A particle is executing SHM according to the equation x`=A cosomegat`. Average speed of the particle during the interval `0larrtlarrpi/(6omega)`

Options

A. `(sqrt(3)Aomega)/2`
B. `(sqrt(3)Aomega)/4`
C. `(sqrt(Aomega)/pi)`
D. None of these

Answer

To find the average speed of a particle executing Simple Harmonic Motion (SHM) described by the equation \( x = A \cos(\omega t) \) during the interval from \( t = 0 \) to \( t = \frac{\pi}{6\omega} \), we can follow these steps:

### Step 1: Determine the position at \( t = 0 \)
At \( t = 0 \):
\[ x(0) = A \cos(0) = A \]
The particle is at its maximum displacement, which is \( A \).

### Step 2: Determine the position at \( t = \frac{\pi}{6\omega} \)
At \( t = \frac{\pi}{6\omega} \):
\[ x\left(\frac{\pi}{6\omega}\right) = A \cos\left(\omega \cdot \frac{\pi}{6\omega}\right) = A \cos\left(\frac{\pi}{6}\right) \]
Using the value of \( \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \):
\[ x\left(\frac{\pi}{6\omega}\right) = A \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}A}{2} \]

### Step 3: Calculate the distance traveled
The distance traveled by the particle during the time interval from \( t = 0 \) to \( t = \frac{\pi}{6\omega} \) is:
\[ \text{Distance} = x(0) - x\left(\frac{\pi}{6\omega}\right) = A - \frac{\sqrt{3}A}{2} \]
This simplifies to:
\[ \text{Distance} = A - \frac{\sqrt{3}A}{2} = \frac{2A}{2} - \frac{\sqrt{3}A}{2} = \frac{(2 - \sqrt{3})A}{2} \]

### Step 4: Calculate the total time
The total time for the interval is:
\[ \Delta t = \frac{\pi}{6\omega} - 0 = \frac{\pi}{6\omega} \]

### Step 5: Calculate the average speed
The average speed \( v_{avg} \) is given by the formula:
\[ v_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{\frac{(2 - \sqrt{3})A}{2}}{\frac{\pi}{6\omega}} \]
This simplifies to:
\[ v_{avg} = \frac{(2 - \sqrt{3})A}{2} \cdot \frac{6\omega}{\pi} = \frac{3\omega A (2 - \sqrt{3})}{\pi} \]

### Final Answer
Thus, the average speed of the particle during the interval \( 0 \) to \( \frac{\pi}{6\omega} \) is:
\[ v_{avg} = \frac{3\omega A (2 - \sqrt{3})}{\pi} \]