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 1: Understand the Motion
The position of a particle in SHM can be described by the equation:
\[ x(t) = A \sin(\omega t) \]
where:
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency given by \( \omega = \frac{2\pi}{T} \),
- \( T \) is the time period.

### Step 2: Determine the Velocity
The velocity \( v(t) \) of the particle is the time derivative of the position:
\[ v(t) = \frac{dx}{dt} = A \omega \cos(\omega t) \]

### Step 3: Calculate the Average Velocity
The average velocity \( V_{avg} \) over a time interval from \( 0 \) to \( \frac{T}{4} \) can be calculated using the formula:
\[ V_{avg} = \frac{1}{\Delta t} \int_{0}^{\Delta t} v(t) \, dt \]
where \( \Delta t = \frac{T}{4} \).

### Step 4: Set Up the Integral
Substituting the expression for velocity into the integral:
\[ V_{avg} = \frac{1}{\frac{T}{4}} \int_{0}^{\frac{T}{4}} A \omega \cos(\omega t) \, dt \]

### Step 5: Solve the Integral
Now, we can compute the integral:
\[ \int A \omega \cos(\omega t) \, dt = A \sin(\omega t) \]
Evaluating this from \( 0 \) to \( \frac{T}{4} \):
\[ \int_{0}^{\frac{T}{4}} A \omega \cos(\omega t) \, dt = A \left[ \sin\left(\omega \cdot \frac{T}{4}\right) - \sin(0) \right] \]
Since \( \omega = \frac{2\pi}{T} \):
\[ \sin\left(\omega \cdot \frac{T}{4}\right) = \sin\left(\frac{2\pi}{T} \cdot \frac{T}{4}\right) = \sin\left(\frac{\pi}{2}\right) = 1 \]
Thus:
\[ \int_{0}^{\frac{T}{4}} A \omega \cos(\omega t) \, dt = A(1 - 0) = A \]

### Step 6: Calculate the Average Velocity
Now substituting back into the average velocity formula:
\[ V_{avg} = \frac{1}{\frac{T}{4}} \cdot A = \frac{4A}{T} \]

### Final Answer
The mean velocity of the particle averaged over a quarter oscillation is:
\[ \boxed{\frac{4A}{T}} \]