- #1
jssamp
- 31
- 3
Homework Statement
A damped harmonic oscillator loses 6.0% of it's mechanical energy per cycle. (a) By what percentage does it's frequency differ from the natural frequency f[itex]_{0} = (\frac{1}{2\pi})\sqrt{\frac{k}{m}}[/itex]? (b) After how many periods will the amplitude have decreased to [itex]\frac{1}{e}[/itex] of it's original value?
Homework Equations
natural frequency
[itex]f_{0} = (\frac{1}{2\pi})\sqrt{\frac{k}{m}}[/itex]
damped frequency
[itex]f' = \frac{1}{2\pi}\sqrt{\frac{k}{m}-\frac{b^{2}}{4m^{2}}}[/itex]
displacement for lightly damped harmonic oscillator
[itex]x = Ae^{(\frac{-b}{2m})t}cos\omega't[/itex]
Total mechanical energy
[itex]E = \frac{1}{2}kA^{2} = \frac{1}{2}mv^{2}_{max}[/itex]
And I know the mean half life, [itex]\frac{2m}{b}[/itex] is the time until oscillations reach 1/e of original.
The Attempt at a Solution
I used the A^2 expression for E and the A decay term, [itex]Ae^{(\frac{-b}{2m})t}[/itex] ,said it loses 6% of E when A^2 = .94A^2 (original) or in other words when [itex]Ae^{(\frac{-b}{2m})t}[/itex] = [itex]\sqrt{0.94}[/itex]A
so, [itex]e^{(\frac{-b}{2m})t}[/itex] = [itex]\sqrt{.94}[/itex]
[itex]\frac{-b}{2m}t[/itex] = [itex]\frac{1}{2}[/itex]ln(.94)
t = [itex]\frac{-m}{b}[/itex]ln(.94)
But this is time and I need it to be one cycle so do I plug the period in for t?
T = 1/f or 2∏ ω?
This is where I'm stuck. The answer is (a) -1.21x10^-3 % and (b) 32.3 periods but I don't see how to clear the unknowns with what is given. I could just copy the answer down but I want to know how to solve it. If anybody can give me hint it would be a great result for my first post here.