Discuss in detail the energy in simple harmonic motion.

Explanation

(i) Expression for potential Energy : for the simple harmonic motion, the force and the displacement are related by Hooke's law
`vecF=- k vecr`
`F=-kx " "...(1)`
the work done by the conservative force field is indepndent of path.
Calculation potential energy :
`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)k x' dx' =(1)/(2)k(x')^(2)]_(0)^(x)=(1)/(2)kx^(2) " "...(3)`
From equation
`omega= sqrt((k)/(m))"rad" s^(-1)`
By substituting the value of force constant `k= m omega^(2)` in equation (3) we get
`U(x)=(1)/(2)momega^(2)x^(2) " "...(4)`
where `omega` is natural frequency of theoscillating system. For the particle executing simple harmonic motion from equation,
`y=A sin omega t`
we get,
`x=A sin omega t`
`U(t)=(1)/(2)m omega^(2)A^(2) sin^(2) omega t " "....(5)`
This variaton of U is shown below.

(ii) Expression for Kinetic Energy : Kinetic energy
`KE=(1)/(2)mv_(x)^(2)=(1)/(2) m((dx)/(dt))^(2) " "...(6)`
Since the particle executes simple harmonic motion, from equation `y=A sin omega t`
`x=A sin omega t`
`:.` Velocity is
`v_(x)=(dx)/(dt)=A omega cos omega t`
`=Aomega sqrt(1-((x)/(A))^(2)) " "...(7)`
`v_(x)= sqrt(A^(2)-x^(2)) " " .....(8)`
`KE=(1)/(2) m omega^(2) A^(2) cos^(2) omega t " ".....(10)`
This variation with time is shown below.

(iii) Exprssion for Total Energy: Total energy is the sum of kinetic energy and potential energy.
`K=KE+U" ".....(11)`
`E=(1)/(2) m omega^(2)(A^(2)-x^(2))+(1)/(2) m omega^(2)x^(2) " ".... (12)`
Hence cancelling `x^(2)` term
`E=(1)/(2) m omega^(2) A^(2)` constant `" "....(13)`
Alternatively from equation (5) and quation (10), we get the total energy as
`E=(1)/(2) m omega^(2)A^(2)sin^(2) omega t+(1)/(2) m omega^(2) A^(2) cos^(2) omegat`
`=(1)/(2) m omega^(2)A^(2)(sin^(2) omegat+cos^(2) omegat)`
From trigonometry identity,
`(sin^(2)omegat+cos^(2) omegat)=1`
`E=(1)/(2) m omega^(2)A^(2)=A` constant
Which gives the law of consevation of total energy