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 find the equation of Simple Harmonic Motion (S.H.M) for a particle with an amplitude of 2 m and a frequency of 50 Hz, starting from the extreme position, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Parameters**:
- Amplitude (A) = 2 m
- Frequency (f) = 50 Hz

2. **Calculate Angular Frequency (ω)**:
- The angular frequency (ω) is given by the formula:
\[ \omega = 2\pi f \]
- Substituting the given frequency:
\[ \omega = 2\pi \times 50 = 100\pi \, \text{rad/s} \]

3. **Determine the Initial Phase (φ)**:
- Since the particle starts from the extreme position, the initial phase (φ) will be:
\[ \phi = \frac{\pi}{2} \]
- This is because at the extreme position, the sine function reaches its maximum value.

4. **Write the General Equation of S.H.M**:
- The general equation for S.H.M can be expressed as:
\[ y(t) = A \sin(\omega t + \phi) \]
- However, since we are starting from the extreme position, we can also express it using cosine:
\[ y(t) = A \cos(\omega t) \]

5. **Substitute the Values into the Equation**:
- Now substituting the values of amplitude and angular frequency into the equation:
\[ y(t) = 2 \cos(100\pi t) \]

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