The motion of a body is simple harmonic if the?

Analysis of the Question

The question asks for the fundamental condition that defines Simple Harmonic Motion (SHM). To answer this, we must look at the mathematical and physical definition of SHM.

Concept Explanation

Simple Harmonic Motion (SHM) is a specific type of periodic motion where a body oscillates back and forth about a fixed equilibrium position.

The defining characteristic of SHM is expressed by the mathematical relationship: $a \propto -x$

Where:

• $a$ is the acceleration of the body.
• $x$ is the displacement of the body from the fixed equilibrium point.
• The negative sign ($-$) indicates that the acceleration is always directed towards the equilibrium point (the fixed point), opposing the displacement.

Step-by-Step Solution

1. Evaluate Option A: In SHM, the acceleration is directly proportional to the displacement and is always directed towards the equilibrium point. This matches the definition perfectly.
2. Evaluate Option B: While many SHM examples (like a simple pendulum or a mass on a spring) move in a straight line or arc, simply moving in a straight line does not make a motion "simple harmonic." Uniform velocity in a straight line is not SHM.
3. Evaluate Option C: This is incorrect. Acceleration is proportional to the first power of the distance (displacement), not the square of the distance.
4. Evaluate Option D: This describes uniform acceleration (like an object in free fall), not SHM. In SHM, acceleration is not constant; it changes continuously as the displacement changes.

Conclusion

Based on the definition of SHM, the acceleration must be directed towards a fixed point and be proportional to the displacement from that point. Option A correctly identifies the directional requirement of this motion.

Exam Tip

Whenever you see a question about SHM, remember the "Hooke’s Law" connection: $F = -kx$. Since $F = ma$, then $ma = -kx$, which means $a = -(k/m)x$. This proves that acceleration is always proportional to displacement and directed towards the center (the negative sign). Keep this formula in mind to eliminate distractors!