A particle oscillating under a force `vecF=-kvecx-bvecv` is a (k and b are constants)

Options

A. Linear oscillation
B. Forces oscillations
C. Damped oscilation
D. SHM

Answer

To solve the question regarding the type of oscillation a particle undergoes when subjected to the force \( \vec{F} = -k\vec{x} - b\vec{v} \), we can break down the components of the force and analyze their implications step-by-step.

### Step-by-Step Solution:

1. **Understanding the Force Components**:
The force acting on the particle is given by:
\[ \vec{F} = -k\vec{x} - b\vec{v} \]
Here, \( -k\vec{x} \) represents the restoring force, which is proportional to the displacement \( \vec{x} \) from the equilibrium position. The term \( -b\vec{v} \) represents a damping force, which is proportional to the velocity \( \vec{v} \) of the particle.

2. **Identifying the Restoring Force**:
The term \( -k\vec{x} \) indicates that the force is directed towards the equilibrium position (where \( \vec{x} = 0 \)). This is characteristic of simple harmonic motion (SHM), where the restoring force always acts in the opposite direction to the displacement.

3. **Identifying the Damping Force**:
The term \( -b\vec{v} \) introduces a damping effect, which opposes the motion of the particle. This damping force is dependent on the velocity of the particle and is a key feature of damped oscillations.

4. **Conclusion on the Type of Oscillation**:
Since the system has both a restoring force (due to the term \( -k\vec{x} \)) and a damping force (due to the term \( -b\vec{v} \)), the oscillation is not purely simple harmonic motion. Instead, it is classified as a damped oscillation. Damped oscillations occur when there is a loss of energy in the system, typically due to friction or resistance.

### Final Answer:
The particle oscillating under the force \( \vec{F} = -k\vec{x} - b\vec{v} \) is undergoing **damped oscillation**.

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