Physics Answer the following in Brief

Derive an expression for period of a simple pendulum.

A simple pendulum of length \(1 m\) has mass \(10 g\), and oscillates freely with amplitude of \(5 cm\). Calculate its potential energy at extreme position.

Define linear S. H. M. Obtain differential equation of linear S. H. M.

From differential equation of linear S.H.M., obtain an expression for acceleration, velocity and displacement of a particle performing S.H.Μ.

Represent graphically the displacement, velocity and acceleration against time for a particle performing linear S.H.M., when it starts from the mean position.

A body of mass \(1 kg\) is made to oscillate on a spring of force constant \(25+10^3\) dyne/cm. Calculate the magnitude of angular velocity and frequency of vibrations of the body.

A clock regulated by seconds pendulum, keeps correct time, During summer, length of pendulum increases to \(1.005 m\). How much will the clock gain or loose in one day? \(\left(9-9.8 m / s ^2\right.\) and \(\left.\pi=3.142\right)\)

State the differential equation of linear S.H.M. Hence, obtain expression for:
(a) acceleration (b) velocity.

State an expression for K.E. (kinetic energy) and P. E. (potential energy) at displacement 'x' for a particle performing linear S.H.M. Represent them graphically. Find the displacement at which K. E. is equal to P. E.

When the length of a simple pendulum is decreased by \(20 cm\), the period changes by \(10 \%\). Find the original length of the pendulum.