The amplitude and time period in SHM are 0.8 cm and 0.2 sec respectively. If the initial phase is `pi//2` radian, then the equation representing SHM is -

Options

A. `y=0.8cos 10pit`
B. `y=0.8sinpit`
C. `y=3xx0.8sinpit`
D. `y=0.8sin 10pit`

Answer

To find the equation representing the Simple Harmonic Motion (SHM) given the amplitude, time period, and initial phase, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Amplitude \( A = 0.8 \) cm
- Time period \( T = 0.2 \) sec
- Initial phase \( \phi = \frac{\pi}{2} \) radians

2. **Convert Amplitude to Meters (if necessary):**
- Since the amplitude is given in cm, we can convert it to meters for standard SI units:
\[ A = 0.8 \, \text{cm} = 0.008 \, \text{m} \]

3. **Calculate Angular Frequency \( \omega \):**
- The angular frequency \( \omega \) is related to the time period \( T \) by the formula:
\[ \omega = \frac{2\pi}{T} \]
- Substituting the given time period:
\[ \omega = \frac{2\pi}{0.2} = 10\pi \, \text{rad/sec} \]

4. **Write the General Equation of SHM:**
- The general equation for SHM can be expressed as:
\[ y(t) = A \sin(\omega t + \phi) \]
- Substituting the values we have:
\[ y(t) = 0.008 \sin(10\pi t + \frac{\pi}{2}) \]

5. **Simplify Using Trigonometric Identity:**
- We can use the trigonometric identity \( \sin\left(x + \frac{\pi}{2}\right) = \cos(x) \):
\[ y(t) = 0.008 \cos(10\pi t) \]

6. **Final Equation:**
- Therefore, the equation representing the SHM is:
\[ y(t) = 0.008 \cos(10\pi t) \]

### Conclusion:
The equation that represents the SHM is \( y(t) = 0.008 \cos(10\pi t) \).