Two ideal springs oscillating, find amplitude and phase difference between them

[scrodger]

Hi, I am repeating first year exams and would really appreciate some help with my study. Just can't seem to get my head around this problem.

Homework Statement

Consider two identical ideal springs with a mass m attached which are harmonically oscillating out of phase relative to each other, with the spring constant k = 100 N/m and the mass m = 1x10-3 kg.
At the time t₀ = +0.1 sec, the displacement of the spring 1 is x₁(t₀) = 10 mm and the displacement of spring 2 is x₂(t₀) = 1 mm.
(i) Calculate the value of the amplitude A of each oscillation.
(ii) Calculate the value of the phase-difference ∅ between the two oscillators.

Homework Equations

ω = $\sqrt{k/m}$
x = Acos(ωt + ∅)
kA² = mv² + kx²

The Attempt at a Solution

So I calculated ω = 316.228 rad/s. In order to find A, I can use either the position equation or the energy equation. But both of these have an unknown variable. I can't seem to figure out how to find one of these. In the case of the equation x = Acos(ωt + ∅), is ∅ included in this as I am given time with the symbol t₀? Any help would really be appreciated. Also for part (ii) of the question, I have never solved a question before asking for the phase difference between two objects. Do you just subtract one from the other? Or is there a specific method?
Thanks

 

[NoPoke]

are you sure that there is no extra piece of information? It seems a little odd that the question would give you displacements at time = 0.1sec without telling you something about time = 0.

 

[scrodger]

Yep, I have posted the entire question. Does the subscript 0 on the t mean anything?

 

[NoPoke]

t0 usually means time zero or initial starting time.

so t0 = 0.1sec or t0 = +0.1sec are both a little odd.

I can't see a way of providing values for A and phi without more information

 

[Nickie]

Hello! It's great to see that you're working hard to understand this problem. Let's break it down step by step.

First, you correctly calculated the value of ω. Next, we can use the given information about the displacements at t0 to find the amplitude A. We know that the displacement at any given time t is given by x = Acos(ωt + ∅). Since we are given the displacements at t0, we can set t = t0 and solve for A. This gives us A1 = 10 mm and A2 = 1 mm for spring 1 and spring 2, respectively.

To find the phase difference ∅, we can use the fact that the displacements of the two springs are out of phase. This means that at t = t0, the displacements are equal but have opposite signs. In other words, x1(t0) = -x2(t0). Substituting in the values we found earlier, we get A1cos(ωt0 + ∅1) = -A2cos(ωt0 + ∅2). Since we know the values of A1 and A2, we can solve for the phase difference ∅1 - ∅2. This gives us the phase difference between the two oscillators at t = t0.

I hope this helps! Keep up the good work and don't be afraid to ask for help if you need it. Good luck on your exams!