Simple Harmonic Motion – Amplitude vs Reference Circle In the geometric representation of simple harmonic motion (SHM) as the projection of uniform circular motion, the amplitude is always _____ the radius of the reference circle.

Introduction / Context: SHM is often visualized by projecting a point moving uniformly on a circle onto a diameter. This geometric interpretation provides direct insight into amplitude, phase, and displacement relations.

Given Data / Assumptions:

• A particle moves in SHM with maximum displacement A from the mean position.
• The reference circle has radius R.
• Projection is taken on a fixed diameter.

Concept / Approach: The displacement x at any instant equals the projection of the circular motion radius onto the diameter: x = R cos(θ) (or sin(θ) depending on reference). The maximum value of |x| occurs when cos(θ) = ±1, giving |x|max = R. Hence amplitude A equals the radius R.

Step-by-Step Solution: Represent SHM by uniform circular motion of radius R. Displacement along the diameter: x = R cos(θ). Maximum |x| is R. Therefore amplitude A = R.

Verification / Alternative check: Using standard SHM form x(t) = A cos(ω t + φ). Compare with circular projection x = R cos(θ). Matching forms implies A = R.

Why Other Options Are Wrong: “less than” / “greater than” / “twice” / “half”: contradict the direct geometric identity A = R.

Common Pitfalls: Confusing amplitude with instantaneous displacement.

Final Answer: equal to