Question

The equation describing acceleration (a) as a function of displacement (x) for four different particles is given below. Which one of these is executing simple harmonic motion?

• A. $a=+2x$
• B. $a=+2x^{2}$
• C. $a=-2x^{2}$
• D. $a=-2x$

Hint:

SHM always requires a negative sign (restoring force) and a linear relationship with displacement ($x^1$).

Answer 1

The Correct Option is D

Step 1: Concept
Simple Harmonic Motion (SHM) is defined by the condition that acceleration is directly proportional to displacement and opposite in direction ($a = -\omega^2 x$).

Step 2: Analysis
We examine each equation for the form $a \propto -x$.

Step 3: Elimination
(A) is positive (not restoring). (B) and (C) involve $x^2$, which is not linear.

Step 4: Conclusion
Option (D) $a = -2x$ fits the requirement where $\omega^2 = 2$. Final Answer: (D)