How to Calculate the Period of Pendulum

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Pendula are fairly common in our lives: you may have seen a grandfather clock with a long pendulum slowly oscillating as time ticks on. The clock needs a functioning pendulum in order to correctly advance the dials on the clock face that display the time. So it is likely a clock-maker needs to understand how to calculate the period of a pendulum.

The pendulum period formula, ​T​, is fairly simple:

T=\sqrt{\frac{L}{g}}

where ​g​ is the acceleration due to gravity and ​L​ is the length of the string attached to the bob (or the mass).

The dimensions of this quantity is a unit of time, such as seconds, hours or days.

Similarly, the frequency of oscillation, ​f​, is 1/​T​, or

f=\sqrt{\frac{g}{L}}

which tells you how many oscillations take place per unit time.

Mass Doesn't Matter

The really interesting physics behind this formula for the period of a pendulum is that the mass doesn't matter! When this period formula is derived from the pendulum equation of motion, the dependence of the mass of the bob cancels out. While it seems counter-intuitive, it is important to remember that the mass of the bob does not affect the period of a pendulum.

...But this Equation Only Works in Special Conditions

It is important to remember that this formula only works for "small angles."

So what is a small angle, and why is that the case? The reason for this comes out from the derivation of the equation of motion. In order to derive this relationship, it is necessary to apply the small angle approximation to the function: sine of ​θ​, where ​θ​ is the angle of the bob with respect to the lowest point in its trajectory (usually the stable point at the bottom of the arc it traces out as it oscillates back and forth.)

The small angle approximation can be made because for small angles, the sine of ​θ​ is almost equal to ​θ​. If the angle of oscillation is very large, the approximation no longer holds, and a different derivation and equation for the period of a pendulum is necessary.

In most cases in introductory physics, the period equation is all that is needed.

Some Simple Examples

Due to the simplicity of the equation, and the fact that of the two variables in the equation, one is a physical constant, there are some easy relationships that you can keep in your back pocket!

The acceleration of gravity is ​9.8 m/s2​, so for a one meter long pendulum, the period is

T=\sqrt{\frac{1}{9.8}}=0.32\text{ seconds}

So now if I tell you the pendulum is 2 meters? Or 4 meters? The convenient thing about remembering this number is that you can simply scale this result by the square root of the numerical factor of the increase because you know the period for a one meter long pendulum.

So for a 1 millimeter long pendulum? Multiply 0.32 seconds by the square root of 10-3 meters, and that's your answer!

Measuring the Period of a Pendulum

You can easily measure the period of a pendulum by doing the following.

Construct your pendulum as desired, simply measure the length of the string from the point it is tied to a support to the center of mass of the bob. You can use the formula to calculate the period now. But we can also simply time an oscillation (or several, and then divide the time you measured by the number of oscillations you measured) and compare what you measured with what the formula gave you.

A Simple Pendulum Experiment!

Another simple pendulum experiment to try is to use a pendulum to measure the local acceleration of gravity.

Instead of using the average value of ​9.8 m/s2​, measure the length of your pendulum, measure the period, and then solve for the acceleration of gravity. Take the same pendulum up to the top of a hill and do your measurements again.

Notice a change? How much of an elevation change do you need to achieve in order to notice a change in the local acceleration of gravity? Try it out!

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