Solved: PHYSICS Question for NEET

NEET PHYSICSEasy

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

$\text{Acceleration} = -k_0x + k_1x^2$

$\text{Acceleration} = -k(x+a)$

$\text{Acceleration} = k(x+a)$

$\text{Acceleration} = kx$

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.
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.

Conclusion: Only the equation $\text{Acceleration} = -k(x+a)$ satisfies all conditions for Simple Harmonic Motion.

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