High School

The period [tex]$T$[/tex] (in seconds) of a pendulum is given by [tex]$T = 2\pi \sqrt{\frac{L}{32}}$[/tex], where [tex]$L$[/tex] stands for the length (in feet) of the pendulum. If [tex]$\pi = 3.14$[/tex] and the period is 1.57 seconds, what is the length?

A. 8 feet
B. 2 feet
C. 20 feet
D. 16 feet

Answer :

To solve the problem, we need to find the length of the pendulum when the period [tex]\( T = 1.57 \)[/tex] seconds using the formula:

[tex]\[ T = 2 \pi \sqrt{\frac{L}{32}} \][/tex]

where [tex]\( \pi \)[/tex] is approximately 3.14, and [tex]\( L \)[/tex] is the length of the pendulum in feet.

Step-by-Step Solution:

1. Substitute the known values into the formula.

We know the period of the pendulum [tex]\( T = 1.57 \)[/tex] seconds, and we approximate [tex]\( \pi \)[/tex] as 3.14. The formula becomes:

[tex]\[
1.57 = 2 \times 3.14 \times \sqrt{\frac{L}{32}}
\][/tex]

2. Isolate the square root term.

Divide both sides of the equation by [tex]\( 2 \times 3.14 \)[/tex]:

[tex]\[
\frac{1.57}{2 \times 3.14} = \sqrt{\frac{L}{32}}
\][/tex]

Simplifying the left side gives us approximately:

[tex]\[
0.25 = \sqrt{\frac{L}{32}}
\][/tex]

3. Eliminate the square root by squaring both sides.

[tex]\[
(0.25)^2 = \frac{L}{32}
\][/tex]

This results in:

[tex]\[
0.0625 = \frac{L}{32}
\][/tex]

4. Solve for [tex]\( L \)[/tex].

Multiply both sides by 32 to isolate [tex]\( L \)[/tex]:

[tex]\[
L = 0.0625 \times 32
\][/tex]

Simplifying this gives:

[tex]\[
L = 2
\][/tex]

Therefore, the length of the pendulum is [tex]\( 2 \)[/tex] feet.

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