Simple Harmonic Motion - Finding the Phase

[jumbogala]

Homework Statement

Here is a position-time graph for a ball on a spring in SHM.

What is the phase of the particle at point a on the graph?

Homework Equations

phase = ωt + ɸ₀

The Attempt at a Solution

First I found the initial phase, ɸ₀. 0.5A / A = 0.5 = cos(ɸ₀). Then ɸ₀ = -π/3. It's negative because the particle is moving to the right.

Now for point a, I figured this is at 0.75T, where T is the period.This is kind of hard to see so it could be wrong. But then:

phase = (2π/T)(0.75T) - (π/3) = (7/6)π or 210 degrees. However, the answer is -120 degrees.

I also tried with points b and c but they were wrong too. Can anyone help?

 

[vela]

Your assumption that t_a=0.75 T is wrong. What's the increase in phase going from the very left to the first maximum? Similarly, what's the increase in phase from the first minimum to point a?

 

[jumbogala]

I'm not sure how to figure out the increase in phase directly... does it depend on knowing time still?

Because I'm confused how my assumption that t at a = 0.75T is wrong. The time from the first crest to the first trough is definitely 0.5T, right? And the time from 0 to a trough or crest is 0.25T. So the time from 0 to a should be 0.5(0.25T) = (1/8)T...

 

[vela]

The time from 0 to the first crest is not 0.25 T. Think about the unit circle. What angles gives you cos x1=0.5 and cos x2=1? What's the difference x2-x1? How much time does that increase correspond to?

 

[jumbogala]

x1 = 60
x2 = 0

x2 - x1 = 60

So... 360 degrees corresponds to one revolution (T), then 60/360 is the amount of time?

But that can't be right either, because (2pi / T)(60/360)T - (pi/3) = 0

 

[vela]

That wasn't meant to give you the answer directly. It was to get you to see that the time from t=0 (-60 degrees) to get to the crest (0 degrees) isn't 0.25 T as you were assuming.

 

[jumbogala]

Ah, I figured it out - I think this is right now:

So I need to solve the equation cos((2pi/T)t - (pi/3)) = -0.5

cos(x) = -0.5 when x = 120 and 240, 480 and 600, etc.

So the answer is 240 degrees (same as -120 degrees).

But if the question didn't show the diagram, how would you know which angle to use (120 or 240?)

 

[vela]

You wouldn't be able to because without additional information, like the diagram, you don't know in which direction the particle is moving.

 

[jumbogala]

Okay, that makes sense :)

Thanks for your help!