Statement-I : A particle is moving along x-axis. The resultant force `F` acting on it is given by `F =- ax - b`, wher `a` and `b` are both positive constants. The motion of this particle is not `S.H.M.`
Statement-II : In `S.H.M.` resultant force must be proportional to the dispacement form mean position.

Options

A. Statement-I is true, Statement-Ii is true, Statement-II is a correct explanation for Statement-I.
B. Statement-I is true, Statement-II is true, Statement-II is NOT a correct explanantion for Statement-I.
C. Statement-I is true, Statement-II is false
D. Statement-I is false, Statement-II is true

Answer

To analyze the statements given in the question, we will break down the information provided in each statement and evaluate their validity step by step.

### Step 1: Understand the Force Equation
The resultant force acting on the particle is given by:
\[ F = -ax - b \]
where \( a \) and \( b \) are positive constants.

### Step 2: Identify the Components of the Force
The force can be separated into two parts:
1. The term \(-ax\) which is proportional to the displacement \(x\) from the mean position.
2. The term \(-b\) which is a constant force that does not depend on \(x\).

### Step 3: Analyze the First Statement
**Statement I** claims that the motion of the particle is not simple harmonic motion (S.H.M.).

In S.H.M., the restoring force must be directly proportional to the displacement from the mean position and must act in the opposite direction. The standard form of the force in S.H.M. is:
\[ F = -kx \]
where \( k \) is a positive constant.

In our case, the presence of the constant term \(-b\) means that the force does not solely depend on \(x\) and introduces a constant force that affects the equilibrium position of the particle. This indicates that the motion is damped and not purely harmonic.

### Step 4: Conclusion on Statement I
Since the force includes a constant term, the motion is not purely S.H.M. Therefore, **Statement I is true**.

### Step 5: Analyze the Second Statement
**Statement II** states that in S.H.M., the resultant force must be proportional to the displacement from the mean position.

This is indeed a defining characteristic of S.H.M. The force should be of the form \( F = -kx \). Since our force equation includes a constant term, it does not meet the strict criteria for S.H.M.

### Step 6: Conclusion on Statement II
Thus, **Statement II is also true** as it correctly describes the condition for S.H.M.

### Final Evaluation
- **Statement I**: True (the motion is not S.H.M. due to the constant force).
- **Statement II**: True (the condition for S.H.M. is correctly stated).

### Final Answer
Both statements are true, but Statement I is correct in its assertion that the motion is not S.H.M. due to the presence of the constant force.

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