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# Explain your reasoning continued 3 phys 2210 5 energy

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Unformatted text preview: tarting at point P) were instead {overdamped, critically damped}. Explain your reasoning CONTINUED –3– PHYS 2210 5. Energy considerations in a 1D simple harmonic oscillator: (a) Consider a simple harmonic oscillator with period τ . Let ￿f ￿ denote the average value of a function f (t) averaged over one complete cycle: ￿ 1τ ￿f ￿ = f (t)dt. (1) τ0 Prove that ￿T ￿=￿U ￿ = E/2, where E is the total energy of the oscillator, T the kinetic energy and U the potential energy. Hint Start by proving the more general, and extremely useful, result that ￿sin2 (ω t − δ )￿ = ￿cos2 (ω t − δ )￿ = 1/2. Explain why these two results are almost obvious, then prove them by using trig identities to rewrite sin2 θ and cos2 θ in terms of cos(2θ). (b) If an additional damping force Fdamp = −bx is added, ﬁnd the rate of change of the energy E = ˙ 1...
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