Video: Simple Pendulums

A pendulum of length 𝑙₁ has a period 𝑇₁. 𝑙₁ is decreased by 9.00%. The pendulum’s new period is 𝑇₂. What percent of 𝑇₁ is 𝑇₂?

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Video Transcript

A pendulum of length 𝑙 one has a period 𝑇 one. 𝑙 one is decreased by 9.00 percent. The pendulum’s new period is 𝑇 two. What percent of 𝑇 one is 𝑇 two?

To start on our solution, we can recall that the period of a pendulum is equal to two πœ‹ times the square root of the pendulum’s length divided by the acceleration due to gravity 𝑔. For this exercise, to find the percent of 𝑇 one that 𝑇 two is, we wanna take the ratio 𝑇 two to 𝑇 one and then multiply it by 100. That will give us that value as a percent.

First, we write out 𝑇 two in terms of 𝑙 two and 𝑇 one in terms of 𝑙 one based on the relationship for pendulum period. Then we’ll divide these equations one by another. And when we do that, we see that the factors of two πœ‹ cancel out, as well as one over the square root of 𝑔. The fraction simplifies to the square root of 𝑙 two over 𝑙 one.

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

To get this result as a percent, we multiply it by 100. And that gives us 95.4 percent. That’s the percent of 𝑇 one that 𝑇 two is.

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