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

Options

A. Translatory
B. Oscillatory
C. SHM
D. All of these

Answer

To determine the type of motion exhibited by a particle under the force \( F = -5 (x - 2)^2 \), we will analyze the characteristics of the force and its relation to displacement.

### Step-by-Step Solution:

1. **Identify the Form of the Force**:
The force acting on the particle is given by:
\[ F = -5 (x - 2)^2 \]
This indicates that the force depends on the square of the displacement from the point \( x = 2 \).

2. **Check for Simple Harmonic Motion (SHM)**:
For a motion to be classified as simple harmonic motion, the restoring force must be directly proportional to the displacement from an equilibrium position. Mathematically, this is expressed as:
\[ F \propto -x \]
In our case, the force is proportional to \((x - 2)^2\), which is not linear in \(x\). Therefore, this does not satisfy the condition for SHM.

**Conclusion**: The motion is not SHM.

3. **Check for Oscillatory Motion**:
Oscillatory motion occurs when a particle moves back and forth around an equilibrium position. For a force to lead to oscillatory motion, it typically should have a restoring nature, which is often characterized by odd powers of displacement.

Here, since the force involves \((x - 2)^2\), which is an even power, it does not lead to oscillatory motion. Instead, it suggests that the force increases as the particle moves away from \(x = 2\) in either direction, pushing it away rather than restoring it back.

**Conclusion**: The motion is not oscillatory.

4. **Determine Translatory Motion**:
Translatory motion refers to the motion of a particle where it moves from one point to another without any oscillation about a point. Since the force does not restore the particle to a specific equilibrium position but rather pushes it away as it moves, we can classify the motion as translatory.

**Conclusion**: The motion is translatory.

### Final Answer:
The motion of the particle is **translatory motion**.

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