Solve Damping & Forced Oscillations Problems: Frequency, Amplitude & More

[k0b3]

I was hoping someone could explain damping and forced oscillations. I had a couple of problems I could not do that revolved around these topics because I couldn't figure out which equations to use. Here's an example.


1. Homework Statement
Damping is negligible for a 0.155 kg object hanging from a light 6.30 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?

Give higher and lower frequencies.


2. Homework Equations
For this I used the forced oscillation equation which is basically

Amplitude = (Fo/m)/(sqrt ((w^2 - wo^2)^2 + (damping)^2))


3. The Attempt at a Solution
So since it said damping was negligible, the damping part is 0. For wo (omega initial), I used sqrt (k/m) and plugged that in. Fo was given at 1.70N and mass was also given. I solved for omega (two different answers because it is squared) and I got 3.96Hz for the upper but it wasn't correct.





I was hoping also someone could basically explain what equation is used for these types of problems. There is another one where mass, an equation for external force (F= 3sin(2pit)) and spring constant are given and it asks to find period, amplitude, etc. How do I incorporate the external force? I think this problem and the org problem are hand in hand so if I can figure out one, I can do the other. Any Help would be greatly appreciated!

 

[jbsteele]

The equation you used is correct for this problem. Remember, since damping is negligible, the damping term in the equation is zero. The equation for the external force is F = F(t) = F0*sin(wt). You can also use this equation to find the period and amplitude of the oscillation. The period is given by T = 2*pi/w and the amplitude is given by A = F0/mw^2.

 

[baba89]

I can explain the concepts of damping and forced oscillations and how to approach these types of problems.

Damping refers to the gradual decrease in the amplitude of an oscillating system over time due to external factors such as friction or air resistance. In the case of negligible damping, the amplitude of the oscillation remains constant over time.

Forced oscillations occur when an external force is applied to an oscillating system, causing it to vibrate at a specific frequency. The amplitude of the oscillation is determined by the amplitude and frequency of the external force.

To solve problems involving damping and forced oscillations, the equation you used is correct. This is the equation for forced oscillations, where the amplitude is dependent on the external force, mass, and damping.

For the example given, you correctly identified that the damping is negligible and used the equation to solve for the frequency that would result in an amplitude of 0.440 m. However, it is important to note that there are two possible frequencies for this scenario, as you have found.

To find the lower frequency, you can use the equation w = sqrt(wo^2 - (damping)^2), where wo represents the initial frequency. Plugging in the values given, you should get a lower frequency of 3.82 Hz. This is because the damping term is subtracted from the initial frequency to get the new frequency.

To incorporate the external force into the equation, you can simply substitute the value for the external force (F) into the equation as Fo. So the final equation would be Amplitude = (F/m)/(sqrt ((w^2 - wo^2)^2 + (damping)^2)). You can then solve for the other variables as needed.

Similarly, for the other problem, you can use the same equation and substitute the values given for mass and spring constant. The external force, F = 3sin(2pit), can be substituted as Fo in the equation.

In summary, to solve problems involving damping and forced oscillations, you can use the equation for forced oscillations and substitute the given values for mass, damping, and external force to solve for the unknown variables. Remember to consider both the upper and lower frequencies for the given scenario.