Answer :
Sure! Let's go through the solution to find the length of the pendulum 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 of the pendulum in seconds.
- [tex]\( L \)[/tex] is the length of the pendulum in feet.
- [tex]\( \pi \)[/tex] is approximately 3.14.
- The constant 32 is the acceleration due to gravity measured in feet per second squared.
We are told that the period [tex]\( T \)[/tex] is 1.57 seconds, and we need to find the length [tex]\( L \)[/tex].
### Steps to Find [tex]\( L \)[/tex]:
1. Start with the given formula:
[tex]\[ T = 2 \pi \sqrt{\frac{L}{32}} \][/tex]
2. Plug in the given values:
[tex]\[ 1.57 = 2 \times 3.14 \times \sqrt{\frac{L}{32}} \][/tex]
3. Solve for [tex]\( \sqrt{\frac{L}{32}} \)[/tex]:
Divide both sides by [tex]\( 2 \times 3.14 \)[/tex]:
[tex]\[ \frac{1.57}{2 \times 3.14} = \sqrt{\frac{L}{32}} \][/tex]
4. Square both sides to eliminate the square root:
[tex]\[ \left(\frac{1.57}{2 \times 3.14}\right)^2 = \frac{L}{32} \][/tex]
5. Solve for [tex]\( L \)[/tex]:
Multiply both sides by 32:
[tex]\[ L = 32 \times \left(\frac{1.57}{2 \times 3.14}\right)^2 \][/tex]
After calculating this expression, we find that [tex]\( L \)[/tex] is 2 feet.
So, the length of the pendulum is 2 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 of the pendulum in seconds.
- [tex]\( L \)[/tex] is the length of the pendulum in feet.
- [tex]\( \pi \)[/tex] is approximately 3.14.
- The constant 32 is the acceleration due to gravity measured in feet per second squared.
We are told that the period [tex]\( T \)[/tex] is 1.57 seconds, and we need to find the length [tex]\( L \)[/tex].
### Steps to Find [tex]\( L \)[/tex]:
1. Start with the given formula:
[tex]\[ T = 2 \pi \sqrt{\frac{L}{32}} \][/tex]
2. Plug in the given values:
[tex]\[ 1.57 = 2 \times 3.14 \times \sqrt{\frac{L}{32}} \][/tex]
3. Solve for [tex]\( \sqrt{\frac{L}{32}} \)[/tex]:
Divide both sides by [tex]\( 2 \times 3.14 \)[/tex]:
[tex]\[ \frac{1.57}{2 \times 3.14} = \sqrt{\frac{L}{32}} \][/tex]
4. Square both sides to eliminate the square root:
[tex]\[ \left(\frac{1.57}{2 \times 3.14}\right)^2 = \frac{L}{32} \][/tex]
5. Solve for [tex]\( L \)[/tex]:
Multiply both sides by 32:
[tex]\[ L = 32 \times \left(\frac{1.57}{2 \times 3.14}\right)^2 \][/tex]
After calculating this expression, we find that [tex]\( L \)[/tex] is 2 feet.
So, the length of the pendulum is 2 feet.
