A particle of mass $m$ kept at origin is subjected to a force $\overrightarrow{\mathit{F}}=\left(\mathrm{p}\mathrm{t}-\mathrm{q}\mathrm{x}\right)\widehat{\mathrm{i}}$ where $\mathrm{t}$ : time elapsed and $\mathrm{x}:\mathrm{ }\mathrm{x}$ -coordinate of position of particle. Particle starts its motion at $\mathrm{t}=0$ with zero initial velocity. If $\mathrm{p}$ and $\mathrm{q}$ are positive constants, then -

A simple pendulum of length $l$ is made to oscillate with amplitude of $45$ degrees. The acceleration due to gravity is $g$. Let $T_0=2\mathrm{\pi }\sqrt{\frac{l}{g}}.$ The time period of oscillation of this pendulum will be

Match List$‐\mathrm{I}$ (Event) with List$‐\mathrm{II}$ (Order of the time interval for the happening of the event) and select the correct option from the options given below the lists.

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 

One end of a spring of negligible unstretched length and spring constant $k$ is fixed at the origin $(0, 0)$. A point particle of mass $m$ carrying a positive charge $q$ is attached at its other end. The entire system is kept on a smooth horizontal surface. When a point dipole $\overrightarrow{p}$ pointing towards the charge $q$ is fixed at the origin, the spring gets stretched to a length $l$ and attains a new equilibrium position (see figure below). If the point mass is now displaced slightly by $\mathrm{\Delta }l\ll l$ from its equilibrium position and released, it is found to oscillate at frequency $\frac{1}{\delta }\sqrt{\frac{k}{m}},$ The value of $\delta$ is _______.

A simple pendulum is attached to a block which slides without friction down an inclined plane $\left(ABC\right)$ having an angle of inclination $\alpha$ as shown while the block is sliding down the pendulum oscillates in such a way that at its mean position the direction of the string 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.]