If the particle in linear S.H.M. starts from the extreme left position, then its equation of motion is given by

Options

A. `x= A "sin" (omega t + pi//2)`
B. `x =A "sin" (omega t - pi//2)`
C. `x= - A "sin" (omega t +pi//2)`
D. `x = A "cos" omega t`

Explanation

To determine the equation of motion for a particle in linear Simple Harmonic Motion (SHM) that starts from the extreme left position, we can follow these steps:

### Step 1: Understand the Position of the Particle
The particle starts from the extreme left position in SHM. In SHM, the extreme positions are defined as the maximum displacements from the mean position (equilibrium position).

### Step 2: Define the Amplitude
Let the amplitude of the motion be denoted by \( A \). The extreme left position can be represented as \( -A \) (since it is to the left of the mean position).

### Step 3: Write the General Equation of SHM
The general equation of motion for a particle in SHM can be expressed as:
\[ x(t) = A \sin(\omega t + \phi) \]
where:
- \( x(t) \) is the displacement at time \( t \),
- \( \omega \) is the angular frequency,
- \( \phi \) is the phase constant.

### Step 4: Determine the Phase Constant
Since the particle starts at the extreme left position, we need to find the appropriate phase constant \( \phi \). At \( t = 0 \), the displacement \( x(0) \) should be \( -A \):
\[ x(0) = A \sin(\phi) = -A \]
This implies:
\[ \sin(\phi) = -1 \]
The angle \( \phi \) that satisfies this condition is \( \phi = -\frac{\pi}{2} \) or \( \phi = \frac{3\pi}{2} \). However, we typically use \( -\frac{\pi}{2} \) for simplicity.

### Step 5: Substitute the Phase Constant into the Equation
Substituting \( \phi = -\frac{\pi}{2} \) into the general equation gives:
\[ x(t) = A \sin(\omega t - \frac{\pi}{2}) \]

### Step 6: Simplify the Equation
Using the trigonometric identity \( \sin(x - \frac{\pi}{2}) = -\cos(x) \), we can rewrite the equation as:
\[ x(t) = -A \cos(\omega t) \]

### Final Equation of Motion
Thus, the equation of motion for a particle in linear SHM starting from the extreme left position is:
\[ x(t) = -A \cos(\omega t) \]