zero and the mass $m$ does not bounce relative to the pan, then the amplitude of vibration is 

An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass $\mathit{\text{M}}$. The piston and the cylinder have equal cross-sectional area $\mathit{\text{A}}$. When the piston is in equilibrium, the volume of the gas is $V_0$ and its pressure is $M_0$. The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frequency
[Assume the system is in space.]

A simple pendulum suspended from the ceiling of a stationary lift has a time period $T_0$. When the lift descends at uniform speed, the time period is $T_1$. When the lift descends with constant acceleration, the time period is $T_2$. Which of the following is correct?

Two point-like objects of masses $20 \mathrm{gm}$ and $30 \mathrm{gm}$ are fixed at the two ends of a rigid massless rod of length $10 \mathrm{cm}$. This system is suspended vertically from a rigid ceiling using a thin wire attached to its center of mass, as shown in the figure. The resulting torsional pendulum undergoes small oscillations. The torsional constant of the wire is $1.2\times {10}^{-8} \mathrm{N} \mathrm{m} {\mathrm{rad}}^{-1}$. The angular frequency of the oscillations in $n\times {10}^{-3} \mathrm{rad} {\mathrm{s}}^{-1}$. The value of $n$ is _____.