Time taken for energy to drop 95% in damped SHO

[Phynos]

There is a post about the same problem here: https://www.physicsforums.com/threads/damped-oscilating-spring.12838/

It was helpful for solving part B.

1. Homework Statement

A 10.6kg object oscillates at the end of a vertical spring that has a spring constant of 2.05x10^4 N/m. The effect of air resistance is represented by the damping coefficient b = 3.00Ns/m.

(a) Calculate the frequency of the damped oscillation.

(b) By what percentage does the amplitude of the oscillation decrease in each cycle?

(c) Find the time interval that elapses while the energy of the system drops to 5.00% of it's initial value.

Homework Equations

1. x = Ae^(bt/2m) cos(wt + phi)

2. (From post mentioned above): %Difference = 1 - e^(bt/2m) * 100%

The Attempt at a Solution

I have completed part (A): frequency is 7Hz and (B): Percent drop is 2%.

For part (C) I tried solving the second equation above for t, then subbing in difference of 0.95.

I end up with t = -(2m/b) ln(1-D)

solving when D = 0.95 gives me:

t = 21.2s

This is exactly double the solution in the back. Why is it not the same? I didn't insert a factor of two anywhere, the two in the equation is meant to be there.

Is the back wrong? Or am I making a stupid mistake somewhere? It's more likely that I've made a mistake but I've been looking at it awhile now and I'm stumped.

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I also tried multiplying the time taken to drop 2% (The period, sqrt(k/m) ) by 47.5 but then I end up with:

t = 6.97s

I realized after this would not work because the decrease in energy is not linear, hence the exponential term in the equation.

 

[gneill]

Hi Phynos, Welcome to Physics Forums.

Looking at your method I can't find fault. t = 21.2s seems like a correct value to me. Of course, I stand ready to be corrected by someone more clever than I

I do have one nit to pick, and that's with the units of the given damping coefficient. I think they should be Ns/m rather than Nm/s. Usually the air resistance force is modeled as being proportional to velocity in this type of problem, so b*V should yield Newtons.

 

[Phynos]

Thanks.

[Ns/m][m/s] = N

The units are in Ns/m in my post. Perhaps you misread, unless you meant they should not be? But I think they are correct.

 

[gneill]

Thanks.

[Ns/m][m/s] = N

The units are in Ns/m in my post. Perhaps you misread, unless you meant they should not be? But I think they are correct.

Ah. My apologies. I think I must have been confused by Klion's post (which you linked to).