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

### Step 1: Identify the given values
- Amplitude (A) = 0.8 cm
- Time period (T) = 0.2 sec
- Initial phase (Φ) = π/2 radian

### Step 2: Write the general equation of SHM
The general equation for SHM can be expressed as:
\[ y(t) = A \sin(\omega t + \Phi) \]
where:
- \( y(t) \) is the displacement at time \( t \)
- \( A \) is the amplitude
- \( \omega \) is the angular frequency
- \( \Phi \) is the initial phase

### Step 3: Calculate the angular frequency (ω)
The angular frequency \( \omega \) is related to the time period \( T \) by the formula:
\[ \omega = \frac{2\pi}{T} \]
Substituting the given time period:
\[ \omega = \frac{2\pi}{0.2} = 10\pi \, \text{radians/second} \]

### Step 4: Substitute the values into the SHM equation
Now we can substitute the values of \( A \), \( \omega \), and \( \Phi \) into the general equation:
\[ y(t) = 0.8 \sin(10\pi t + \frac{\pi}{2}) \]

### Step 5: Simplify the equation using trigonometric identities
Using the trigonometric identity \( \sin\left(\frac{\pi}{2} + \theta\right) = \cos(\theta) \), we can rewrite the equation:
\[ y(t) = 0.8 \cos(10\pi t) \]

### Final Equation
Thus, the equation representing the SHM is:
\[ y(t) = 0.8 \cos(10\pi t) \]