The angle made by the string of a simple pendulum with the vertical depends on time as `theta=pi/90sin[(pis^-1)t]`. Find the length of the pendulum if `g=pi^2ms^-2`

Explanation

To find the length of the pendulum given the angle made by the string with the vertical as \(\theta = \frac{\pi}{90} \sin\left(\pi t^{-1}\right)\) and \(g = \pi^2 \, \text{m/s}^2\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the angular frequency (\(\omega\))**:
The angle \(\theta\) is given in the form \(\theta = A \sin(\omega t)\), where \(A = \frac{\pi}{90}\) and \(\omega = \pi \, \text{s}^{-1}\).

2. **Determine the time period (T)**:
The angular frequency is related to the time period by the formula:
\[ \omega = \frac{2\pi}{T} \]
Substituting \(\omega = \pi\):
\[ \pi = \frac{2\pi}{T} \]
Rearranging gives:
\[ T = 2 \, \text{s} \]

3. **Use the formula for the time period of a simple pendulum**:
The time period \(T\) of a simple pendulum is given by:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where \(L\) is the length of the pendulum and \(g\) is the acceleration due to gravity.

4. **Substituting known values**:
We know \(T = 2 \, \text{s}\) and \(g = \pi^2 \, \text{m/s}^2\). Substitute these values into the formula:
\[ 2 = 2\pi \sqrt{\frac{L}{\pi^2}} \]

5. **Simplifying the equation**:
Dividing both sides by \(2\):
\[ 1 = \pi \sqrt{\frac{L}{\pi^2}} \]
This simplifies to:
\[ 1 = \sqrt{\frac{L}{\pi}} \]

6. **Squaring both sides**:
Squaring both sides gives:
\[ 1 = \frac{L}{\pi} \]

7. **Solving for \(L\)**:
Multiplying both sides by \(\pi\):
\[ L = \pi \, \text{m} \]

8. **Conclusion**:
The length of the pendulum is:
\[ L = 1 \, \text{m} \]

### Final Answer:
The length of the pendulum is \(L = 1 \, \text{m}\).

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