For a particle showing motion under forces `F= - 5 ( x-2)` , the motion is

Options

A. Translatory
B. Oscillatory
C. SHM
D. Both (2) & (3)

Answer

To analyze the motion of a particle under the force \( F = -5(x - 2) \), we can follow these steps:

### Step 1: Identify the Force Equation
The force acting on the particle is given by:
\[ F = -5(x - 2) \]
This indicates that the force is dependent on the position \( x \) of the particle.

### Step 2: Determine the Equilibrium Position
To find the equilibrium position, we set the force \( F \) to zero:
\[ 0 = -5(x - 2) \]
Solving for \( x \):
\[ x - 2 = 0 \implies x = 2 \]
Thus, the equilibrium position is at \( x = 2 \).

### Step 3: Define Displacement from Equilibrium
Let \( y \) be the displacement from the equilibrium position:
\[ y = x - 2 \]
This means that the displacement \( y \) is the distance from the equilibrium position.

### Step 4: Rewrite the Force in Terms of Displacement
Substituting \( y \) into the force equation, we have:
\[ F = -5y \]
This shows that the force is directly proportional to the displacement \( y \) and acts in the opposite direction.

### Step 5: Identify the Type of Motion
The equation \( F = -5y \) is characteristic of simple harmonic motion (SHM), where the restoring force is proportional to the displacement from the equilibrium position. Therefore, the motion of the particle is:
- **Simple Harmonic Motion (SHM)**: The particle oscillates about the equilibrium position \( x = 2 \).

### Step 6: Conclusion
Since SHM is a type of oscillatory motion, we conclude that the motion of the particle is both:
- **Translatory** (as it moves back and forth)
- **Oscillatory** (as it oscillates around the equilibrium position).

Thus, the correct answer is that the motion is both 2 (SHM) and 3 (Oscillatory).

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