To determine the length of the string in the pendulum system, given the period (T) of 1.36 seconds, we use the formula for the period of a simple pendulum:
[tex]\[ T = 2 \pi \sqrt{\frac{L}{g}} \][/tex]
Here:
- [tex]\( T \)[/tex] is the period
- [tex]\( \pi \)[/tex] is a constant approximately equal to 3.14159
- [tex]\( L \)[/tex] is the length of the string
- [tex]\( g \)[/tex] is the acceleration due to gravity, which is approximately [tex]\( 9.8 \, \text{m/s}^2 \)[/tex]
We are given [tex]\( T = 1.36 \, \text{s} \)[/tex] and [tex]\( g = 9.8 \, \text{m/s}^2 \)[/tex], and we need to solve for [tex]\( L \)[/tex].
First, let's isolate [tex]\( L \)[/tex] in the equation.
Start by dividing both sides of the equation by [tex]\( 2\pi \)[/tex]:
[tex]\[ T = 2 \pi \sqrt{\frac{L}{g}} \][/tex]
[tex]\[ \frac{T}{2\pi} = \sqrt{\frac{L}{g}} \][/tex]
Next, square both sides to eliminate the square root:
[tex]\[ \left( \frac{T}{2\pi} \right)^2 = \frac{L}{g} \][/tex]
Then, multiply both sides by [tex]\( g \)[/tex]:
[tex]\[ L = g \left( \frac{T}{2\pi} \right)^2 \][/tex]
Now, plug in the values [tex]\( T = 1.36 \, \text{s} \)[/tex] and [tex]\( g = 9.8 \, \text{m/s}^2 \)[/tex]:
[tex]\[ L = 9.8 \left( \frac{1.36}{2 \pi} \right)^2 \][/tex]
Calculate the value inside the parentheses first:
[tex]\[ \frac{1.36}{2 \pi} \approx \frac{1.36}{6.28318} \approx 0.216 \][/tex]
Next, square this result:
[tex]\[ (0.216)^2 \approx 0.0466 \][/tex]
Finally, multiply by [tex]\( 9.8 \)[/tex]:
[tex]\[ L = 9.8 \times 0.0466 \approx 0.459 \][/tex]
Therefore, the length of the string is approximately [tex]\( 0.459 \)[/tex] meters, to three decimal places.
