How Does Damping Affect Oscillation Amplitude and Frequency?

[ThirdEyeBlind]

Homework Statement

The drawing to the left shows a mass m= 1.8 kg hanging from a spring with spring constant k = 7 N/m. The mass is also attached to a paddle which is emersed in a tank of water with a total depth of 27 cm. When the mass oscillates, the paddle acts as a damping force given by -b(dx/dt) where b= 270 g/sec. Suppose the mass is pulled down a distance 1.1 cm and released.

a) What is the time required for the amplitude of the resulting oscillations to fall to one third of its initial value?
ΔT = 14.64816 sec

b) How many oscillations are made by the block in this time?
?

Homework Equations

[tex]A(t)=A(0)e^{-bt/2m}[/tex]
?

The Attempt at a Solution

I solved part a) and the computer verified my answer so I know its correct. I am just stuck on part b. I think I need to somehow find the period and then take the time in a) and divide it by the period. My book is confusing and I am not sure which equation to use.


EDIT: Finally found equation for period. It was T= 2pi sqrt (k/m) and then I took my answer from a) and divided it by the period to get the answer.

 

[anathema]

Just in case someone else has this problem.

Great job on solving part a)! To solve part b), you are correct in needing to find the period of oscillation. The period is the time it takes for one complete oscillation, so if you divide the total time (14.64816 sec) by the period, you will get the number of oscillations made in that time.

The equation for the period of a mass-spring system is T = 2π√(m/k), where m is the mass and k is the spring constant. Plugging in the given values, we get T = 2π√(1.8/7) = 0.910 sec.

Dividing the total time from part a) by this period, we get 14.64816 sec / 0.910 sec = 16.09 oscillations. So, the block will make approximately 16 oscillations in the given time period.

Hope this helps!