# Question Video: Comparing the Periods of Oscillation of Two Pendulums

A pendulum takes 5 seconds to complete one oscillation. A shorter pendulum takes only 1 second to complete an oscillation. What is the ratio of the shorter pendulum’s period to the longer pendulum’s period?

02:35

### Video Transcript

A pendulum takes five seconds to complete one oscillation. A shorter pendulum takes only one second to complete an oscillation. What is the ratio of the shorter pendulum’s period to the longer pendulum’s period?

Okay, so in this question, we’re dealing with two pendulums. We’ve got the longer one and the shorter one. We’ve been told that the longer pendulum takes five seconds to complete one oscillation. In other words, if we were to displace this pendulum slightly, for example, by pulling it to the right and then letting go until the pendulum oscillated back to its original position, went all the way to the other side, stopped, and then returned back to its initial position. Then, that entire process of one entire cycle or one oscillation would take five seconds.

And we’ve also been told a similar sort of thing for the shorter pendulum. If we were to displace the shorter pendulum and then release it, then it would oscillate until it moves to the other side and then returns back to its original position. It would take one second for the shorter pendulum to return back to its original position or, in other words, to complete one oscillation.

Now, we’ve been asked in the question to find the ratio of the shorter pendulum’s period to the longer pendulum’s period. At which one, we can recall that the period of an oscillation is simply the time taken for one oscillation to be completed. And we’ve been given this information already for both pendulums. In other words, we can say that the period of the longer pendulum, which we will label 𝑇 subscript 𝑙, is equal to five seconds. And the period of the shorter pendulum, 𝑇 subscript 𝑠, is equal to one second.

We’ve been asked to find the ratio between the period of the shorter pendulum and the period of the longer pendulum. In other words, we’ve been asked to find 𝑇 subscript 𝑠 divided by 𝑇 subscript 𝑙 because we can recall that the ratio between one quantity and another can be given by dividing that first quantity by the second quantity.

For example, if we’d been asked to find the ratio between some quantities 𝑎 and 𝑏, then this would be equal to 𝑎 divided by 𝑏. And in this case, we’ve been asked to find the ratio between the shorter pendulum’s period and the longer pendulum’s period. So, we find 𝑇 subscript 𝑠 divided by 𝑇 subscript 𝑙.

This simply ends up being one second divided by five seconds. And we can see that we’ve got a unit of seconds in the numerator and the denominator. So, dividing seconds by seconds means that those cancel to give us a unitless quantity, which is important because this ratio is going to have no units whatsoever since it’s a time period divided by a time period.

And so, we find that the ratio that we’re looking for is simply one-fifth, which is equal to 0.2 in decimal notation. We can choose to give either of these answers, which means we found that the ratio of the shorter pendulum’s period to the longer pendulum’s period is one-fifth or 0.2.