Forced and damped oscillators

[Mikey]

I have two problems, the second of which I think I might be solving right. The web program we use to do our homework isn't accepting my answer. It might be the program's fault, but I'm not sure, so I'd like to check.
Here's my first problem:

Damping is negligible for a 0.131-kg object hanging from a light 6.50-N/m spring. A sinusoidal force with an amplitude of 1.70 N drives the system. At what frequency will the force make the object vibrate with an amplitude of 0.440 m? There are two possible solutions.

I would think I know how to answer the problem if there weren't two possible solutions. That completely throws me. It also confuses me that amplitude is given in Newtons and in meters. The only thing I can think of to solve the problem is to use the equation for amplitude of a driven oscillator:
A = (F₀/m)/((w²-w₀²)² + ((bw)/m)²)^1/2
w₀² = k/m
F₀ = 1.70 N?
bw/m = 0, because damping is negligible
I got 8.79 Hz when I solved for w. That's still only one answer, so I guess it's not right.

Problem two:
Consider a damped oscillator that is an object hanging vertically from a spring and submersed in a viscous liquid. Assume the mass is 382 g, the spring constant is 105 N/m, and b = 0.119 N*s/m. (a) How long does it take for the amplitude to drop to half its initial value? (b) How long does it take for the mechanical energy to drop to half its initial value?

I used the equation for position of a damped oscillator:
x = Ae^(-b/(2m))tcos(wt + theta)
I used .5A as x, took the natural long of both sides and solved for t. I ignored cos(wt + theta) because the cos at the maximum amplitude is equal to one. I got 4.45 s. The time it takes for the mechanical energy to drop to half its initial value would be 8.90 s (twice as much), because E = .5kA². What am I doing wrong?

Thanks for your help!

 

[Nate810]

For the first problem, you need to use a parametric equation, such as x(t) = A sin(wt + θ). Plug in the given values for A and solve for w. This will give you the two possible solutions for w. For the second problem, you are correct in your approach. You should use the equation for the position of the damped oscillator and take the natural logarithm of both sides to solve for t. However, you should note that the time it takes for the mechanical energy to drop to half its initial value is twice the time it takes for the amplitude to drop to half its initial value. Therefore, you should get 8.90 s as your answer.

 

[wais]

Hi there,

It sounds like you have a good understanding of the equations for forced and damped oscillators. The issue with the first problem might be with the units. Amplitude is usually measured in meters, so the amplitude of the driving force should also be in meters. This could be why you are getting a different answer than expected. I would recommend double checking your units and trying again.

For the second problem, your approach seems correct. However, when taking the natural log, you should include the cos(wt+theta) term because it is not equal to one at all times. This could be why you are getting a different answer than expected. Also, make sure to use the correct value for the initial amplitude (A) in the equation for energy, which is 0.382 kg.

If you are still having trouble, I would suggest checking with your instructor or a classmate to see if they are getting the same answers as you. It's always helpful to have someone else double check your work. And if the web program is still not accepting your answer, it might be worth contacting the program's support team to see if there is an issue with the system.

I hope this helps and good luck with your homework!