# I need to find the drag coefficient of a pendulum bob [closed]

Is it possible to find the drag coefficient of a pendulum bob from the damping caused on it during swinging. I will be able to measure its displacement from the point of origin and plot it against time. I know the $$F_D=\frac 12 \rho v^2AC_D$$ equation could be used if I had more information, but I only have displacement from origin over time.

## closed as off-topic by Jon Custer, Chair, Chris, stafusa, ZeroTheHeroOct 23 '18 at 15:15

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• Do you expect your bob to reach terminal velocity fairly quickly? – Aaron Stevens Oct 17 '18 at 18:45
• I don't expect it to, it's only a small pendulum with roughly a 30cm long cord. – CM92 Oct 17 '18 at 19:08
• If you aren't measuring the force, you would need to basically determine which drag coefficient gives you a solution to your differential equation that best fits the data – Aaron Stevens Oct 17 '18 at 20:30
• Wouldn't it perhaps be less troublesome to account for the linear drag ($-6 \pi \eta r \mathbf v$) on the bob, instead of the quadratic one? Then you would get a nice second order homogeneous differential equation for the damped oscillatory motion of the pendulum. – Vinícius Peixoto Oct 17 '18 at 21:28
• If the drag is small so you have slowly decaying sinusoidal oscillations, calculate how much energy is lost during one period of oscillation; then integrate the work done by the drag force over the period. This will be enough to determine the constant in the force equation, assuming it is known that the force is quadratic (or linear) with the pendulum bob speed. An interesting question is how to determine from these experimental data the exponent in the friction force F, assuming it is a power law. – Maxim Umansky Oct 17 '18 at 21:49

Welcome to the stack exchange! Good question.

Absolutely.

But since there are losses other than drag you'll need to run two sets of tests:

1. Pendulum swing in the air and
2. Pendulum swing in a vacuum

To be as precise as possible you should use the nonlinear model (differential equation) of a pendulum

derived here which applies to the pendulum in vacuum, and without losses that might occur in whatever is suspending the pendulum.

The hard part at this point is determining a functional model that accounts for the non-drag energy losses. But assume you can do this - perhaps by trial and error - by proposing a loss model and fitting to data.

In any event this gives you the drag-free model.

You'll then need to modify this drag free model (which assumes a point mass) with a tangential force that's equal to the drag force on the projected area, $$A$$

$$\rho$$ is your calculated or measured air density , $$v$$ is the tangential velocity, and $$C_D$$ is the drag coefficient of the bob - what you are trying to solve for.

This gives you a total loss model

Between the measurements in vacuum and air, and the total and drag-free loss models you can back out, calculate $$C_D$$

• Caveat: If somehow you can determine that loss due to drag forces >> losses due to the bob suspension mechanism then you don't need to run the vacuum experiment. – docscience Oct 18 '18 at 1:11