Oscillations and Waves
Energy Changes During Simple
Harmonic Motion
Energy in SHM
energy
velocity
Energy-time graphs
KE
PE
Total
Note: For a spring-mass system:
KE = ½ mv2  KE is zero when v = 0
PE = ½ kx2  PE is zero when x = 0 (i.e. at vmax)
Energy–displacement graphs
energy
KE
PE
Total
-xo
displacement
+xo
Note: For a spring-mass system:
KE = ½ mv2  KE is zero when v = 0 (i.e. at xo)
PE = ½ kx2  PE is zero when x = 0
Kinetic energy in SHM
We know that the velocity at any time is given by…
v = ω √ (xo2 – x2)
So if Ek = ½ mv2 then kinetic energy at an instant is
given by…
Ek = ½ mω2 (xo2 – x2)
Potential energy in SHM
If
a = - ω2 x
then the average force applied trying to pull the
object back to the equilibrium position as it moves
away from the equilibrium position is…
F = - ½ mω2x
Work done by this force must equal the PE it gains
(e.g in the springs being stretched). Thus..
Ep = ½ mω2x2
Total Energy in SHM
Clearly if we add the formulae for KE and PE in
SHM we arrive at a formula for total energy in SHM:
ET = ½ mω2xo2
Summary:
Ek = ½ mω2 (xo2 – x2)
Ep = ½ mω2x2
ET = ½ mω2xo2
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