A particle execute SHM and its position varies with time as `x = A sin omega t`. Its average speed during its motion from mean position to mid-point of mean and extreme position is

Options

A. zero
B. `(3 A omega)/(pi)`
C. `(A omega)/(2pi)`
D. `(2A omega)/(pi)`

Answer

To find the average speed of a particle executing simple harmonic motion (SHM) from the mean position to the midpoint of the mean and extreme position, we can follow these steps:

### Step 1: Understand the SHM Equation
The position of the particle in SHM is given by:
\[ x = A \sin(\omega t) \]
where \( A \) is the amplitude and \( \omega \) is the angular frequency.

### Step 2: Identify the Mean and Extreme Positions
In SHM:
- The mean position is at \( x = 0 \).
- The extreme positions are at \( x = A \) and \( x = -A \).

### Step 3: Determine the Midpoint
The midpoint between the mean position and the extreme position \( A \) is:
\[ \text{Midpoint} = \frac{0 + A}{2} = \frac{A}{2} \]

### Step 4: Calculate the Distance Covered
The distance covered by the particle from the mean position (0) to the midpoint (\( A/2 \)) is:
\[ \text{Distance} = \frac{A}{2} - 0 = \frac{A}{2} \]

### Step 5: Find the Time Taken to Reach Midpoint
To find the time taken to reach the midpoint, we need to find the angle \( \phi \) corresponding to \( x = \frac{A}{2} \):
Using the sine function:
\[ \sin(\phi) = \frac{\frac{A}{2}}{A} = \frac{1}{2} \]
This gives:
\[ \phi = \frac{\pi}{6} \]

Now, since the angular displacement \( \omega t = \phi \), we can find the time \( t \):
\[ t = \frac{\phi}{\omega} = \frac{\frac{\pi}{6}}{\omega} = \frac{\pi}{6\omega} \]

### Step 6: Calculate the Average Speed
The average speed \( v_{avg} \) is given by the formula:
\[ v_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} \]
Substituting the values we have:
\[ v_{avg} = \frac{\frac{A}{2}}{\frac{\pi}{6\omega}} \]
\[ v_{avg} = \frac{A}{2} \cdot \frac{6\omega}{\pi} \]
\[ v_{avg} = \frac{3A\omega}{\pi} \]

### Final Answer
Thus, the average speed of the particle during its motion from the mean position to the midpoint of the mean and extreme position is:
\[ \boxed{\frac{3A\omega}{\pi}} \]

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