Pendulum Period

Clocks with quartz movements keep time more accurately than pendulum. As a result, quartz has largely replaced pendulums in modern clocks. But in their day, pendulum clocks were profoundly important. The first pendulum clocks were produced in the mid 17th century. They use ushered in a new era of accurate time keeping.

The reliability of pendulum clocks is based on the predictable relationship between the length of a pendulum and the time it takes the pendulum to complete one full swing. The illustration below explores the relationship between a pendulums period* as the force of gravity and its length is varied.

Length: 1 m

In the absence of outside forces, a weight** on a string will hang directly below the point from which the string is suspended. This is called the weight's resting or equilibrium position. If the weight is pulled to one side and let go, gravity will pull it back towards it’s equilibrium position. The momentum built up as the weight falls causes it to overshoot and swing up in the opposite direction. Gravity then acts to slow the weight until it stops and begins falling back down again. This back and forth motion repeats itself over and over as the momentum of the weight pulls it past the equilibrium position on each swing. This is the motion of a pendulum and is an example of simple harmonic motion.

A pendulum weight is called a bob. The point from which the bob is hanging is called the pivot point. The distance between the pivot point and the bob is the pendulum's length (L). The time it takes a bob displaced from equilibrium to to complete one full swing is the pendulum's period. The period of a pendulum is proportional to to the square root of its length and is described by the equation:

P = 2π × √ L / g   

where pi is 3.1415 and g is the force of gravity.

One thing to note about this equation is how few variables are involved. If the force of gravity (9.8 m/s2 on Earth) and the length of the pendulum is known, the pendulum can be used to tell time. The mass* of the bob does not matter, nor does the distance bob is displaced from the equilibrium position*. More detailed studies of pendulum physics requires knowledge of the bob’s mass, but only the distance between the pivot point and the bob is needed to use a pendulum for time keeping.

Take a look at the pendulum period practice problems pendulum period practice problems to test your understanding of the concepts covered in this illustration.

*This is only true for small angles of displacement. When the bob is displaced by larger angles, the angle of displacement needs to be taken into consideration.

Video Overview



In giving us the simple formula for the period* of a pendulum, the word proportional is seriously wrong. At this level students should know exactly what it means, e.g. Y = CX. The pendulum period is in fact proportional to the square root of the length, Y = C x sqr(X).

Moreover, this is easily tested in a very easy and interesting grade-school experiment: e.g. double the length, and the period increases by the square root of 2, i.e. 1.414.

"Proportional" is not the same as "a function of". Get it right, please. But thanks for the formula, which I had remembered as being very simple.

The text has been modified. It now reads that the peroid of a pendulum is proportionlal to the square root of its length. 

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