The maximum speed of a particle in SHM is given by `mA=V_(m)`. The average speed of the particle in terms of maximum speed is given by

Options

A. `(V_(m))/(2)`
B. `(pi V_(m))/(2)`
C. `(2V_(m))/(pi)`
D. `(pi)/(2V_(m))`

Explanation

To find the average speed of a particle in Simple Harmonic Motion (SHM) in terms of its maximum speed, we can follow these steps:

### Step 1: Understand the Maximum Speed
The maximum speed \( V_{max} \) of a particle in SHM is given by the formula:
\[ V_{max} = \omega A \]
where \( \omega \) is the angular frequency and \( A \) is the amplitude of the motion.

### Step 2: Calculate the Total Distance Covered in One Cycle
In one complete cycle of SHM, the particle moves from the mean position to the maximum amplitude in one direction, back to the mean position, and then to the maximum amplitude in the opposite direction, and back again to the mean position. Therefore, the total distance covered in one cycle is:
\[ \text{Total Distance} = 4A \]

### Step 3: Determine the Time Period
The time taken to complete one full cycle is known as the time period \( T \). The time period can be expressed in terms of angular frequency as:
\[ T = \frac{2\pi}{\omega} \]

### Step 4: Relate Angular Frequency to Maximum Speed
From the maximum speed formula, we can express \( \omega \) in terms of \( V_{max} \):
\[ \omega = \frac{V_{max}}{A} \]

### Step 5: Substitute \( \omega \) into the Time Period Formula
Substituting the expression for \( \omega \) into the time period formula gives:
\[ T = \frac{2\pi}{\frac{V_{max}}{A}} = \frac{2\pi A}{V_{max}} \]

### Step 6: Calculate the Average Speed
The average speed \( V_{avg} \) is defined as the total distance covered divided by the time taken to cover that distance:
\[ V_{avg} = \frac{\text{Total Distance}}{T} = \frac{4A}{T} \]
Substituting the expression for \( T \):
\[ V_{avg} = \frac{4A}{\frac{2\pi A}{V_{max}}} \]

### Step 7: Simplify the Expression
Now, simplify the expression for average speed:
\[ V_{avg} = \frac{4A \cdot V_{max}}{2\pi A} = \frac{4V_{max}}{2\pi} = \frac{2V_{max}}{\pi} \]

### Final Answer
Thus, the average speed of the particle in terms of maximum speed is:
\[ V_{avg} = \frac{2V_{max}}{\pi} \]

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