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 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 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.
