# How Do You Calculate the Damping Coefficient of a Pendulum?

• Punchlinegirl
In summary: AZON (1) In summary, a 86.0 cm pendulum was released from a small angle and after 107 oscillations, the amplitude decreased to half its original value. The damping of the pendulum is proportional to the speed of the bob. The value of the damping coefficient \alpha, in Hz, can be found by using the differential equation for a small angle pendulum and solving for the period using the given information.
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?

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..

Punchlinegirl said:
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

## What is a damping coefficient problem?

A damping coefficient problem is a phenomenon that occurs in systems with oscillatory motion, where the energy of the system gradually decreases due to the presence of a damping force. This results in a decrease in the amplitude of the oscillations over time.

## How is the damping coefficient calculated?

The damping coefficient is calculated by dividing the damping force by the velocity of the system, or by using the equation C = 2*m*w, where C is the damping coefficient, m is the mass of the system, and w is the natural frequency of the system.

## What factors affect the damping coefficient?

The damping coefficient can be affected by various factors such as the material properties of the system, the damping force applied, and the frequency and amplitude of the oscillations. It can also be influenced by external factors such as temperature and humidity.

## Why is the damping coefficient important?

The damping coefficient is important because it affects the stability and performance of systems with oscillatory motion. A high damping coefficient can reduce the amplitude of oscillations and make the system more stable, while a low damping coefficient can result in large oscillations and decrease the performance of the system.

## How can the damping coefficient problem be solved?

The damping coefficient problem can be solved by adjusting the damping force or by changing the material properties of the system. It can also be mitigated by using different damping techniques such as active or passive damping methods.

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