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Question
Deduce the expression for kinetic energy, potential energy, and total energy of a particle performing S.H.M. State the factors on which total energy depends.
Solution
- Expression for kinetic energy:
a. Consider a particle of mass m, performing a linear S.H.M. along the path MN about the mean position O as shown in the figure.
Energy in an S.H.M.
b. At a given instant, let the particle be at P, at a distance x from O.
c. Velocity of the particle in S.H.M. is given as
v = ω`sqrt("A"^2 - "x"^2)` = Aω cos(ωt + Φ)
where x is the displacement of the particle performing S.H.M. and A is the amplitude of S.H.M.
d. Thus, the kinetic energy,
Ek = `1/2"mv"^2`
= `1/2"m"ω^2("A"^2 - "x"^2)` ............(1)
= `1/2"k"("A"^2 - "x"^2)`
This is the kinetic energy at displacement x.
e. Also, at time t, kinetic energy is,
Ek = `1/2"mv"^2 = 1/2`mA2ω2cos2(ωt + Φ)
= `1/2`kA2cos2(ωt + Φ)
Thus, with time, it varies as cos2θ. - Expression for potential energy:
a. The restoring force acting on the particle at point P is given by, f = –kx where k is the force constant.
b. Suppose that the particle is displaced further by an infinitesimal displacement ‘dx’ against the restoring force ‘f’.
c. The external work done (dW) during this displacement is dW = f(–dx) = –kx(–dx) = kxdx
d. The total work done on the particle to displace it from O to P is given by,
W = `int_0^x "dW" = int_0^x "kx" "dx" = 1/2"kx"^2`
e. This work done is stored as the potential energy (P.E.) Ep of the particle at displacement x.
∴ Ep = `1/2 "kx"^2 = 1/2 "m"omega^2"x"^2` .....(2)
f. At time t,
∴ Ep = `1/2 "kx"^2 = 1/2"kA"^2 sin^2(omega"t" + phi) = 1/2"mA"^2omega^2cos^2(omega"t" + phi)`Thus, with time, it varies as sin2θ. - Expression for total energy:
a. The total energy of the particle is the sum of its kinetic energy and potential energy.
∴ E = Ek + Ep
b. Using equation (1) and equation (2), we get
E = `1/2`mω2(A2 - x2) + `1/2`mω2x2
E = `1/2` mω2A2 = `1/2`kA2 = `1/2`m(vmax)2
This expression gives the total energy of the particle at point P. - The total energy in S.H.M. is
a. directly proportional to
1. the mass of the particle
2. the square of the amplitude
3. the square of the frequency
4. the force constant
b. inversely proportional to the square of the period.
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