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Answer :
Let's solve the problem step by step:
We are given the formula for the period [tex]\( T \)[/tex] of a pendulum:
[tex]\[ T = 2 \pi \sqrt{\frac{L}{32}} \][/tex]
where:
- [tex]\( T \)[/tex] is the period in seconds,
- [tex]\( L \)[/tex] is the length of the pendulum in feet.
We know:
- [tex]\( T = 1.57 \)[/tex] seconds,
- [tex]\( \pi = 3.14 \)[/tex].
We are asked to find [tex]\( L \)[/tex].
1. Rearrange the formula to solve for [tex]\( L \)[/tex]:
[tex]\[
T = 2 \pi \sqrt{\frac{L}{32}}
\][/tex]
Divide both sides by [tex]\( 2 \pi \)[/tex]:
[tex]\[
\frac{T}{2 \pi} = \sqrt{\frac{L}{32}}
\][/tex]
2. Square both sides to eliminate the square root:
[tex]\[
\left(\frac{T}{2 \pi}\right)^2 = \frac{L}{32}
\][/tex]
3. Multiply both sides by 32 to solve for [tex]\( L \)[/tex]:
[tex]\[
L = 32 \times \left(\frac{T}{2 \pi}\right)^2
\][/tex]
4. Substitute in the known values of [tex]\( T \)[/tex] and [tex]\( \pi \)[/tex]:
[tex]\[
L = 32 \times \left(\frac{1.57}{2 \times 3.14}\right)^2
\][/tex]
5. Calculate inside the parentheses first:
[tex]\[
\frac{1.57}{2 \times 3.14} = \frac{1.57}{6.28} \approx 0.25
\][/tex]
6. Square the result:
[tex]\[
0.25^2 = 0.0625
\][/tex]
7. Multiply by 32:
[tex]\[
L = 32 \times 0.0625 = 2
\][/tex]
Therefore, the length of the pendulum is [tex]\( 2 \)[/tex] feet.
We are given the formula for the period [tex]\( T \)[/tex] of a pendulum:
[tex]\[ T = 2 \pi \sqrt{\frac{L}{32}} \][/tex]
where:
- [tex]\( T \)[/tex] is the period in seconds,
- [tex]\( L \)[/tex] is the length of the pendulum in feet.
We know:
- [tex]\( T = 1.57 \)[/tex] seconds,
- [tex]\( \pi = 3.14 \)[/tex].
We are asked to find [tex]\( L \)[/tex].
1. Rearrange the formula to solve for [tex]\( L \)[/tex]:
[tex]\[
T = 2 \pi \sqrt{\frac{L}{32}}
\][/tex]
Divide both sides by [tex]\( 2 \pi \)[/tex]:
[tex]\[
\frac{T}{2 \pi} = \sqrt{\frac{L}{32}}
\][/tex]
2. Square both sides to eliminate the square root:
[tex]\[
\left(\frac{T}{2 \pi}\right)^2 = \frac{L}{32}
\][/tex]
3. Multiply both sides by 32 to solve for [tex]\( L \)[/tex]:
[tex]\[
L = 32 \times \left(\frac{T}{2 \pi}\right)^2
\][/tex]
4. Substitute in the known values of [tex]\( T \)[/tex] and [tex]\( \pi \)[/tex]:
[tex]\[
L = 32 \times \left(\frac{1.57}{2 \times 3.14}\right)^2
\][/tex]
5. Calculate inside the parentheses first:
[tex]\[
\frac{1.57}{2 \times 3.14} = \frac{1.57}{6.28} \approx 0.25
\][/tex]
6. Square the result:
[tex]\[
0.25^2 = 0.0625
\][/tex]
7. Multiply by 32:
[tex]\[
L = 32 \times 0.0625 = 2
\][/tex]
Therefore, the length of the pendulum is [tex]\( 2 \)[/tex] feet.
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