---
title: "A physical pendulum consists of a uniform rigid rod of length \\(L\\) and mass \\(M\\) pivoted frictionlessly at one end, with a small object of mass \\(m\\) affixed to its free end.  For small-amplitude oscillations, the period of the system is given by \\[ T = 2\\pi \\sqrt{\\dfrac{2M + 6m}{3M + 6m}\\dfrac{L}{g}} \\]  Which of the following statements correctly describes the limiting behavior of the period \\(T\\) and provides the correct physical justification?"
description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]"
url: "https://nerd-notes.com/ubq/121020/"
date_modified: "2026-08-23T04:45:23+00:00"
---

# A physical pendulum consists of a uniform rigid rod of length \(L\) and mass \(M\) pivoted frictionlessly at one end, with a small object of mass \(m\) affixed to its free end.

For small-amplitude oscillations, the period of the system is given by
\[ T = 2\pi \sqrt{\dfrac{2M + 6m}{3M + 6m}\dfrac{L}{g}} \]

Which of the following statements correctly describes the limiting behavior of the period \(T\) and provides the correct physical justification?

A physical pendulum consists of a uniform rigid rod of length \(L\) and mass \(M\) pivoted frictionlessly at one end, with a small object of mass \(m\) affixed to its free end.

For small-amplitude oscillations, the period of the system is given by
\[ T = 2\pi \sqrt{\dfrac{2M + 6m}{3M + 6m}\dfrac{L}{g}} \]

Which of the following statements correctly describes the limiting behavior of the period \(T\) and provides the correct physical justification?

![A schematic of a physical pendulum. At the top is a small circular pivot point attached to a horizontal fixed ceiling support represented by a horizontal line with short hatching above it. Extending vertically downward from the pivot is a straight uniform rod of length \(L\) and mass \(M\). At the lowest tip of the rod is a solid filled circle representing an attached point mass \(m\). A vertical dashed line extends downward along the rod. A double-headed arrow to the left indicates the length \(L\) spanning from the pivot to the center of the point mass. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-diagram-1-1787460323-sxSrEZ.jpg)

- **A.** In the limit \(m \gg M\), the period approaches infinity (\(T \to \infty\)) because the rotational inertia of the system increases without bound while the length \(L\) remains constant.
- **B.** In the limit \(m \to 0\), the period approaches \(2\pi\sqrt{\dfrac{L}{2g}}\) because the physical pendulum behaves as if its entire mass were concentrated at the center of mass of the rod.
- **C.** In the limit \(m \gg M\), the period approaches \(2\pi\sqrt{\dfrac{L}{g}}\) because both the rotational inertia and restoring torque become dominated by the point mass, converging to a simple pendulum of length \(L\).
- **D.** In the limit \(m \to 0\), the period approaches \(2\pi\sqrt{\dfrac{L}{3g}}\) because the rotational inertia of the rod about its pivot is \(\dfrac{1}{3}ML^2\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121020/*