A simple pendulum in oscillation has a maximum angular displacement of `(pi)/(72) rad`, and a time period of `2 s`, then angular speed at the instant when it's angular displacement is `(3 pi)/(360)` is `(k pi^2)/(360)`. Find k.

Answer

To solve the problem step by step, we will follow the outlined approach:

### Step 1: Identify Given Values
- Maximum angular displacement, \( \theta_0 = \frac{\pi}{72} \) rad
- Time period, \( T = 2 \) s
- Angular displacement at the instant, \( \theta = \frac{3\pi}{360} = \frac{\pi}{120} \) rad

### Step 2: Calculate Angular Frequency
The angular frequency \( \omega \) can be calculated using the formula:
\[ \omega = \frac{2\pi}{T} \]
Substituting the value of \( T \):
\[ \omega = \frac{2\pi}{2} = \pi \text{ rad/s} \]

### Step 3: Use the Equation of Motion for Angular Displacement
The equation for angular displacement in simple harmonic motion is:
\[ \theta = \theta_0 \sin(\omega t) \]
We need to find \( \sin(\omega t) \) when \( \theta = \frac{\pi}{120} \):
\[ \frac{\pi}{120} = \frac{\pi}{72} \sin(\omega t) \]
Dividing both sides by \( \frac{\pi}{72} \):
\[ \sin(\omega t) = \frac{\frac{\pi}{120}}{\frac{\pi}{72}} = \frac{72}{120} = \frac{3}{5} \]

### Step 4: Find \( \cos(\omega t) \)
Using the Pythagorean identity:
\[ \sin^2(\omega t) + \cos^2(\omega t) = 1 \]
Substituting \( \sin(\omega t) = \frac{3}{5} \):
\[ \left(\frac{3}{5}\right)^2 + \cos^2(\omega t) = 1 \]
\[ \frac{9}{25} + \cos^2(\omega t) = 1 \]
\[ \cos^2(\omega t) = 1 - \frac{9}{25} = \frac{16}{25} \]
Thus,
\[ \cos(\omega t) = \frac{4}{5} \]

### Step 5: Calculate Angular Speed
The angular speed \( \omega \) at any time \( t \) is given by:
\[ \omega = \frac{d\theta}{dt} = \theta_0 \cdot \omega \cdot \cos(\omega t) \]
Substituting the known values:
\[ \omega = \frac{\pi}{72} \cdot \pi \cdot \frac{4}{5} \]
\[ \omega = \frac{4\pi^2}{360} \]

### Step 6: Compare with Given Expression
We are given that the angular speed can be expressed as:
\[ \omega = \frac{k \pi^2}{360} \]
From our calculation, we have:
\[ \frac{4\pi^2}{360} = \frac{k \pi^2}{360} \]
This implies:
\[ k = 4 \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{4} \]