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Answer :
To find the length of the pendulum when the period is given, you can use the formula for the period of a pendulum:
[tex]\[ T = 2 \pi \sqrt{\frac{L}{32}} \][/tex]
Here, [tex]\( T \)[/tex] is the period (1.57 seconds), [tex]\(\pi\)[/tex] is approximately 3.14, and the gravitational acceleration is 32 ft/s² (as given in the formula). We need to solve for [tex]\( L \)[/tex], the length of the pendulum.
Let's solve for [tex]\( L \)[/tex] step by step:
1. Isolate the square root term:
[tex]\[
\sqrt{\frac{L}{32}} = \frac{T}{2\pi}
\][/tex]
2. Substitute the values for [tex]\( T \)[/tex] and [tex]\(\pi\)[/tex]:
[tex]\[
\sqrt{\frac{L}{32}} = \frac{1.57}{2 \times 3.14}
\][/tex]
3. Calculate the right side:
[tex]\[
\frac{1.57}{6.28} \approx 0.25
\][/tex]
So,
[tex]\[
\sqrt{\frac{L}{32}} = 0.25
\][/tex]
4. Remove the square root by squaring both sides:
[tex]\[
\frac{L}{32} = 0.25^2 = 0.0625
\][/tex]
5. Solve for [tex]\( L \)[/tex] by multiplying both sides by 32:
[tex]\[
L = 0.0625 \times 32
\][/tex]
6. Calculate [tex]\( L \)[/tex]:
[tex]\[
L = 2.0
\][/tex]
Therefore, the length [tex]\( L \)[/tex] of the pendulum is 2 feet.
[tex]\[ T = 2 \pi \sqrt{\frac{L}{32}} \][/tex]
Here, [tex]\( T \)[/tex] is the period (1.57 seconds), [tex]\(\pi\)[/tex] is approximately 3.14, and the gravitational acceleration is 32 ft/s² (as given in the formula). We need to solve for [tex]\( L \)[/tex], the length of the pendulum.
Let's solve for [tex]\( L \)[/tex] step by step:
1. Isolate the square root term:
[tex]\[
\sqrt{\frac{L}{32}} = \frac{T}{2\pi}
\][/tex]
2. Substitute the values for [tex]\( T \)[/tex] and [tex]\(\pi\)[/tex]:
[tex]\[
\sqrt{\frac{L}{32}} = \frac{1.57}{2 \times 3.14}
\][/tex]
3. Calculate the right side:
[tex]\[
\frac{1.57}{6.28} \approx 0.25
\][/tex]
So,
[tex]\[
\sqrt{\frac{L}{32}} = 0.25
\][/tex]
4. Remove the square root by squaring both sides:
[tex]\[
\frac{L}{32} = 0.25^2 = 0.0625
\][/tex]
5. Solve for [tex]\( L \)[/tex] by multiplying both sides by 32:
[tex]\[
L = 0.0625 \times 32
\][/tex]
6. Calculate [tex]\( L \)[/tex]:
[tex]\[
L = 2.0
\][/tex]
Therefore, the length [tex]\( L \)[/tex] of the pendulum is 2 feet.
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