What Are the Angular Frequency and Amplitude of a Particle Undergoing SHM?

[NewtonsHead]

Homework Statement

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.

Homework Equations

x = A sin (ωt + ∅ )

v = A ω cos ( ωt + ∅ )

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?

 

[TSny]

x = A sin (ωt + ∅ )

v = A ω cos ( 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?

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?

 

[NewtonsHead]

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

 

[haruspex]

x² = A² sin² (ωt + ∅ )

(v/ω)² = A² cos² ( ωt + ∅ )

I could say

A = √ (v/ω)² + x²

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

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

You just did.

 

[NewtonsHead]

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) ) - ( (x_2)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

 

[TSny]

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) ) - ( (x_2)²) (v1)²) / ( (v₂)²) - (v₁)² ) ]

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

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.