The displacement of a particle executing SHM is given by
`Y=5 " sin "(4t+(pi)/(3))`
If T is the time period and the mass of the particle is 2g, the kinetic energy of the particle When t=`(T)/(4)` is given by-

Options

A. 0.4J
B. 0.5J
C. 3J
D. 0.3J

Explanation

To find the kinetic energy of a particle executing simple harmonic motion (SHM) at a specific time, we can follow these steps:

### Step 1: Identify the given parameters
The displacement of the particle is given by:
\[ Y = 5 \sin(4t + \frac{\pi}{3}) \]
From this equation, we can identify:
- Amplitude \( A = 5 \)
- Angular frequency \( \omega = 4 \) rad/s

The mass of the particle is given as \( m = 2 \) g, which we will convert to kg:
\[ m = 2 \times 10^{-3} \text{ kg} \]

### Step 2: Calculate the time period \( T \)
The time period \( T \) is related to the angular frequency \( \omega \) by the formula:
\[ T = \frac{2\pi}{\omega} \]
Substituting the value of \( \omega \):
\[ T = \frac{2\pi}{4} = \frac{\pi}{2} \text{ seconds} \]

### Step 3: Find the velocity \( v \)
The velocity \( v \) of the particle in SHM is given by the derivative of the displacement \( Y \) with respect to time \( t \):
\[ v = \frac{dY}{dt} \]
Calculating the derivative:
\[ v = \frac{d}{dt}[5 \sin(4t + \frac{\pi}{3})] = 5 \cdot 4 \cos(4t + \frac{\pi}{3}) = 20 \cos(4t + \frac{\pi}{3}) \]

### Step 4: Evaluate the velocity at \( t = \frac{T}{4} \)
We need to find the velocity when \( t = \frac{T}{4} \):
\[ t = \frac{T}{4} = \frac{\pi}{2} \cdot \frac{1}{4} = \frac{\pi}{8} \]
Substituting \( t = \frac{\pi}{8} \) into the velocity equation:
\[ v = 20 \cos\left(4 \cdot \frac{\pi}{8} + \frac{\pi}{3}\right) \]
\[ v = 20 \cos\left(\frac{\pi}{2} + \frac{\pi}{3}\right) \]
Using the cosine addition formula:
\[ v = 20 \cos\left(\frac{\pi}{2} + \frac{\pi}{3}\right) = 20 \cdot \left(-\sin\left(\frac{\pi}{3}\right)\right) = 20 \cdot \left(-\frac{\sqrt{3}}{2}\right) = -10\sqrt{3} \]

### Step 5: Calculate the kinetic energy \( KE \)
The kinetic energy \( KE \) is given by:
\[ KE = \frac{1}{2} m v^2 \]
Substituting the values of \( m \) and \( v \):
\[ KE = \frac{1}{2} \cdot (2 \times 10^{-3}) \cdot (-10\sqrt{3})^2 \]
Calculating \( v^2 \):
\[ v^2 = (10\sqrt{3})^2 = 100 \cdot 3 = 300 \]
Now substituting back into the kinetic energy formula:
\[ KE = \frac{1}{2} \cdot (2 \times 10^{-3}) \cdot 300 \]
\[ KE = (10^{-3}) \cdot 300 = 0.3 \text{ Joules} \]

### Final Answer
The kinetic energy of the particle when \( t = \frac{T}{4} \) is:
\[ \boxed{0.3 \text{ Joules}} \]