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.
factors on which total energy depends.
State the​

Answer 1

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11th

Physics

Oscillations

Energy in SHM

Obtain expressions for kine...

PHYSICS

Obtain expressions for kinetic energy, potential energy and total energy of a particle performing linear S.H.M.

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ANSWER

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

2

y

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

now, workdone by particle =

F

.

d

y

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

so, W=−mω

2

ydy

where m is the mass of the particle.

W=−mω

2

∫ydy

W=−

2

1

mω

2

y

2

so, potential energy = -W

=

2

1

mω

2

y

2

we know, ω=2πη

so, P.E=2π

2

η

2

my

2

......(1)

velocity of particle , v=ωAcosωt

or, v=ω

A

2

−y

2

so, kinetic energy of particle, K.E=

2

1

mv

2

hence, K.E=

2

1

mω

2

(A

2

−y

2

)

but ω=2πη

so, K.E=2π

2

η

2

m(A

2

−y

2

) ....(2)

so, total mechanical energy = K.E + P.E

=2π

2

η

2

mA

2

......(3)