This problem has me confused..

Question

The function x(T)=9.8(T/[2π])^2 gives the length x in meters for a pendulum to swing for a period of T seconds. Write the inverse function to find the period of a pendulum in seconds. The period of a pendulum is the time it takes the time to complete one back and forth swing. Find the period of a pendulum with length of 5 meters.

Answer 1

Since x, the length (in meters) of the pendulum is given as a function of its period T (in seconds), we can find the inverse function in order to write T(x). This means that we want to re-write the expression below so that only T is on one side and the other terms are on the other side.

 

x = 9.8(T/2π)²

 

Divide both sides by 9.8,

 

x/9.8 = (T/2π)²

 

Take the square root of both sides:

 

√(x/9.8) = T/2π

 

Multiply both sides by 2π:

 

2π√(x/9.8) = T

 

This means that T(x), which mean the period T as a function of the length of the pendulum x is:

 

T(x) = 2π√(x/9.8).

 

So, the period of a pendulum of length 5 meters is:

 

T(5) = 2*π*√(5/9.8) = 4.5 seconds

Answer 2

x(t) = 9.8(T/2pi)²

 

To find the inverse function, just solve for T:

 

x = 9.8(T/2pi)²

 

x/9.8 = (T/2pi)²

 

√(x/9.8) = T/2pi

 

(2pi)√(x/9.8) = T

 

To find the period of a pendulum with length of 5 meters, just plug in x = 5 and solve for T.