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1.3 a) Variation with time of energy in simple harmonic motion
If the variation with time of displacement is as shown, then the energies should be drawn as shown.
recalling Energy formula
PE = ½ k x2
in terms of time t,
x = x0 sin(ωt)
differentiating with t gives
v = v0 cos (ωt)
therefore, KE = ½ m v2= ½ m (v0 cos (ωt))2= ½ m (x02ω2)cos (ωt))2similarly
PE = ½ k x2= ½ (mω2 )(x0 sin (ωt))2= ½ m (x02ω2 )sin (ωt))2
therefore total energy is a constant value in the absence of energy loss due to drag (resistance)
TE = KE + PE = ½ m (x02ω2 )[cos2(ωt)
+ sin2(ωt))] = ½ m (x02ω2)
this is how the x vs t looks together of the energy vs t graphs
1.3.1 Summary
the table shows some of the common valuesgeneral energy formula | SHM energy formula | when t = 0 | when t = T/4 | when t = T/2 | when t = 3T/4 | when t = T |
KE = ½ m v2 | ½ m (x02ω2)cos (ωt))2 | ½ m (x02ω2) | 0 | ½ m (x02ω2) | 0 | ½ m (x02ω2) |
PE = ½ k x2 | ½ m (x02ω2)cos (ωt))2 | 0 | ½ m (x02ω2) | 0 | ½ m (x02ω2) | 0 |
TE = KE + PE | TE = ½ m (x02ω2) | ½ m (x02ω2) | ½ m (x02ω2) | ½ m (x02ω2) | ½ m (x02ω2) | ½ m (x02ω2) |
1.3.2 Model:
Translations
Code | Language | Translator | Run | |
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- Parent Category: 02 Newtonian Mechanics
- Category: 09 Oscillations
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