# Damped Harmonic Motion

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#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....
0
17 years ago
#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?
0
#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...
0
17 years ago
#4
I would start by finding dy/dx = 0 to find where the turning points are.
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#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.
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#6
Sorry to bump this but I need a hand 0
17 years ago
#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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