When something is hanging loosely from a ceiling, and you give it a nudge, it will start swinging back and forth. But how fast will it swing, and why? This is something we can actually answer, and there is a pretty simple formula to figure it out. These questions are related to a property called the period of a pendulum.
The meaning of the period of a pendulum
To understand what the period of a pendulum is, we need to know the meaning of two things: a period and a pendulum.
A pendulum is a system that consists of an object with a certain mass that hangs by a rod or cord from a fixed pivot. The hanging object is called a bob.
A pendulum will swing back and forth, and the maximum value that the angle θ of the cord with the vertical takes on is called the amplitude. This situation is actually quite complicated, and in this article, we will only talk about a simple version of a pendulum.
A simple pendulum is a pendulum in which the rod or cord is massless and the pivot is frictionless.
See the figure below for an illustration of a simple pendulum.
Figure 1: A simple pendulum.
In this article, whenever we talk about a pendulum, we have in mind a simple pendulum with a small amplitude. Now that we understand what we mean by a pendulum, we need one more bit of information, namely, what we mean by a period.
The period of a pendulum is the duration of one full swing of the bob.
For example, the time duration between two successive situations in which the bob of a pendulum is all the way to the right is one period of the pendulum.
The impact of length on the period of a pendulum
The length of the cord of a pendulum has an impact on the period of the pendulum it belongs to. This statement is pretty convincing if we just look at some everyday examples.
Some Christmas tree decorations are pretty good examples of a pendulum. These small decorations have a small cord length of a couple of centimeters and small periods of less than half a second (they wobble quickly).
A playground swing is an example of a pendulum with a cord length of multiple meters. The period of these swings is often more than 3 seconds.
A set of swings, of which the left will have a shorter period than the right.
Thus, the longer the cord, the larger the period of the pendulum.
Other factors that affect the period of a pendulum
There are two other factors that affect the period of a pendulum: the gravitational acceleration and the amplitude of the pendulum. As we are only talking about pendulums with small amplitudes, the only other factor we have to take into account is gravitational acceleration. With a very small gravitational acceleration, we can imagine things playing out in slow motion. Thus, we expect that the larger the gravitational acceleration, the faster the swing of the pendulum and the smaller the period of the pendulum.
But hang on, why doesn’t the mass of the bob affect the period of a pendulum? This is very similar to the fact that the mass of an object doesn’t affect how fast it falls down: if the mass doubles, the gravitational force on it doubles as well, but the acceleration remains the same:![]()
. The bob of our pendulum experiences the same thing: the force on bob 1 that is twice as massive as that on bob 2 is twice as large, but the bob itself is also twice as heavy as bob 2. Bob 1 is, therefore, twice as hard to displace as bob 2, and so the acceleration of both bobs will be the same (again by![]()
). Hence the period of a pendulum does not depend on the mass of the bob.
You can experimentally test this by going to a swing on a playground and measuring the period of the swing when someone is on it and when no one is on it. The two periods measured will turn out to be the same: the mass of the bob has no influence on the period of the swing.
The time period formula for a pendulum
If![]()
is the length of the cord of the pendulum and g is the gravitational acceleration, the formula for the period T of a pendulum is:
![]()
We see that we were right about our predictions. A larger pendulum cord length and a smaller gravitational acceleration both cause a larger period of the pendulum, and the mass of the bob does not affect the period of the pendulum at all.
It is a good short exercise to check that the units of this equation are correct.
A diagram of a small-amplitude simple pendulum with relevant quantities shown.
With a bit of calculus, we can derive the formula for the period of a pendulum. We need to measure angles in radians, such that for small angles, we have roughly sin(θ) = θ. The only net forces on a bob with mass m are horizontal forces, and the only horizontal force we can find is the horizontal part of the tension in the cord.
The total tension in the cord is roughly the vertical component of the tension because the amplitude of the pendulum is small. This vertical component is equal to the downward force on the bob (because there is no net vertical force on the bob), which is its weight mg.
The horizontal part of the tension is then -mgsin(θ) (with the minus sign because the acceleration is in the direction opposite to its position, which we take to be positive). This is roughly -mgθ because of the small amplitude of the pendulum. So, the acceleration of the bob is![]()
.
The acceleration is also measured as the second time derivative of its horizontal position, which is roughly![]()
. But![]()
is constant, so the equation is now![]()
, where we have to solve for the angle θ as a function of time t. The solution to this equation (as you can check) is![]()
, where A is the amplitude of the pendulum. We see that θ is equal to A every![]()
units of time, and so the period of the pendulum is given by![]()
. This derivation shows explicitly where all the factors affecting the period of a pendulum come from.
We conclude that on Earth, the only factor influencing the period of a pendulum is the length of the cord of the pendulum.
Calculating the period of a pendulum
Suppose we can regard a playground swing as a simple pendulum. What is the period of a swing which has its seat 4 m below its pivot if we only let it swing gently, i.e., with a small amplitude?
We know that g = 10 m/s2 and that![]()
. The period T of this pendulum is then calculated as:
![]()
.
This is indeed what we know from our own experience.
Suppose we can regard an earring as a simple pendulum. If someone walks, it nudges the earring only a little bit, causing a small amplitude. What is the period of such an earring if the length of the cord is 1 cm?
The period of this pendulum is calculated as follows:
![]()
.
This is also what we know from experience: a small pendulum wobbles very quickly.
The frequency of a pendulum
The frequency (often denoted by f ) of a system is always the inverse of the period of that system.
Therefore, the frequency of a pendulum is given by:
![]()
.
Remember that the standard unit of frequency is hertz (Hz), which is the inverse of a second.
Period of Pendulum - Key takeaways
A pendulum is a system that consists of an object with a certain mass that hangs by a rod or cord from a fixed pivot. The hanging object is called a bob. The maximum angle of the cord with the vertical is called the amplitude.
A simple pendulum is a pendulum in which the rod or cord is massless and the pivot is frictionless.
The period of a pendulum is the duration of one full swing of the bob.
The only factors influencing the period of a pendulum are the gravitational acceleration and the length of the cord. Thus, on Earth, only the length of the cord influences the period of a pendulum.
The formula for the period of a pendulum is![]()
.
The frequency of a pendulum is the inverse of the period, so it is given by![]()
.
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