# Harmonic motion with damping

## Homework Statement

An object of mass 0.2kg is hung from a spring whose spring constant is 80N/m. The object is subject to a resistive force given by -bv, where v is it's velocity in meters per second.

If the damped frequency is √(3)/2 of the undamped frequency, what is the value of b?

F=ma
ω=√k/m

## The Attempt at a Solution

I tried to write the sum of the forces of the system and got ∑F=-kx-bv=ma
I rewrote it as -kx=b(dx/dt)+m(d^2x/dt^2)

Now I don't have much experience with differential equations but I know the solution is x(t)=Ae^(γt)cos(ωt) where γ=(-b/2m). I also know that the damped frequency is (√(3)/2)√k/m given from the problem. I not sure where to go from here. I am supposed to use the solution and solve for b? Any help would be appreciated.

rude man
Homework Helper
Gold Member
Now I don't have much experience with differential equations but I know the solution is x(t)=Ae^(γt)cos(ωt) where γ=(-b/2m).
What is the value of ω in terms of k, m and/or b? You know γ(b,k,m) but you don't know ω(b,k,m)?

What is the value of ω in terms of k, m and/or b? You know γ(b,k,m) but you don't know ω(b,k,m)?

So ω in terms of k,m, and b would be √((ω^2)-(γ^2)) right?

rude man
Homework Helper
Gold Member
Going back to your 1st post, you are mixing up two ω's. One is the natural frequency ωn = √(k/m). the other is the damped frequency which is ωd. The ω in your cos argument should be the latter. The idea is that ωd < ωn as your problem statement gives. Don't use ω again, use the two above.
So ω in terms of k,m, and b would be √((ω^2)-(γ^2)) right?
is correct IF you use the right omegas. You can't say x = x + a, a ≠ 0, can you?

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Going back to your 1st post, you are mixing up two ω's. One is the natural frequency ωn = √(k/m). the other is the damped frequency which is ωd. The ω in your cos argument should be the latter. The idea is that ωd < ωn as your problem statement gives. Don't use ω again, use the two above.
Having said that, to answer your question is correct IF you use the right omegas. You can't say x = x + a, a ≠ 0, can you?

Yeah I agree with that. I seem to be getting the 2 omega's confused with each other. So my solution should actually be expressed as x(t)=Ae^(γt)cos(ωdt), where ωd refers to the damped frequency. Sorry for making it look messy but I wasn't sure how to write the subscript d. Plugging in √((ω^2)-(γ^2)) for ω damping I can solve for my value of b. Is that the right idea?

rude man
Homework Helper
Gold Member
Yeah I agree with that. I seem to be getting the 2 omega's confused with each other. So my solution should actually be expressed as x(t)=Ae^(γt)cos(ωdt), where ωd refers to the damped frequency. Sorry for making it look messy but I wasn't sure how to write the subscript d. Plugging in √((ω^2)-(γ^2)) for ω damping I can solve for my value of b. Is that the right idea?
1. You're still using ω instead of wd or ωn. Don't.
2 . Picking the correct omegas, rewrite your equation; this time make it a real equation with an = sign and everything.
3. It's easy to make subscripts or superscripts. See the "x2" and the "x2" on the toolbar where you got your ω?

Sorry for the late reply. Yes I found it, thanks for pointing that out.
3. It's easy to make subscripts or superscripts. See the "x2" and the "x2" on the toolbar where you got your ω?