Question

Represent the Potential energy U(t), Kinetic energy K(t) and the total energy E
as functions of time t and as functions of position x for a linear harmonic oscillator.

Answer 1

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)

Answer 2

Answer:

sorry I don't know the answer