Simple harmonic motion (SHM) — identify the correct statement Which of the following statements about a particle executing simple harmonic motion (SHM) is correct? Choose the most accurate definition or property.

Introduction / Context:
This question checks your conceptual understanding of simple harmonic motion (SHM) — a foundational topic in engineering mechanics and physics. Knowing which definitions and properties are correct helps when analyzing springs, pendulums, and vibration problems.

Given Data / Assumptions:

• A particle performs SHM about a mean (equilibrium) position.
• Standard SHM relations apply: x = r sin(ωt + φ), v = ωr cos(ωt + φ), a = −ω^2 x.
• Angular frequency ω and amplitude r are constants for the motion considered.

Concept / Approach:
The periodic time (or period) T is the time taken to complete one full oscillation and return to the same dynamic state. For ideal SHM, T = 2π / ω. Velocity and acceleration vary sinusoidally: velocity is maximum at the mean position and zero at extremes; acceleration is zero at the mean and maximum in magnitude at extremes.

Step-by-Step Solution:

Verification / Alternative check:
Using energy: total energy E = (1/2) k r^2 is constant; kinetic energy is maximum at the mean where speed is maximum, consistent with the kinematics above.

Why Other Options Are Wrong:

• Directly proportional to ω: Incorrect; T ∝ 1/ω.
• Velocity zero at mean: Opposite; velocity is maximum at mean.
• Acceleration maximum at mean: Acceleration is zero at mean and maximal at extremes.
• Kinetic energy maximum at extremes: It is maximum at mean, not extremes.

Common Pitfalls:
Confusing where velocity versus acceleration are maximum; mixing up period and frequency relations.

Final Answer:
The periodic time of a particle in SHM is the time taken for one complete oscillation.