The amplitude and the time period in a S.H.M. is 0.5 cm and 0.4 sec respectively. If the initial phase is `pi//2` radian, then the equation of S.H.M. will be

Options

A. y=0.5 sin 5`pit`
B. y=0.5 sin 4`pit`
C. y=0.5 sin 2.5`pit`
D. y=0.5 cos `5pit`

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 Given Values:**
- Amplitude (A) = 0.5 cm
- Time Period (T) = 0.4 seconds
- Initial Phase (φ) = π/2 radians

2. **Convert Amplitude to Meters (if necessary):**
- Since the amplitude is given in centimeters, we can convert it to meters:
\[ A = 0.5 \text{ cm} = 0.005 \text{ m} \]

3. **Calculate Angular Frequency (ω):**
- The angular frequency (ω) is related to the time period (T) by the formula:
\[ \omega = \frac{2\pi}{T} \]
- Substituting the given time period:
\[ \omega = \frac{2\pi}{0.4} = 5\pi \text{ rad/s} \]

4. **Write the Standard Equation of S.H.M.:**
- The standard equation for S.H.M. can be written as:
\[ x(t) = A \sin(\omega t + \phi) \]
- Substituting the values of A, ω, and φ:
\[ x(t) = 0.005 \sin(5\pi t + \frac{\pi}{2}) \]

5. **Simplify Using Trigonometric Identity:**
- We know that:
\[ \sin\left(\frac{\pi}{2} + \theta\right) = \cos(\theta) \]
- Therefore, we can rewrite the equation as:
\[ x(t) = 0.005 \cos(5\pi t) \]

6. **Final Equation:**
- The final equation of S.H.M. is:
\[ x(t) = 0.005 \cos(5\pi t) \]
- If we convert back to centimeters:
\[ x(t) = 0.5 \cos(5\pi t) \text{ cm} \]

### Conclusion:
The equation of S.H.M. is:
\[ x(t) = 0.5 \cos(5\pi t) \text{ cm} \]
Thus, the correct option is \( y = 0.5 \cos(5\pi t) \).

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