Obtain an expression for the kinetic energy of a particle performing linear SHM. State how it depends upon the amplitude, frequency and period of SHM.

Explanation

Consider a particle of mass m performing linear SHM with amilitude A. The restoring force acting on the particle is `F=-kx`, where k is the force constant and x is the displacement of the particle from its mean position.
As `omega=2pif=(2pi)/(T)`, where f and T are respectively the frequency and period of SHM,
`KE=(1)/(2)momega^(2)(A^(2)-x^(2))=(1)/(2)m(4pi^(2)f^(2))(A^(2)-x^(2))`
`=2 m pi^(2)f^(2)(A^(2)-x^(2))=(2mpi^(2)(A^(2)-x^(2)))/(T^(2))`
Thus, `KE prop A^(2) prop f^(2)` and `KE prop (1)/(T^(2))`
[Note : Total energy : The total energy of a particle performing linear SHM is
`E=PE+KE=(1)/(2)kx^(2)+(1)/(2)k(A^(2)-x^(2))=(1)/(2)kx^(2)+(1)/(2)kA^(2)-(1)/(2)kA^(2)`
`=(1)/(2)momega^(2)=2mpi^(2)f^(2)A^(2)=(2mpi^(2))/(T^(2))A^(2)`
`:. E prop A^(2), E prop f^(2), E prop (1)/(T^(2))`]