One More hard Oscillations Problem

[mihir871]

Homework Statement

An oscillator with a mass of 420 g and a period of 1.00 s has an amplitude that decreases by 1.20% during each complete oscillation.

If the initial amplitude is 8.20 cm, what will be the amplitude after 50.0 oscillations?
If the initial amplitude is 8.20 , what will be the amplitude after 50.0 oscillations?

Homework Equations

x(t)= Ae$$^{-bt/2m}$$cos($$\omega$$t)

The Attempt at a Solution

I have no idea :(

 

[G01]

What term in the equation you gave is always making the amplitude smaller as time increases?

 

[mihir871]

Time or the dampening coeffecient?

 

[G01]

Time or the dampening coeffecient?

Your part of the way there. It is the damping coefficient term. Read my post in your other thread. It may clear some things up on this problem as well.

 

[mihir871]

this doesn't make sense i hate physics :(

 

[G01]

Don't give up yet!

Your amplitude starts at a value A and decreases due to the damping. The amplitude at any time then, is given by:

$$A(t)=Ae^{-bt/2m}$$

How can you use this equation to find what you are looking for?

 

[mihir871]

i think you can solve for the time but i don't undersand what b would be in this equation

 

[G01]

In order to find b you have to use the information for how much A decreases in a certain amount of time. They tell us, in one second, A decreases by 1.2%: We can say this mathematically like so:

A(1)=.012A

But from the equation in my above post, what is A(1) in terms of b? Using these two equations then, you should be able to solve for b.

 

[mihir871]

I got b= -3.7132 but i don't know if that is correct/ makes sense

 

[G01]

I get positive 3.7 . I think you just forgot a negative sign somewhere.

 

[mihir871]

ah i see it now, now with this b i find time?

 

[mihir871]

repost sorry

 

[G01]

ah i see it now, now with this b i find time?

Yep that's the plan

 

[mihir871]

im such an idiot... i got the first part already :\ =4.48cm

i meant this one: At what time will the energy be reduced to 18.0% of its initial value?

 

[G01]

im such an idiot... i got the first part already :\ =4.48cm

i meant this one: At what time will the energy be reduced to 18.0% of its initial value?

Remember that the energy of a wave is proportional to the amplitude squared. I can't seem to find the actual equation describing that, but it should be simple enough to look up. So, HINT: You want to find the time at which the amplitude is such that it makes the energy 18% of its initial value.

Use the percents like we did above with the amplitude, but this time work with the energy:

$$.18E_{initial}= E_{at A(t)}$$

Then use this energy to find A. Then, you should be able to find t from A.

 

[mickles]

ah i see it now, now with this b i find time?

i got b as well, but how do i find time at this point