The phase (at a time t) of a particle in simple harmonic motion tells

Options

A. only the position of the particle at time t
B. only the direction of motion of the particle at time t
C. both the position and direction of motion of the paraticle at time t
D. neither the position of the particle nor its direction of motion at time t

Explanation

To solve the question, we need to understand what the phase of a particle in simple harmonic motion (SHM) represents.

### Step-by-Step Solution:

1. **Understanding Simple Harmonic Motion (SHM)**:
- In SHM, the position of a particle can be described by the equation:
\[ y(t) = A \sin(\omega t + \phi) \]
where:
- \(y(t)\) is the position of the particle at time \(t\),
- \(A\) is the amplitude,
- \(\omega\) is the angular frequency,
- \(\phi\) is the phase constant.

2. **Defining Phase**:
- The term \(\omega t + \phi\) is referred to as the phase of the motion. It indicates the state of the oscillation at any given time \(t\).

3. **Analyzing the Phase**:
- The sine function can take on values from -1 to 1, which means:
- When \(\sin(\omega t + \phi) > 0\), the particle is moving in one direction.
- When \(\sin(\omega t + \phi) < 0\), the particle is moving in the opposite direction.
- When \(\sin(\omega t + \phi) = 0\), the particle is at the equilibrium position.

4. **Position and Direction**:
- The position of the particle is given by \(y(t) = A \sin(\omega t + \phi)\). Thus, the magnitude of the sine function multiplied by the amplitude gives the position.
- The sign of the sine function indicates the direction of motion (positive for one direction and negative for the opposite).

5. **Conclusion**:
- Therefore, the phase at a time \(t\) tells us both the position and the direction of the motion of the particle.

### Final Answer:
The correct option is **option 3**, which states that the phase at a time \(t\) tells both the position and the direction of motion of the particle.

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