Question

Deduce the expressions for the kinetic
energy and potential energy of a particle
executing S.H.M. Hence obtain the
expression for total energy of a particle
performing S.H.M and show that the
total energy is conserved. state that
factors on which total energy depends.​

Answer 1

Answer:

Acceleration of the particle , performing S.H.M is given by α=−ω2y

where ω is the angular velocity, and y is the displacement of particle. 

now, workdone by particle = F.dy 

as we know, acceleration and displacement are in opposite directions in case of S.H.M

so, W=−mω2ydy 

where m is the mass of the particle. 

W=−mω2∫ydy

W=−21mω2y2

so, potential energy = -W 

=21mω2y2

we know, ω=2πη 

so, P.E=2π2η2my2 ......(1) 

velocity of particle , v=ωAcosωt

or, v=ωA2−y2 

so, kinetic energy of particle, K.E=21mv2 

hence, K.E=21mω2(A2−y2)

but ω=2πη