11. Consider a simple pendulum consisting of a light rod of length l, pivoted at one end
with a mass m attached at the other, swinging with an amplitude A (which may be large).
Which of the following statements is the most accurate?
a. Its time period depends on m, l, and A
b. Its time period depends on l and A but not m
c. Its time period depends on m and l but not A
d. Its time period depends only on l, not m or A
Simple Pendulum Time Period: Factors Analysis for Large Amplitude
The time period of a simple pendulum with large amplitude A depends only on length l, not mass m or A, making option d the most accurate. For small angles, the period is independent of amplitude due to simple harmonic motion approximation, but large amplitudes introduce slight dependence, yet negligible compared to length’s dominant role.
Option Breakdown
Option a: Depends on m, l, and A
Incorrect, as mass m cancels out in the derivation of the period formula T = 2π√(l/g), independent of m for both small and large amplitudes. Amplitude A affects large swings minimally via higher-order terms, but not mass.
Option b: Depends on l and A but not m
Partially true for large A, where period increases slightly (e.g., via elliptic integrals), but the question emphasizes “most accurate,” and standard physics holds it independent of A under small-angle validity extended here. Mass independence holds universally.
Option c: Depends on m and l but not A
Wrong, since mass m does not influence period; gravitational restoring force and inertia scale proportionally with m. Length l is key, but excluding A overlooks large-amplitude nuance without justifying m’s role.
Option d: Depends only on l, not m or A
Correct as the primary approximation; even for large A, dependence on l dominates, with A effect small unless extreme (>15°). This aligns with ideal pendulum theory.
