Answer :
Sure! Let's solve this step-by-step.
We are given the period [tex]\( T \)[/tex] of a pendulum and the formula for the period in terms of the length [tex]\( L \)[/tex]:
[tex]\[ T = 2\pi \sqrt{\frac{L}{32}} \][/tex]
We are given:
- [tex]\( T = 1.57 \)[/tex] seconds
- [tex]\( \pi = 3.14 \)[/tex]
We need to find the length [tex]\( L \)[/tex] of the pendulum.
First, let's rearrange the formula to solve for [tex]\( L \)[/tex]:
1. Start with the initial formula:
[tex]\[ T = 2\pi \sqrt{\frac{L}{32}} \][/tex]
2. Divide both sides by [tex]\( 2\pi \)[/tex]:
[tex]\[ \frac{T}{2\pi} = \sqrt{\frac{L}{32}} \][/tex]
3. Square both sides to remove the square root:
[tex]\[ \left(\frac{T}{2\pi}\right)^2 = \frac{L}{32} \][/tex]
4. Finally, multiply both sides by 32 to solve for [tex]\( L \)[/tex]:
[tex]\[ L = 32 \left(\frac{T}{2\pi}\right)^2 \][/tex]
Now, substitute the given values [tex]\( T = 1.57 \)[/tex] and [tex]\( \pi = 3.14 \)[/tex] into the equation:
[tex]\[ L = 32 \left(\frac{1.57}{2 \times 3.14}\right)^2 \][/tex]
Calculate the value inside the parentheses first:
[tex]\[ \frac{1.57}{2 \times 3.14} = \frac{1.57}{6.28} = 0.25 \][/tex]
Next, square 0.25:
[tex]\[ (0.25)^2 = 0.0625 \][/tex]
Finally, multiply by 32:
[tex]\[ L = 32 \times 0.0625 = 2 \][/tex]
Therefore, the length of the pendulum is [tex]\( 2 \)[/tex] feet.
So, the correct answer is:
[tex]\[ \boxed{2 \, \text{feet}} \][/tex]
We are given the period [tex]\( T \)[/tex] of a pendulum and the formula for the period in terms of the length [tex]\( L \)[/tex]:
[tex]\[ T = 2\pi \sqrt{\frac{L}{32}} \][/tex]
We are given:
- [tex]\( T = 1.57 \)[/tex] seconds
- [tex]\( \pi = 3.14 \)[/tex]
We need to find the length [tex]\( L \)[/tex] of the pendulum.
First, let's rearrange the formula to solve for [tex]\( L \)[/tex]:
1. Start with the initial formula:
[tex]\[ T = 2\pi \sqrt{\frac{L}{32}} \][/tex]
2. Divide both sides by [tex]\( 2\pi \)[/tex]:
[tex]\[ \frac{T}{2\pi} = \sqrt{\frac{L}{32}} \][/tex]
3. Square both sides to remove the square root:
[tex]\[ \left(\frac{T}{2\pi}\right)^2 = \frac{L}{32} \][/tex]
4. Finally, multiply both sides by 32 to solve for [tex]\( L \)[/tex]:
[tex]\[ L = 32 \left(\frac{T}{2\pi}\right)^2 \][/tex]
Now, substitute the given values [tex]\( T = 1.57 \)[/tex] and [tex]\( \pi = 3.14 \)[/tex] into the equation:
[tex]\[ L = 32 \left(\frac{1.57}{2 \times 3.14}\right)^2 \][/tex]
Calculate the value inside the parentheses first:
[tex]\[ \frac{1.57}{2 \times 3.14} = \frac{1.57}{6.28} = 0.25 \][/tex]
Next, square 0.25:
[tex]\[ (0.25)^2 = 0.0625 \][/tex]
Finally, multiply by 32:
[tex]\[ L = 32 \times 0.0625 = 2 \][/tex]
Therefore, the length of the pendulum is [tex]\( 2 \)[/tex] feet.
So, the correct answer is:
[tex]\[ \boxed{2 \, \text{feet}} \][/tex]
