The equation of S.H.M. with amplitude 4m and time period 1/2 s with initial phase `pi//3` is x =

Options

A. `4sin(2pit+pi//3)m`
B. `4sin(4pit+pi//3)m`
C. `4sin(pit+pi//3)m`
D. `4sin(2pit+pi//6)m`

Explanation

To find the equation of simple harmonic motion (S.H.M.) given the amplitude, time period, and initial phase, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Parameters**:
- Amplitude \( A = 4 \, \text{m} \)
- Time period \( T = \frac{1}{2} \, \text{s} \)
- Initial phase \( \phi = \frac{\pi}{3} \)

2. **Calculate Angular Frequency \( \omega \)**:
The angular frequency \( \omega \) is calculated using the formula:
\[ \omega = \frac{2\pi}{T} \]
Substituting the value of \( T \):
\[ \omega = \frac{2\pi}{\frac{1}{2}} = 2\pi \times 2 = 4\pi \, \text{rad/s} \]

3. **Write the Equation of S.H.M.**:
The general equation of S.H.M. can be expressed in terms of sine or cosine. Here, we will use the sine function:
\[ x(t) = A \sin(\omega t + \phi) \]
Substituting the values of \( A \), \( \omega \), and \( \phi \):
\[ x(t) = 4 \sin(4\pi t + \frac{\pi}{3}) \]

4. **Final Equation**:
Thus, the equation of S.H.M. is:
\[ x(t) = 4 \sin(4\pi t + \frac{\pi}{3}) \]