Chapter 6.7, Problem 13CYU

(a).

To determine

Find the time taken by the pendulum to take a complete swing.

Answer to Problem 13CYU

about 9.5 seconds

Explanation of Solution

Given:

The time T in seconds that it takes a pendulum to make a complete swing back and forth is given by the formula $T=2\pi \sqrt{\frac{L}{g}}$ , where L is the length of the pendulum in feet and g is the acceleration due to gravity , 32 feet per second squared.

In Tokyo , Japan , a huge pendulum in the Shinjuku building measures 73 feet 9.75 inches

Formula Used:

1 inch = 0.083333 inches

Calculation:

Given:

  $T=2\pi \sqrt{\frac{L}{g}}$

9.75 inches =0.812496

Substitute

L = 73.8125

g = 32

  $\begin{array}{l}T=2\pi \sqrt{\frac{L}{g}}\\ \Rightarrow T=2\pi \sqrt{\frac{73.8125}{32}}\\ \Rightarrow T=2\pi \sqrt{2.3}\\ \Rightarrow T=2\pi (1.52)\\ \Rightarrow T=9.550441\approx 9.5\end{array}$

The pendulum takes about 9.5 seconds to complete the swing.

(b).

To determine

Find the length of the pendulum.

Answer to Problem 13CYU

The length of the pendulum is about 324 feet.

Explanation of Solution

Given:

A clockmaker wants to build a pendulum that takes 20 seconds to swing back and forth .

Calculation:

Given:

  $T=2\pi \sqrt{\frac{L}{g}}$

Substitute

T = 20 seconds

g = 32

  $\begin{array}{l}T=2\pi \sqrt{\frac{L}{g}}\\ \Rightarrow 20=2\pi \sqrt{\frac{L}{32}}\\ \Rightarrow \frac{20}{2\pi }=\sqrt{\frac{L}{32}}\mathrm{..........................}[\text{divide each side by 2}\pi \text{]}\\ \Rightarrow \frac{L}{32}={\left(\frac{20}{2\pi }\right)}^2\mathrm{..............}[\text{square each side]}\\ \Rightarrow L=32\cdot \frac{400}{4{\pi }^2}\mathrm{..............}[\text{multiply 32 each side]}\\ \Rightarrow L=324.22778\approx 324\end{array}$

So, the length of the pendulum is about 324 feet.