Ratio of Damped to Initial Oscillation Amplitudes - 20 Cycles

In summary, to find the ratio of the amplitude of the damped oscillations to the initial amplitude at the end of 20 cycles, we need to first find the period of the oscillator, which is 0.353 seconds. Then, using the formulas for damping and oscillation, we can substitute the given values and calculate the ratio. This is done by taking the amplitude of the damped oscillations (xme-bt/2m) and dividing it by the initial amplitude (xm). This gives us the ratio of Adamped / Ainitial.
  • #1
bclark23
2
0
In the figure below, a damped simple harmonic oscillator has mass m = 300 g, k = 95 N/m, and b = 70 g/s. Assume all other components have negligible mass. What is the ratio of the amplitude of the damped oscillations to the initial amplitude at the end of 20 cycles (Adamped / Ainitial)?

I know I need to find the period (T), which is 2πsqrt(m/k).
T=2πsqrt[(.0kg)/(95nN/m)]=.353 s

Also, there are 20 cycles, so the final time would be (20 cycles)(.353s)=7.062s

The formula for damping (Adamped?) is x(t)=xme-bt/2mcos(wt+rho)
The formula for oscillation (Ainital?) is x(t)=xmcos(wt+rho)

I'm pretty sure I need to use these two equations, and put the answers in a ratio, but I'm not sure how to go about doing that.
 
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  • #2
bclark23 said:
The formula for damping (Adamped?) is x(t)=xme-bt/2mcos(wt+rho)
Don't you just substitute for b, t and m in e-bt/2m to find the attenuation?
 
  • #3
Yes, that was exactly it! Thank you
 

1. What is the ratio of damped to initial oscillation amplitudes?

The ratio of damped to initial oscillation amplitudes is a measure of how much the amplitude of a damped oscillation decreases over time compared to its initial amplitude. It is typically denoted as "A/A0" and is calculated by dividing the final amplitude (A) by the initial amplitude (A0).

2. What causes oscillations to be damped?

Oscillations can be damped by various factors such as friction, air resistance, and electrical resistance. These external forces act to decrease the amplitude of the oscillation over time, resulting in a damped oscillation.

3. How is the ratio of damped to initial oscillation amplitudes related to the number of cycles?

The ratio of damped to initial oscillation amplitudes is directly related to the number of cycles. As the number of cycles increases, the amplitude of a damped oscillation will decrease at a faster rate, resulting in a larger ratio of damped to initial oscillation amplitudes.

4. What is the significance of 20 cycles in this ratio?

The number of cycles used in the calculation of the ratio of damped to initial oscillation amplitudes is arbitrary and can vary depending on the specific system being studied. In some cases, 20 cycles may be a sufficient number to accurately determine the ratio, while in other cases a larger or smaller number may be needed.

5. How is the ratio of damped to initial oscillation amplitudes used in scientific research?

The ratio of damped to initial oscillation amplitudes is commonly used in the study of damped oscillations in various fields such as physics, engineering, and biology. It allows researchers to quantify the rate at which oscillations are damped and can provide valuable insights into the behavior of a system. It can also be used to compare different systems and determine which one has a stronger damping effect.

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