Period of simple harmonic motion (relationship with angular frequency): How does the time period of an SHM vary with its angular velocity ω?

Introduction / Context:
The time period T of a simple harmonic oscillator specifies the time for one complete oscillation. Connecting T with the angular frequency ω is central to vibration analysis, pendulum approximations, and spring–mass systems.

Given Data / Assumptions:

• Pure SHM with angular frequency ω.
• Linear restoring law; no damping or forcing.
• Standard kinematic relations of SHM hold.

Concept / Approach:
By definition, angular frequency ω relates to the time period by ω = 2π / T, or equivalently T = 2π / ω. Thus, T varies inversely with ω: doubling ω halves T, etc.

Step-by-Step Solution:

Verification / Alternative check:
For a spring–mass system, ω = sqrt(k/m) ⇒ T = 2π sqrt(m/k), again showing inverse relation to ω.

Why Other Options Are Wrong:

• Direct or square relations contradict T = 2π/ω.
• Inverse square would imply T ∝ 1/ω^2, which is incorrect.

Common Pitfalls:
Confusing angular frequency ω (rad/s) with cyclic frequency f (Hz); note that T = 1/f and ω = 2πf.

Final Answer:
inversely proportional to its angular velocity