Homework Help: A Particle Undergoes SHM

1. Jan 16, 2013

1. The problem statement, all variables and given/known data

A particle undergoes simple harmonic motion. It has velocity v1 when the displacement is x1 and velocity v2 when the displacement is x2. Find the angular frequency ω and amplitude A in terms of the given quantities.

2. Relevant equations

x = A sin (ωt + ∅ )

v = A ω cos ( ωt + ∅ )

3. The attempt at a solution

I tried starting x1 and v1 at t=0 s. This yields

x = A sin ( ∅ )

v = A ω cos ( ∅ )
The equations for position 2 included the ωt. I have four equations: two position and two velocity and four unknowns: A, ω, ∅, and t. I just need A and ω. Is this the right direction?

2. Jan 16, 2013

TSny

Welcome to PF! Your approach is fine. (Another approach is to consider energy relations.) What do you get if you divide your velocity equation above by ω and then square the equation? How does that compare with squaring the x equation? Can you see how to eliminate ∅ and t in one fell swoop?

3. Jan 16, 2013

x2 = A2 sin2 (ωt + ∅ )

(v/ω)2 = A2 cos2 ( ωt + ∅ )

I could say

A = √ (v/ω)2 + x2

I don't see how I can eliminate t and ∅ though

4. Jan 16, 2013

haruspex

Right. And you have two such equations, one for x1 and v1, and one for x2 and v2.
You just did.

5. Jan 16, 2013

Okay guys, I believe I have the right answer.

I solved for A in each set of x and v.

A = √ (v1/ω)2 + (x1)2 and A = √ (v2/ω)2 + (x2)2

Then I solved for ω in the second equation ( 2 )

I substituted this into the first equation. After a bunch of algebra, I obtained an answer in terms of the given values which makes me happy.

A = √ [ ( (x1)2 (v2)2) ) - ( (x2)2) (v1)2) / ( (v2)2) - (v1)2 ) ]

Then I just had to substitute this value for A in equation 1 to find ω.

Thanks for the help TSny and haruspex

6. Jan 16, 2013

TSny

Looks very good! You can save some effort by not taking the square roots. You have

A2 = (v1/ω)2 + (x1)2

A2 = (v2/ω)2 + (x2)2

Subtracting these two equations should allow you to fairly easily find ω. Then you can find A.