### Video Transcript

A pendulum of length π one has a
period π one. π one is decreased by 9.00
percent. The pendulumβs 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
pendulumβs 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 weβll 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.

Weβre 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 oneβs length. When we substitute that expression
in for π two in our square root, the factors of π one cancel out. And weβre left with the square root
of 0.91. To three significant figures,
thatβs 0.954.

To get this result as a percent, we
multiply it by 100. And that gives us 95.4 percent. Thatβs the percent of π one that
π two is.