Question

the period of a pendulum is the time it takes the pendulum to make one full swing back and forth. the period t, in seconds, is given by the formula. l is the length, in feet, of the pendulum. suppose a pendulum used to show the rotation of the earth has a period of approximately 14.8 seconds. what is its length? use 3.14 for π. the length of a pendulum that has a period of 14.8 seconds is feet (round to two decimal places as needed.)

Explanation:

Step1: Write down the pendulum - period formula

The formula for the period of a pendulum is $T = 2\pi\sqrt{\frac{L}{32}}$, where $T$ is the period in seconds and $L$ is the length in feet. We know that $T = 14.8$ seconds and $\pi=3.14$.

Step2: Isolate $L$

First, divide both sides of the equation $T = 2\pi\sqrt{\frac{L}{32}}$ by $2\pi$:
$\frac{T}{2\pi}=\sqrt{\frac{L}{32}}$.
Then square both sides of the equation: $(\frac{T}{2\pi})^2=\frac{L}{32}$.
Finally, multiply both sides by 32 to solve for $L$: $L = 32\times(\frac{T}{2\pi})^2$.

Step3: Substitute the values of $T$ and $\pi$

Substitute $T = 14.8$ and $\pi = 3.14$ into the formula for $L$:
$L=32\times(\frac{14.8}{2\times3.14})^2$.
First, calculate $\frac{14.8}{2\times3.14}=\frac{14.8}{6.28}\approx2.3567$.
Then, $(2.3567)^2\approx5.554$.
Finally, $L = 32\times5.554 = 177.728\approx177.73$.

Answer:

$177.73$