(a) For the simple harmonic motion thr force and the displacement are realated by Hooke.s law `vecF=-kvecr`
Since force is a vertor 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 sca,lar funtion which has only one component. In one dimensional case `F=-kx`
We know that the work done by the conversation force field is independent of path . The potential energy U can br calulated from the following expression. `F=-(dU)/(dx)`
Comparing (1) and (2) we get `-(dU)/(dx)=-kx`
`dU=kx dx`
The 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)`
Where `omega` is the naturak frequency of the oscillating system. Force the particle executing simple harmonic motion from equation `y=Asin omegat`,
`U(t)=(1)/(2)momega^(2)A^(2)sin^(2)omegat` U.
(b) Expression for kinectic energy
`KE=(1)/(2)mv_(x)^(2)=(1)/(2)m((dx)/(dt))^(2)`
Since the particle is executing simple harmonic motion , from equation
`y=Asinomegat`
`x=Asinomegat`
Therefore, velocity is
`v_(x)=(dx)/(dt)=Aomega cos omegat`..
`Aomegasqrt1-(x/A)^(2)`
`v_(x)=omegasqrtA^(2)-x^(2)`
`KE=(1)/(2)momega^(2)A^(2)cos^(2)omegat`
(c) Expression for Total energy.
Total energy is the sum of kinetic energy and potential energy
`E=KE+U`
`E=(1)/(2)momega^(2)(A^(2)-x^(2))+(1)/(2)momega^(2)x^(2)` . . (11)
Hence, cancelling `x^(2)` term,
`E=(1)/(2)momega^(2)A^(2)=`Constant
. . (12) ,br> 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))`
