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

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:** The defining characteristic of Simple Harmonic Motion is that the restoring force (and thus acceleration) must be directly proportional to the displacement from the mean position and directed towards it. Mathematically, $F = -kX$ or $a \propto -X$, where $X$ is the displacement from the mean position .
2. **Analyze Option 1:** $a = -k_0x + k_1x^2$. This depends on $x^2$, so it is non-linear and not SHM.
3. **Analyze Option 2:** $a = -k(x+a)$. Let $X = x+a$. Then $a = -kX$. This represents a linear restoring force directed towards the position $X=0$ (i.e., $x=-a$). This represents SHM about the mean position $x=-a$.
4. **Analyze Option 3:** $a = k(x+a)$. The positive sign indicates the force is away from the mean position, leading to unstable equilibrium, not SHM.
5. **Analyze Option 4:** $a = kx$. The positive sign indicates the force is in the direction of displacement (away from origin), which is not a restoring force.
6. **Conclusion:** Only the equation $a = -k(x+a)$ satisfies the condition for SHM.

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