Discuss in detail the energy in simple harmonic motion.

Explanation

Expression for potential Energy For the simple harmonic motion , the force and the displacement are related by Hooke's law
`vec(F)=-k bar(r)`
Since Force is a vector quantity it has three compounds.
(i) The force in the above equation is a conservation force field. such a force can be derived from a scalar function which has only one component from the following expression.
`F=-kx" "...(1)`
(ii) The work done by the conservative force field in 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`
(iii) This work done by the force During a small displacement dx stores as potential energy
`U(x)=int_(0)^(x)kx'dx'`
`=1/2k(x') |_(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)=1/2msquare^(2)x^(2)`
where `omega` is the natural frequency of the oscillating system . For the particle executing simple harmonic motion from equation `y=asin square t,`
We get `x=Asinsquare t`
`U(t) =1/2 msquare^(2)A^(2)sin^(2)omegat" "...(4)`
this variation of U is shown below.
Variation of potential energy with time t

(b) Expression for Kinetic Energy
Kinetic Energy `=1/2mv_(2)^(2)=1/2m((dx)/(dt))^(2)`
(i) Since the particle is executing simple harmonic motion `x=A sinsquare t`
`v_(x)=dx/dt=Asquarecossquaret=Aomegasqrt(1-((x)/(A))^(2))`
`v_(x)=omegasqrt(A^(2)-x^(2))" "...(5)`
Hence,
`KE=1/2mv_(x)^(2)=1/2momega^(2)(A^(2)-x^(2))" "...(6)`
`KE=1/2momega^(2)A^(2)cos^(2)omegat" "...(7)`

Variation of kinetic energy with time t
(c) Expression for Total Energy
(i) Total energy is the sum of kinetic energy and potential energy
`E=KE+U" "..(8)`
`E=1/2momega^(2)(A^(2)-x^(2))+(1)/(2)momega^(2)x^(2)`
Hence canceling `x^(2)` term,
`E=1/2momega^(2)A^(2)= "constant " ...(9)`
Alternatively , from equation (4) and equation (7), we get the total energy as
`E=1/2momega^(2)A^(2)sin^(2)omegat+(1)/(2)momega^(2)A^(2)cos^(2)omegat`
`1/2msquare^(2)A^(2)(sin^(2)squaret+cos^(2)squaret)`
which gives the law of conservation of total energy.

(iv) Thus, the amplitude of simple harmonic oscillator, can be expressed in terms of total energy .
`A=sqrt((2E)/(momega^(2)))=sqrt((2E)/(k))`