Question Video: Calculating the Period of a Pendulum | Nagwa Question Video: Calculating the Period of a Pendulum | Nagwa

# Question Video: Calculating the Period of a Pendulum Physics

A pendulum has a length of 0.500 m. What is the period of the pendulum? Use a value of 9.81 m/s² for the local acceleration due to gravity. Give your answer to 3 significant figures.

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### Video Transcript

A pendulum has a length of 0.500 meters. What is the period of the pendulum? Use a value of 9.81 meters per second squared for the local acceleration due to gravity. Give your answer to three significant figures.

So in this question, we’re considering a pendulum, and we’re being asked to find its period. For any oscillating system, the period refers to the amount of time taken to complete a full cycle. For a pendulum, that means the time taken for it to swing from its starting position to the other side and then back to its starting position. In this case, we’re told that our pendulum has a length of 0.500 meters. And we’re also told that the local acceleration due to gravity is 9.81 meters per second squared, which is the standard gravitational acceleration for an object near the surface of the Earth.

Now, it may not seem like we’ve been given much information about this pendulum. For example, we have no idea what its mass is. However, this is all the information we need to calculate its period. In fact, we can recall that the period of a pendulum is given by this formula. 𝑇 is approximately equal to two 𝜋 times the square root of 𝐿 over 𝑔, where 𝑇 is the period of the pendulum, 𝐿 is the length of the pendulum, and 𝑔 is the gravitational acceleration experienced by the pendulum.

Note that in this formula, 𝑇 is only approximately equal to the expression on the right-hand side. This is because the formula is derived using a number of approximations. Importantly, it uses a small-angle approximation, which means it’s only accurate if the initial angle of the pendulum from the vertical, 𝜃 zero, is much smaller than one radian. The bigger the initial angle is, the more 𝑇 will vary from the expression given on the right. So in our answer to this question, we’re assuming that the pendulum only swings over a small angle.

In this question, we’re told that the length of the pendulum is 0.500 meters and the local acceleration due to gravity is 9.81 meters per second squared. So all we need to do to calculate the period of the pendulum is substitute these values into this formula. Substituting them in gives us two 𝜋 times the square root of 0.500 meters divided by 9.81 meters per second squared.

At this point, it’s worth mentioning that this formula only works when we use the magnitude of the acceleration. It’s fairly common in physics problems to define upward acceleration as positive and downward acceleration as negative. But when we’re using this formula, it’s important that we don’t use a negative value for acceleration.

If we type all of this into our calculator, we obtain a value of 1.418 and so on, which, rounded to three significant figures, is 1.42. Since the period is a measure of time, we would expect our answer to have units of time. And if we look at the units we’ve used in our calculation, we can see that this is the case. We’re dividing the length, which is a quantity in meters, by an acceleration, measured in meters per second squared. And we’re then taking the square root of this quantity. We can see that in this fraction, the factor of meters on the top and the bottom will cancel, leaving us with the square root of one over one over second squared.

Now, when we have one over one over a quantity, in other words, the reciprocal of the reciprocal of a quantity, this actually just leaves us with whatever quantity it was that we started with, in this case second squared. And the square root of second squared is of course seconds, which shows us that our answer is 1.42 seconds. And this is the answer to the question. If a pendulum has a length of 0.500 meters and the local acceleration due to gravity is 9.81 meters per second squared, then the period of the pendulum is 1.42 seconds.

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