Answer :
To find the length [tex]\( L \)[/tex] of the pendulum, we start with the formula for the period of a pendulum:
[tex]\[ T = 2\pi \sqrt{\frac{L}{32}} \][/tex]
We're given that the period [tex]\( T \)[/tex] is 1.57 seconds, and we can use [tex]\( \pi \approx 3.14 \)[/tex]. Now, let's solve for [tex]\( L \)[/tex]:
1. Substitute the given values into the formula:
[tex]\[ 1.57 = 2 \times 3.14 \times \sqrt{\frac{L}{32}} \][/tex]
2. Simplify the right side:
[tex]\[ 1.57 = 6.28 \sqrt{\frac{L}{32}} \][/tex]
3. To isolate the square root, divide both sides by 6.28:
[tex]\[ \frac{1.57}{6.28} = \sqrt{\frac{L}{32}} \][/tex]
4. Simplify the left side:
[tex]\[ 0.25 = \sqrt{\frac{L}{32}} \][/tex]
5. Remove the square root by squaring both sides:
[tex]\[ (0.25)^2 = \frac{L}{32} \][/tex]
[tex]\[ 0.0625 = \frac{L}{32} \][/tex]
6. Solve for [tex]\( L \)[/tex] by multiplying both sides by 32:
[tex]\[ L = 32 \times 0.0625 \][/tex]
[tex]\[ L = 2 \][/tex]
So, the length of the pendulum is 2 feet.
[tex]\[ T = 2\pi \sqrt{\frac{L}{32}} \][/tex]
We're given that the period [tex]\( T \)[/tex] is 1.57 seconds, and we can use [tex]\( \pi \approx 3.14 \)[/tex]. Now, let's solve for [tex]\( L \)[/tex]:
1. Substitute the given values into the formula:
[tex]\[ 1.57 = 2 \times 3.14 \times \sqrt{\frac{L}{32}} \][/tex]
2. Simplify the right side:
[tex]\[ 1.57 = 6.28 \sqrt{\frac{L}{32}} \][/tex]
3. To isolate the square root, divide both sides by 6.28:
[tex]\[ \frac{1.57}{6.28} = \sqrt{\frac{L}{32}} \][/tex]
4. Simplify the left side:
[tex]\[ 0.25 = \sqrt{\frac{L}{32}} \][/tex]
5. Remove the square root by squaring both sides:
[tex]\[ (0.25)^2 = \frac{L}{32} \][/tex]
[tex]\[ 0.0625 = \frac{L}{32} \][/tex]
6. Solve for [tex]\( L \)[/tex] by multiplying both sides by 32:
[tex]\[ L = 32 \times 0.0625 \][/tex]
[tex]\[ L = 2 \][/tex]
So, the length of the pendulum is 2 feet.
