Adding three SHM oscillations of equal frequency?

[applestrudle]

Homework Statement

A particle is simultaneously subjected to three SHM, all of the same frequency, and in the x direction. If the amplitudes are 0.25, 0.20, 0.15mm, respectively, and the phase difference between the 1st and 2nd is 45 degrees, and between the 2nd and 3rd is 30 degrees, find the amplitude of the resultant displacement and it's phase relative to the first (0.25mm amplitude) component.

Homework Equations

z = x + iy

z1 = x1 + iy1

x1 = 0.25 cos(wt +0)
x2 = 0.20 cos(wt +45)
x3 = 0.15 cos(wt +75)

z1 = 0.25 [x1 + i sin(wt +0)]
z2 = 0.2 [x2 + i sin(wt +45)]
z3 = 0.15 [x3 + i sin(wt +75)]

The Attempt at a Solution

Firstly, I assumed there was no initial phase angle for z1 said at t=0

x1 = 0.25
x2 = 0.20 cos(45)
x3 = 0.15 cos(75)

therefore the Amplitude must be x = x1 + x2 + x3 (as at t=0 it must be at maximum displacement so x = A)

I got x = 0.430244mm
the answers says 52mm

then i decided maybe at t=0 x is not equal to A (although I don't know why ) so i tried to calculate the magnitude of z.

z = (x^2 +y^2)^0.5

i added up all the y components at t =0 and got y = 0.286310mm

therefore z = 0.516

now for the angle:

I used arctan(y/x) = 56 degrees, the answer is ~33.5 degrees

 

[tiny-tim]

hi applestrudle!

x1 = 0.25 cos(wt +0)
x2 = 0.20 cos(wt +45)
x3 = 0.15 cos(wt +75)

expand, and add: you should get Acosωt + Bsinωt …

then convert that into amplitude*cos(ωt + phase)

 

[haruspex]

z = x + iy

I'm not sure whether it helps to introduce an iy component. Maybe.

z1 = 0.25 [x1 + i sin(wt +0)]
z2 = 0.2 [x2 + i sin(wt +45)]
z3 = 0.15 [x3 + i sin(wt +75)]

The x1, x2, x3 already have the amplitude factor. I think you mean
z1 = [x1 + 0.25 i sin(wt +0)]
z2 = [x2 + 0.2 i sin(wt +45)]
z3 = [x3 + 0.15 i sin(wt +75)]

x1 = 0.25
x2 = 0.20 cos(45)
x3 = 0.15 cos(75)

therefore the Amplitude must be x = x1 + x2 + x3 (as at t=0 it must be at maximum displacement so x = A)

That's only the instantaneous magnitude of x at time 0. It might achieve a greater magnitude at some other time. The amplitude is the peak magnitude.

so i tried to calculate the magnitude of z.

Same problem.
As tiny-tim posted, you can just expand each cos(ωt+ψ) as cos(ωt)cos(ψ) - sin(ωt)sin(ψ).

 

[gneill]

The frequencies are all the same so you can treat the components as phasors and add accordingly.

 

[rude man]

I would go with post #2 method.

Add the coefficients of sin(wt), add the coeff. of cos(wt), giving
a sin(wt) + b cos (wt) = c sin(wt + phi).