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 for a particle under the force \( F = -5(x - 2)^2 \), we can analyze the given force equation step by step.

### Step 1: Identify the form of the force equation
The force is given as:
\[ F = -5(x - 2)^2 \]
This can be compared to the general form of force:
\[ F = -k(x - x_0)^n \]
where \( k \) is a positive constant, \( x_0 \) is the equilibrium position, and \( n \) indicates the nature of the force.

### Step 2: Determine the parameters
From the given force equation, we can identify:
- \( k = 5 \) (which is positive)
- \( x_0 = 2 \) (the equilibrium position)
- \( n = 2 \) (since the force is proportional to the square of the displacement)

### Step 3: Analyze the value of \( n \)
The value of \( n \) helps us determine the type of motion:
- If \( n = 1 \), the motion is Simple Harmonic Motion (SHM).
- If \( n \) is odd and greater than 1, the motion is oscillatory.
- If \( n \) is even, the motion is translatory.

Since in our case \( n = 2 \) (which is even), we conclude that the motion is not SHM and not oscillatory.

### Step 4: Conclude the type of motion
Given that \( n \) is even, the motion of the particle is classified as translatory motion. Therefore, the correct answer is:

**Translatory motion.**

### Final Answer:
The motion is translatory.

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