The equation of S.H.M of a particle whose amplitude is 2 m and frequency 50 Hz. Start from extreme position is

Options

A. `y=(2sin100pit)m`
B. `y=(2cos100pit)m`
C. `y=(2sin50pit)m`
D. `y=(2cos50pit)m`

Answer

To derive the equation of Simple Harmonic Motion (S.H.M) for a particle with given parameters, we will follow these steps:

### Step 1: Identify the parameters
We are given:
- Amplitude (A) = 2 m
- Frequency (f) = 50 Hz

### Step 2: Write the general equation of S.H.M
The general equation for S.H.M can be expressed as:
\[ y(t) = A \sin(2 \pi f t + \phi) \]
where:
- \( y(t) \) is the displacement at time \( t \),
- \( A \) is the amplitude,
- \( f \) is the frequency,
- \( \phi \) is the phase constant.

### Step 3: Substitute the amplitude and frequency
Substituting the values of amplitude and frequency into the equation:
\[ y(t) = 2 \sin(2 \pi \times 50 t + \phi) \]

### Step 4: Determine the phase constant (\( \phi \))
Since the particle starts from the extreme position, the initial phase constant \( \phi \) is \( \frac{\pi}{2} \). This is because at the extreme position, the sine function reaches its maximum value.

### Step 5: Substitute the phase constant into the equation
Now, substituting \( \phi = \frac{\pi}{2} \):
\[ y(t) = 2 \sin(2 \pi \times 50 t + \frac{\pi}{2}) \]

### Step 6: Simplify using the sine addition formula
Using the sine addition formula \( \sin(a + b) = \sin a \cos b + \cos a \sin b \):
\[ y(t) = 2 \left( \sin(100 \pi t) \cos\left(\frac{\pi}{2}\right) + \cos(100 \pi t) \sin\left(\frac{\pi}{2}\right) \right) \]
Since \( \cos\left(\frac{\pi}{2}\right) = 0 \) and \( \sin\left(\frac{\pi}{2}\right) = 1 \):
\[ y(t) = 2 \cdot 1 \cdot \cos(100 \pi t) \]
Thus, we have:
\[ y(t) = 2 \cos(100 \pi t) \]

### Final Equation
The equation of S.H.M for the particle is:
\[ y(t) = 2 \cos(100 \pi t) \, \text{meters} \]