Lesson 32: Pendulums

Pendulums are a great example of periodic motion because they are simple to make and follow a pretty well established pattern of going back and forth the same way.

The equilibrium point of an object is the place it will come to rest if nothing is being done to it.

The periodic motion of something like a pendulum is also sometimes called Simple Harmonic Motion. "SHM" needs a force that causes an object to return to its equilibrium point. In the case of a pendulum this force is supplied by gravity.

One of the first people to realize how physics could explain pendulums was Galileo.

Although we have come a long way from the work that Galileo started, we still have a pretty simple equation that we can use to figure out how long it takes for a pendulum to swing…

T = period of one swing (s)
ℓ = length of wire (m)
g = gravity (m/s2)

Notice that the mass of the bob (the weight at the end of a pendulum) is not in the formula.

Be careful when you are using this formula. Remember three things:

  1. The period is the time it takes to complete one full swing… that means if you let go of the bob (the weight on the end), it will swing away from your hand and back to your hand. That’s one complete swing. Back to where it started and ready to move in the original direction again.
  2. You can call the equilibrium point the equilibrium position or even rest position if you want.
  3. This formula only works well if the pendulum is held at an angle of less than 15° from equilibrium position. As the angle gets further past 15° errors start to creep in.
  4. Be careful if you need to solve it for “ℓ” or “g”. You’ll see in the following examples how to do this.

Example 1: What is the period of a pendulum that is 12.5 m long?

We will assume that it is on Earth…

T = 2 π √(l / g)
= 2 π √(12.5 / 9.81)
T = 7.09 s

Note: I used 3.14 for pi , which gave me three sig digs. Do not use the pi button on your calculator.

Example 2: We decide to measure gravity in a particular location on Earth. I use a 2.75m long pendulum and find that it has a period of 3.33 s. What is the acceleration due to gravity in this area.

T = 2 π √(l / g)
Do you get this same formula as this? Double check!

g = 9.78m/s2

Example 3: What length of pendulum would give me a period of one minute?

We can already make a pretty good guess that this is going to be a pretty long pendulum to take that long to swing.

l = 895m

Yikes! Almost one kilometre long. And before you get grumpy about sig digs, notice that in the question I said “one minute”, which is by definition exactly 60 seconds, so I had an infinite number of sig digs there. Only the measurement of the acceleration due to gravity and pi have sig digs I need to worry about (3 on each).