# simple harmonic motion

by Eric_meyers
Tags: harmonic, motion, simple
 P: 68 1. The problem statement, all variables and given/known data "Two light springs have spring constants k1 and k2, respectively, and are used in a vertical orientation to support an object of mass m. Show that the angular frequency of small amplitude oscillations about the equilibrium state is [(k1 + k2)/m]^1/2 if the springs are in parallel, and [k1 k2/(k1 + k2)m]^1/2 if the springs are in series. " 2. Relevant equations f = -kx w = (k/m)^1/2 frequency = w/(2 * pi) 3. The attempt at a solution So I set up my fnet for the parallel one m * x'' = - (k1x + k2x) since in the parallel feature there are two distinct springs exerting two distinct forces however dividing through I'm left with x'' = - (k1x + k2x)/m I'm not quite sure how to use x'' to get to angular frequency or how to remove the negative sign.
 P: 595 For a simple harmonic oscillation, x"=-(w^2)x
 P: 1,345 Solutions to x'' = -[(k1+k2)/m]x can be written in the form x(t) = A*sin(wt+phi) so from what you know about the sine function if you can find the period, you can find the frequency, which will enable you to find the angular frequency...
P: 68

## simple harmonic motion

Oh I think I got it!

x'' + [(k1 + k2)/m]x = 0

And using the characteristic polynomial to solve this second order DE, gives me the solution

x(t) = A cos(wt + phi) if and only if w = [(k1 + k2)/m]^1/2

now for setting up the DE in the series case I'm having some difficulty would it be:

m * x'' = -k1 * k2 x ? Since you could treat both k1 and k2 acting as one k? errr..
P: 595
 Quote by Eric_meyers now for setting up the DE in the series case I'm having some difficulty would it be: m * x'' = -k1 * k2 x ? Since you could treat both k1 and k2 acting as one k? errr..
Sure you got the effective k for springs in series correct?
 P: 68 oh wait, if I want to combine the k in both springs into another constant I'm going to have to take the "center of mass" sort of speak for the spring stiffness - I forget the correct terminology... center of stiffness? m * x'' = -[(k1 * k2)/(k1 + k2)] x x '' + (k1 * k2)/[(k1 + k2) * m]x = 0 Using the characteristic equation I again get x(t) = A cos (wt - phi) if and only if w = {(k1 * k2)/[(k1 + k2) * m]}^1/2 Which of course is the answer. Thanks

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