Obtain the expressions of kinetic energy potential energy and total energy in simple harmonic motion.

Explanation

Kinetic energy : Displacement of SHM particle at any instant
`x= A cos (omega t +phi)`
where A = amplitude, `omega`= angular frequency
velocity `v= (d(x))/(dt)= (d)/(dt)[ A cos (omega t+ phi)]`
`therefore v= -A omega sin (omega t+phi)`
Now kinetic energy,
`K= (1)/(2) mv^(2)`
`=(1)/(2) mA^(2) omega^(2) sin^(2) (omega t+phi)`
`therefore K= (1)/(2) kA^(2) sin^(2) (omega t+phi)" ""......"(1) [therefore m omega^(2) = k]`
Hence, it is a periodic function of time, being zero when the displacement is maximuml and maximuml when the particle is at the mean position.
The period of kinetic energy is `(T)/(2)`
Potential energy : Potential energy is possible only for conservative forces and restoring force is the conservative force in SHM.
If the restoring force exerted on a particle of SHM from its mean position is x then
`F= -kx`
The work opposite to restoring force for small displacement dx of a particle is
`dW= -F dx`
`therefore dW = -kx dx`
Now work done for displacement of particle from `x=0` to x=x,
`W = int dW`
`= int_(0)^(x) -kx dx`
`= -k[(x^2)/(2)]_(0)^(x)`
`= -(kx^2)/(2)`
`= -(1)/(2) kx^(2)`
The work done opposite to restoring force is stored as potential energy in particle.
Potential energy in displacement x of particle is
`U = -W`
`=(1)/(2) kx^(2)`
`= (1)/(2)m omega^(2) x^(2)`
`=(1)/(2) m omega^(2) A^(2) cos^(2) (omega t+ phi)`
`therefore U= (1)/(2) kA^(2) cos^(2) (omega t +phi)`
Hence, the potential energy of a particle executing simple harmonic motion is also periodic.
At mean position potential energy is zero and at extreme points it is maximum. Hence, the period of potential energy is `(T)/(2)`.
Total energy : The sum of kinetic and potential energy of SHM is knownf as total energy E.
`therefore E= K+U`
`therefore E= (1)/(2) kA^(2) sin^(2)( omega t+phi) +(1)/(2) kA^(2) cos^(2) (omega t+phi)`
`therefore E= (1)/(2) kA^(2) [sin^(2) (omega t+phi)+ cos^(2) (omega t+phi)]`
`=(1)/(2) kA^(2) [therefore sin^(2) (omega t+phi)+ cos^(2) (omega t+phi)=1]`
or `E= (1)/(2) m omega^(2) A^(2)`
Hence, `E propto m, E propto omega^(2)" and "E propto A^(2)` Thus, mechanical energy is proportional to mass and square of angular frequency and varies directly to the square of amplitude but mechnical energy does not depend on the displacement and time.