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 4pit`
C. `y=0.5 sin2.5pit`
D. `y=0.5 cos5pit`

Answer

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 1: Identify the given parameters
- Amplitude (A) = 0.5 cm
- Time period (T) = 0.4 sec
- Initial phase (φ) = π/2 radians

### Step 2: Calculate the angular frequency (ω)
The angular frequency (ω) is calculated using the formula:
\[ \omega = \frac{2\pi}{T} \]
Substituting the value of T:
\[ \omega = \frac{2\pi}{0.4} = 5\pi \text{ rad/sec} \]

### Step 3: Write the general equation of S.H.M.
The general equation of S.H.M. can be expressed as:
\[ x(t) = A \sin(\omega t + \phi) \]
Substituting the values of A, ω, and φ:
\[ x(t) = 0.5 \sin(5\pi t + \frac{\pi}{2}) \]

### Step 4: Simplify the equation using the sine addition formula
Using the sine addition formula:
\[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]
We can simplify:
\[ x(t) = 0.5 \left( \sin(5\pi t) \cos\left(\frac{\pi}{2}\right) + \cos(5\pi t) \sin\left(\frac{\pi}{2}\right) \right) \]
Since \(\cos\left(\frac{\pi}{2}\right) = 0\) and \(\sin\left(\frac{\pi}{2}\right) = 1\):
\[ x(t) = 0.5 \cdot \cos(5\pi t) \]

### Step 5: Finalize the equation
Thus, the equation of S.H.M. is:
\[ x(t) = 0.5 \cos(5\pi t) \]

### Final Answer
The equation of S.H.M. is:
\[ x(t) = 0.5 \cos(5\pi t) \]

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