A particle executing SHM with time period T and amplitude A. The mean velocity of the particle averaged over quarter oscillation, is

Options

A. `(A)/(4T)`
B. `(2A)/(T)`
C. `(3A)/(T)`
D. `(4A)/(T)`

Explanation

To find the mean velocity of a particle executing Simple Harmonic Motion (SHM) averaged over a quarter of its oscillation, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Motion**:
The displacement of a particle in SHM can be described by the equation:
\[ x(t) = A \sin(\omega t) \]
where \(A\) is the amplitude and \(\omega\) is the angular frequency.

2. **Find the Velocity**:
The velocity \(v(t)\) is the derivative of displacement with respect to time:
\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}(A \sin(\omega t)) = A \omega \cos(\omega t) \]

3. **Determine the Time for Quarter Oscillation**:
The time period \(T\) of the oscillation is related to the angular frequency by:
\[ \omega = \frac{2\pi}{T} \]
For a quarter oscillation, the time interval is:
\[ \text{Quarter time} = \frac{T}{4} \]

4. **Calculate Mean Velocity**:
The mean velocity \(v_{mean}\) over the time interval from \(0\) to \(\frac{T}{4}\) is given by:
\[ v_{mean} = \frac{1}{\frac{T}{4}} \int_0^{\frac{T}{4}} v(t) \, dt \]
Substituting the expression for velocity:
\[ v_{mean} = \frac{4}{T} \int_0^{\frac{T}{4}} A \omega \cos(\omega t) \, dt \]

5. **Integrate the Velocity Function**:
The integral of \(\cos(\omega t)\) is:
\[ \int \cos(\omega t) \, dt = \frac{1}{\omega} \sin(\omega t) \]
Thus, we evaluate:
\[ \int_0^{\frac{T}{4}} A \omega \cos(\omega t) \, dt = A \left[ \sin(\omega t) \right]_0^{\frac{T}{4}} = A \left( \sin\left(\frac{\pi}{2}\right) - \sin(0) \right) = A(1 - 0) = A \]

6. **Substitute Back into Mean Velocity**:
Now substituting back into the mean velocity formula:
\[ v_{mean} = \frac{4}{T} \cdot A = \frac{4A}{T} \]

7. **Final Result**:
Therefore, the mean velocity of the particle averaged over a quarter oscillation is:
\[ v_{mean} = \frac{4A}{T} \]

### Conclusion:
The correct answer is \( \frac{4A}{T} \).