Which one of the following equations of motion represents simple harmonic motion where $k$, $k_0$, $k_1$, and

Question

Which one of the following equations of motion represents simple harmonic motion where $k$, $k_0$, $k_1$, and $a$ are all positive?

Accepted Answer

1. **Identify the Condition for SHM:** For a particle to execute Simple Harmonic Motion, its acceleration $a$ must be directly proportional to its displacement $X$ from the mean (equilibrium) position and directed towards it. This is mathematically expressed as $a = -\omega^2 X$, where $\omega^2$ is a positive constant.
2. **Analyze the Given Options:**
   - **Option A ($a = -k_0x + k_1x^2$):** The acceleration depends on $x^2$, meaning it is not linearly proportional to the displacement. Thus, it does not represent SHM.
   - **Option B ($a = -k(x+a)$):** If we consider the mean position to be at $x = -a$, the displacement from the mean position is $X = x - (-a) = x + a$. Substituting this into the equation gives $a = -kX$. Since $k$ is given as positive, this matches the standard SHM equation $a = -\omega^2 X$ with $\omega = \sqrt{k}$. This represents SHM.
   - **Option C ($a = k(x+a)$):** The positive sign indicates that the acceleration is in the same direction as the displacement (the force is repulsive, not restoring). Thus, it is not SHM.
   - **Option D ($a = kx$):** Similar to Option C, the lack of a negative sign means the force is not a restoring force. It does not represent SHM.
3. **Conclusion:** Only the equation $	ext{Acceleration} = -k(x+a)$ satisfies all conditions for Simple Harmonic Motion.

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