Energy in Simple Harmonic Motion Video Lessons

Concept

# Problem: An object attached to a spring experiences simple harmonic motion (define the potential energy of the object + spring system to be zero when it is in equilibrium). (a) When the displacement of the object is one-half the maximum amplitude A, what fraction of the total energy is kinetic and what fraction is potential? (b) At what displacement x is the total energy half kinetic and half potential?

###### FREE Expert Solution

The expression for the total energy of a simple harmonic oscillator is:

$\overline{){\mathbf{E}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{k}}{{\mathbf{A}}}^{{\mathbf{2}}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{m}}{{{\mathbf{v}}}_{\mathbf{m}\mathbf{a}\mathbf{x}}}^{{\mathbf{2}}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{k}}{{\mathbf{x}}}^{{\mathbf{2}}}{\mathbf{+}}\frac{\mathbf{1}}{\mathbf{2}}{\mathbf{m}}{{\mathbf{v}}}^{{\mathbf{2}}}}$

We are told that at the equilibrium position, U = (1/2)kx2 = 0

Consequently, U = 0 when K = Kmax (Equilibrium position) and K = 0 when U = Umax (Maximum displacement).

K is maximum at the equilibrium position where x = 0 and v = vmax.

U is maximum at the turning points where v = 0 and x = A.

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###### Problem Details

An object attached to a spring experiences simple harmonic motion (define the potential energy of the object + spring system to be zero when it is in equilibrium).

(a) When the displacement of the object is one-half the maximum amplitude A, what fraction of the total energy is kinetic and what fraction is potential?

(b) At what displacement x is the total energy half kinetic and half potential?