Question Video: Simple Pendulums | Nagwa Question Video: Simple Pendulums | Nagwa

# Question Video: Simple Pendulums

A pendulum of length πβ has a period πβ. πβ is decreased by 9.00%. The pendulumβs new period is πβ. What percent of πβ is πβ?

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### Video Transcript

A pendulum of length π one has a period π one. π one is decreased by 9.00 percent. The pendulumβs new period is π two. What percent of π one is π two?

To start on our solution, we can recall that the period of a pendulum is equal to two π times the square root of the pendulumβs length divided by the acceleration due to gravity π. For this exercise, to find the percent of π one that π two is, we wanna take the ratio π two to π one and then multiply it by 100. That will give us that value as a percent.

First, we write out π two in terms of π two and π one in terms of π one based on the relationship for pendulum period. Then weβll divide these equations one by another. And when we do that, we see that the factors of two π cancel out, as well as one over the square root of π. The fraction simplifies to the square root of π two over π one.

Weβre told in the problem statement that π two, the shorter pendulum arm length, is equal to π one minus nine percent of π one. Mathematically, this is π one times the quantity one minus 0.09. Or π two is equal to 0.91 π one, that is, 91 percent of π oneβs length. When we substitute that expression in for π two in our square root, the factors of π one cancel out. And weβre left with the square root of 0.91. To three significant figures, thatβs 0.954.

To get this result as a percent, we multiply it by 100. And that gives us 95.4 percent. Thatβs the percent of π one that π two is.

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