A loaded vertical spring executes $\mathrm{SHM}$ with period of $4 \mathrm{s}.$ The difference between the kinetic energy and the potential energy of this system oscillates with a period of

The coefficient of friction between block of mass $m$ and $2\mathrm{m}$ is $\mu =2\tan \left(\theta \right)$. There is no friction between a block of mass $2\mathrm{m}$ and inclined plane. The maximum amplitude of two block system for which there is no relative motion between both the blocks-

A constant force produces maximum velocity $v$ on the block connected to the spring of force constant $k$ as shown in the figure. When the force constant of spring becomes $4\mathrm{k},$ the maximum velocity of the block is (block is at rest when spring is relaxed):

A system is shown in the figure. The time period for small oscillations of the two blocks will be

A block of mass $\text{'}m\text{'}$ is attached with two springs of spring constant $\text{'}k\text{'}$ is performing $\mathrm{SHM}$ on a smooth horizontal surface. One of the spring is cut when the block is at the extreme position. Find the ratio of the amplitude of new $\mathrm{SHM}$ and old $\mathrm{SHM}.$

A block of mass $\text{'}m\text{'}$ is suspended from a spring and executes vertical $\mathrm{SHM}$ of time period $T$ as shown in figure. The amplitude of the $\mathrm{SHM}$ is $A$ and spring is never in compressed state during the oscillation. The magnitude of the minimum force exerted by spring on the block is

A horizontal rod of mass $m$ and length $L$ is pivoted smoothly at one end. The rod's other end is supported by a spring of force constant $k.$ The rod is rotated (in vertical plane) by a small angle $\theta$ from its horizontal equilibrium position and released.

The angular frequency of the subsequent simple harmonic motion is:

A simple pendulum $50 \mathrm{cm}$ long is suspended from the roof of a cart accelerating in the horizontal direction with constant acceleration $\sqrt{3} g \mathrm{m} {\mathrm{s}}^{-2}.$ The period of small oscillations of the pendulum about its equilibrium position is $\left(g={\mathrm{\pi }}^2 \mathrm{m} {\mathrm{s}}^{-2}\right)$: