# Simple harmonic motion - damping introduced

• Jozefina Gramatikova
In summary, the homework equation is ## A e^{-\frac{b}{2m} t}=\frac{1}{2} Ae^{-\frac{b}{2m} 0} ##, and the amplitude halves after 3.036 seconds.f

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## The Attempt at a Solution

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The damping coefficient is ## b ## in your formula. Do you know how to solve ## e^{-\frac{b}{2m}t}=\frac{1}{2} ## for ## t ## ? . ## \\ ## Edit: A google of the topic calls ## \gamma=\frac{b}{2m} ## the damping coefficient, but it is really a choice of terminology. By the units they give you, they are giving you ## b ##.

The damping coefficient is ## b ## in your formula. Do you know how to solve ## e^{-\frac{b}{2m}t}=\frac{1}{2} ## for ## t ## ? .
I don't know how much is x(t)

I don't know how much is x(t)
The sinusoidal oscillation is assumed to happen at a much higher frequency with small damping, so that the period of the oscillation ## T ## is quite short, and you don't need to consider the term ## \cos(\omega t ) ##. The amplitude is ## A e^{- \frac{b}{2m} t} ##.

Jozefina Gramatikova
The sinusoidal oscillation is assumed to happen at a much higher frequency with small damping, so that the period of the oscillation ## T ## is quite short, and you don't need to consider the term ## \cos(\omega t ) ##. The amplitude is ## A e^{- \frac{b}{2m} t} ##.
Ok, thank you and what about x(t)

Your (first equation) that you posted in part 2 of the OP for ## x(t) ## is correct. (You didn't list an equation for ## \omega' ## though). And your second equation, which I think is a damping ratio number is really not of prime interest here. It would help if you would state what the parameter is. I had to go googling for the second equation you posted, to see if it was correct.## \\ ## When they say, how long does it take for the amplitude to "halve" that means ## A e^{-\frac{b}{2m} t}=\frac{1}{2} Ae^{-\frac{b}{2m} 0} ##, so that ## e^{-\frac{b}{2m} t}=\frac{1}{2} ##.

Jozefina Gramatikova
Oh, I am sorry I didn't think an equation for w' would be helpful here. Thank you very much for your help!