A particle of mass m is acted upon by a force `F= t^2-kx` . Initially , the particle is at rest at the origin. Then

Options

A. its displacement will be in simple harmonic
B. its velocity will be in simple harmonic
C. its acceleration will be in simple harmonic
D. None of the above

Answer

To solve the problem, we need to analyze the given force acting on the particle and determine whether the motion of the particle can be classified as simple harmonic motion (SHM).

### Step-by-Step Solution:

1. **Identify the Force**: The force acting on the particle is given by:
\[ F = t^2 - kx \]
where \( k \) is a constant, \( x \) is the displacement, and \( t \) is time.

2. **Understand the Conditions for SHM**: For a motion to be classified as simple harmonic motion, the restoring force must be directly proportional to the negative of the displacement:
\[ F = -kx \]
This means that the force should only depend on the displacement \( x \) and not on time \( t \).

3. **Compare the Given Force with SHM Condition**: In the given force equation, we have an additional term \( t^2 \) which depends on time. This means that the force is not solely dependent on the displacement \( x \).

4. **Conclusion about SHM**: Since the force \( F = t^2 - kx \) includes a time-dependent term, it cannot satisfy the condition for simple harmonic motion. Therefore, the motion of the particle cannot be classified as SHM.

5. **Determine the Motion Characteristics**: Since the particle does not exhibit SHM, we can conclude:
- The displacement of the particle will not be in SHM.
- The velocity of the particle will not be in SHM.
- The acceleration of the particle will not be in SHM.

6. **Final Answer**: The correct answer is "none of the above" as the particle does not exhibit simple harmonic motion in any of the aspects mentioned.