How Do You Calculate the Damping Coefficient of a Pendulum?

[Punchlinegirl]

A 86.0 cm pendulum is released from a small angle. After 107 oscillations the amplitude is one half of it's original value. The damping is proportional to the speed of the pendulum bob. Find the value of the damping coefficient $$\alpha$$, in Hz.

I think the period is .009 s, using 1/107, but I'm not sure if that is even right. I don't really have any idea what to do. Can someone please help?

 

[whozum]

You can find the original period of the pendulum since you are given the string length. From there, the period has decreased by half after 107 oscillations, and there's a simple relationship between the number of oscillations and the time that's passed..

 

[Andrew Mason]

A 86.0 cm pendulum is released from a small angle. After 107 oscillations the amplitude is one half of it's original value. The damping is proportional to the speed of the pendulum bob. Find the value of the damping coefficient $$\alpha$$, in Hz.

I think the period is .009 s, using 1/107, but I'm not sure if that is even right. I don't really have any idea what to do. Can someone please help?

Tricky question. Here's is how I would approach it:

The small angle pendulum differential equation:

$$\ddot\theta + \alpha\dot\theta + \frac{g}{L}\theta = 0$$

has solution:

$$\theta = \theta_0e^{-\alpha t/2}sin(\omega t)$$ where

(1)$$\omega = 2\pi/T = \sqrt{g^2/L^2 - \alpha^2/4}$$

And you are told that:

(2)$$\theta = \theta_0e^{-\alpha t/2} = .5\theta_0$$ where $t = 107T$

(1) and (2) give you two equations for T in terms of $\alpha$ so you should be able to find both.

AM