What are periodic motion, Oscillatory motions and simple harmonic motion? Explain that all oscillatory motions are periodic but all periodic motions are not oscillatory.

A particle is executing SHM. If ${\mathrm{v}}_{1 ,} {\mathrm{v}}_2$ are the velocities of particle and ${\mathrm{y}}_{1,} {\mathrm{y}}_2$ are the distances from the mean position, then prove that its time period is given by $\mathrm{T}=2\mathrm{\pi }\sqrt{\frac{{\mathrm{y}}_1^2 -{\mathrm{y}}_2^2}{{\mathrm{v}}_1^2-{\mathrm{v}}_2^2}}$

A body of mass $\mathrm{m}$ is attached to the one end of an ideal spring of force constant $\mathrm{k}$ is executing SHM. Establish that the time period of oscillation is $\mathrm{T}=2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{\mathrm{k}}}$.

Derive the equation for the kinetic energy and potential energy of a body executing SHM and show that the total energy of a particle is proportional to the square of the amplitude

What do you understand by simple harmonic motion? Obtain an expression for the time period of simple pendulum. Show that the time period of a simple pendulum of infinite length is limited to $84.6 \min .$