The type of graph between length of a simple pendulum and time period 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,

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 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

The position co-ordinates of a particle moving in a $3D$ coordinate system is given by

$x=a\cos \mathrm{\omega }\mathrm{t}$

$y=a\sin \omega t$

and $z=a\omega t$

The speed of the particle is:

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