Discuss in detail the energy in simple harmonic motion.

Explanation

(a) Expression for Potential Energy
For the simple harmonic motion, the force and the displacement are related by Hooke.s law
`vecF=-kvecr`
Since force is a vector quantity, in three dimensions it has three components. Further, the force in the above equation is a conservative force field, such a force can be derived from a scalar function which has only component. In one dimensional case
`F=-kx" "...(1)`
The work done by the conservative force fieldis independent of path. The potential energy U can be calculated from the following expression.
`F=-(dU)/(dx)" "...(2)`
Comparing (1) and (2), we get
`-(dU)/(dx)=-kx`
`dU=-kx dx`

This work done by the force F during a small displacement dx stores as potential energy
`U(x)=int_(0)^(x)kx.dx.=(1)/(2)k(x.)^(2)|_(0)^(x)=(1)/(2)kx^(2)" "...(3)`
From equation `omega=sqrt((k)/(m))`, we can substitute the value of force constant `k=momega^(2)` in equation (3),
`U(x)=(1)/(2)momega^(2)x^(2)" "...(4)`
where `omega` is the natural frequency of the oscillating system. For the particle executing simple harmonic motion from equation `y=Asinomegat`, we get
`x=Asinomegat`
`U(t)=(1)/(2)momega^(2)A^(2)sin^(2)omegat" "...(5)`
This variation of U is shown in figure.
(b) Expression for Kinetic Energy
Kinetic energy
`KE=(1)/(2)mv_(x)^(2)=(1)/(2)m((dx)/(dt))^(2)" "...(6)`
Since the particle is executing simple harmonic motion, from equation
`y=Asinomegat`
`x=Asinomegat`
Therefore, velocity is
`v_(x)=(dx)/(dt)=Aomegacosomegat" "...(7)`
`=Aomegasqrt(1-((x)/(A))^(2))`
`v_(x)=omegasqrt(A^(2)-x^(2))" "...(8)`
Hence, `KE=(1)/(2)mv_(x)^(2)=(1)/(2)momega^(2)(A^(2)-x^(2))" "...(9)`
`KE=(1)/(2)momega^(2)A^(2)cos^(2)omegat" "...(10)`
This variation with time is shown in figure.
(c ) Expression for Total Energy
Total energy is the sum of kinetic energy and potential energy
`E=KE+U" "...(11)`

`E=(1)/(2)momega^(2)(A^(2)-x^(2))+(1)/(2)momega^(2)x^(2)`
Hence, cancelling `x^(2)` term,
`E=(1)/(2)momega^(2)A^(2)=` constant `" "...(12)`
Alternatively, from equation (5) and equation (10), we get the total energy as
`E=(1)/(2)momega^(2)A^(2)sin^(2)omegat+(1)/(2)momega^(2)A^(2)cos^(2)omegat`
`=(1)/(2)momega^(2)A^(2)(sin^(2)omegat+cos^(2)omegat)`
From trigonometry identify, `(sin^(2)omegat+cos^(2)omegat)=1`
`E=(1)/(2)momega^(2)A^(2)` = constant
which gives the law of conservation of total energy
Thus the amplitude of simple harmonic oscillator, can be expressed in terms of total energy.
`A=sqrt((2E)/(momega^(2)))=sqrt((2E)/(k))" "...(13)`