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# Damped Harmonic Motion watch

1. Need any good physicists/mathematicians to give me a hand here...

(i) What is the general solution to d^2y/dx^2 + 2k dy/dx + x = 0 when 0 < k < 1?

This bit I can do, I get y = e^-k (A cos(1-k^2)^(1/2) x + B sin(1-k^2)^(1/2) x).

(ii) We know that the above differential equation represents damped simpleharmonic motion with dampening factor k. Also we are given that x(0) = 0 (from this I deduced A = 0).

Let x1, x2, x3 .... xn be the successive turning points such that if xn is a maximum, xn+1 is the following minimum.

Show that the ratio |x(n+1) / xn| takes a value a, independent of n, and furthermore that we can show that k^2 = (ln a)^2/(pi^2 + (ln a)^2), where a is independent.

For this last bit I've completely stuffed, I don't do A-Level physics, and havn't done M4 yet, which has damped SHM in, so I don't really know how to start....
2. (Original post by theone)
Need any good physicists/mathematicians to give me a hand here...

(i) What is the general solution to d^2y/dx^2 + 2k dy/dx + x = 0 when 0 < k < 1?

This bit I can do, I get y = e^-k (A cos(1-k^2)^(1/2) x + B sin(1-k^2)^(1/2) x).

(ii) We know that the above differential equation represents damped simpleharmonic motion with dampening factor k. Also we are given that x(0) = 0 (from this I deduced A = 0).

Let x1, x2, x3 .... xn be the successive turning points such that if xn is a maximum, xn+1 is the following minimum.

Show that the ratio |x(n+1) / xn| takes a value a, independent of n, and furthermore that we can show that k^2 = (ln a)^2/(pi^2 + (ln a)^2), where a is independent.

For this last bit I've completely stuffed, I don't do A-Level physics, and havn't done M4 yet, which has damped SHM in, so I don't really know how to start....
where did you get this question if you don't do physics or P4?
3. (Original post by Unreg)
where did you get this question if you don't do physics or P4?
I do F Maths, and this is a STEP III Pure maths question...
4. I would start by finding dy/dx = 0 to find where the turning points are.
5. (Original post by Unreg)
I would start by finding dy/dx = 0 to find where the turning points are.
But from what I've done it doesn't get anywhere.
6. Sorry to bump this but I need a hand
7. (Original post by theone)
Sorry to bump this but I need a hand
If i had a pen and paper i would help you. plus im going in 5 minutes to the cinema for you-know-what.

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