Compound Pendulum – Centers of Suspension and Oscillation For a compound pendulum, the center of suspension and the center of oscillation are interchangeable. Is this statement true?

Introduction / Context:
A compound pendulum (physical pendulum) is any rigid body swinging about a horizontal axis. An important property, known as the reciprocity or interchangeability of centers, relates the center of suspension to the center of oscillation.

Given Data / Assumptions:

• Small-angle oscillations (standard pendulum approximation).
• Rigid body with a fixed pivot (center of suspension).

Concept / Approach:
Bouguer’s theorem states that the center of suspension and the center of oscillation are interchangeable: if the pendulum is inverted and hung from the center of oscillation, the period remains the same and the original pivot becomes the new center of oscillation.

Step-by-Step Solution:
Define the equivalent simple pendulum length: L_eq = I_p / (m * h), where I_p is mass moment of inertia about pivot and h is distance from pivot to center of mass. The point along the line below the pivot at distance L_eq is the center of oscillation. By reciprocity, swapping the pivot to that point yields the same L_eq and hence the same period.

Verification / Alternative check:
Derivations from rotational dynamics about different axes using the parallel-axis theorem confirm the equality of equivalent lengths.

Why Other Options Are Wrong:
The property holds (within the small-angle model) without requiring point-mass concentration or drag-related assumptions.

Common Pitfalls:
Confusing physical pendulum with simple pendulum; the interchangeability is unique to the compound pendulum model.

Final Answer:
True