Answer :
To solve the problem of finding the length [tex]\( L \)[/tex] of a pendulum when the period [tex]\( T \)[/tex] is given, along with an approximation for [tex]\(\pi\)[/tex], we follow these steps:
1. Understand the Formula: The period [tex]\( T \)[/tex] of a pendulum is given by the formula:
[tex]\[
T = 2 \pi \sqrt{\frac{L}{32}}
\][/tex]
where [tex]\( T \)[/tex] is the period in seconds, and [tex]\( L \)[/tex] is the length in feet.
2. Rearrange the Formula: We need to solve for [tex]\( L \)[/tex]. Let's rearrange the formula:
Start by isolating the square root:
[tex]\[
\sqrt{\frac{L}{32}} = \frac{T}{2\pi}
\][/tex]
Square both sides to eliminate the square root:
[tex]\[
\frac{L}{32} = \left(\frac{T}{2\pi}\right)^2
\][/tex]
Solve for [tex]\( L \)[/tex] by multiplying both sides by 32:
[tex]\[
L = 32 \times \left(\frac{T}{2\pi}\right)^2
\][/tex]
3. Insert Known Values: We know:
- [tex]\( T = 1.57 \)[/tex] seconds
- [tex]\(\pi = 3.14\)[/tex]
Substitute these values into the rearranged formula:
[tex]\[
L = 32 \times \left(\frac{1.57}{2 \times 3.14}\right)^2
\][/tex]
4. Calculate: Let's calculate the value inside the brackets first:
- [tex]\( 2 \times 3.14 = 6.28 \)[/tex]
- [tex]\( \frac{1.57}{6.28} \approx 0.25 \)[/tex]
Now, square [tex]\( 0.25 \)[/tex]:
- [tex]\( (0.25)^2 = 0.0625 \)[/tex]
Finally, multiply by 32 to find [tex]\( L \)[/tex]:
- [tex]\( L = 32 \times 0.0625 = 2 \)[/tex]
Thus, the length of the pendulum is 2 feet.
1. Understand the Formula: The period [tex]\( T \)[/tex] of a pendulum is given by the formula:
[tex]\[
T = 2 \pi \sqrt{\frac{L}{32}}
\][/tex]
where [tex]\( T \)[/tex] is the period in seconds, and [tex]\( L \)[/tex] is the length in feet.
2. Rearrange the Formula: We need to solve for [tex]\( L \)[/tex]. Let's rearrange the formula:
Start by isolating the square root:
[tex]\[
\sqrt{\frac{L}{32}} = \frac{T}{2\pi}
\][/tex]
Square both sides to eliminate the square root:
[tex]\[
\frac{L}{32} = \left(\frac{T}{2\pi}\right)^2
\][/tex]
Solve for [tex]\( L \)[/tex] by multiplying both sides by 32:
[tex]\[
L = 32 \times \left(\frac{T}{2\pi}\right)^2
\][/tex]
3. Insert Known Values: We know:
- [tex]\( T = 1.57 \)[/tex] seconds
- [tex]\(\pi = 3.14\)[/tex]
Substitute these values into the rearranged formula:
[tex]\[
L = 32 \times \left(\frac{1.57}{2 \times 3.14}\right)^2
\][/tex]
4. Calculate: Let's calculate the value inside the brackets first:
- [tex]\( 2 \times 3.14 = 6.28 \)[/tex]
- [tex]\( \frac{1.57}{6.28} \approx 0.25 \)[/tex]
Now, square [tex]\( 0.25 \)[/tex]:
- [tex]\( (0.25)^2 = 0.0625 \)[/tex]
Finally, multiply by 32 to find [tex]\( L \)[/tex]:
- [tex]\( L = 32 \times 0.0625 = 2 \)[/tex]
Thus, the length of the pendulum is 2 feet.
