Simple pendulum — identify the statement that is NOT applicable: For small oscillations of a simple pendulum, which one of the following statements is NOT valid?

Introduction / Context:
The simple pendulum is a classic model used to illustrate simple harmonic motion for small angles. Its period depends primarily on pendulum length and gravitational acceleration. This question asks you to detect a false statement among commonly quoted properties for small oscillations.

Given Data / Assumptions:

• Small angular displacement approximation (sin θ ≈ θ in radians).
• Simple pendulum: point mass, massless string, no air resistance, frictionless pivot.
• Standard time period formula applies.

Concept / Approach:

For a simple pendulum under the small-angle approximation, the time period is T = 2π √(l / g). From this, T ∝ √l and T ∝ 1/√g, and T does not depend on mass. It is also commonly stated that T is independent of amplitude for small angles. Any claim that contradicts these proportionalities is not applicable.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional analysis: [T] must scale as √(length/acceleration). A linear dependence on length violates dimensional consistency unless g changes accordingly.

Why Other Options Are Wrong:

(a) and (e) are standard properties under small-angle assumptions. (b) and (c) directly match the formula. Only (d) contradicts the square-root dependence.

Common Pitfalls:

Applying the conclusions to large amplitudes where T increases slightly with amplitude; ignoring air drag or mass distribution (compound pendulum effects).

Final Answer:

The time period is proportional to l (directly proportional to length)