---
title: "A physical pendulum consists of a uniform thin rod of mass \\(M\\) and length \\(L\\) pivoted frictionlessly at one end, and an adjustable point mass \\(m\\) clamped at a distance \\(x\\) from the pivot, where \\(0 \\le x \\le L\\).  For small-angle oscillations, the period of the system as a function of the clamp position is given by \\[ T(x) = 2\\pi \\sqrt{\\dfrac{\\dfrac{1}{3}ML^2 + mx^2}{g\\left(\\dfrac{1}{2}ML + mx\\right)}} \\]  Which of the following correctly identifies the period of oscillation in the limit \\(x \\to 0\\) and explains whether the period initially increases or decreases as the point mass is moved slightly away from the pivot?"
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url: "https://nerd-notes.com/ubq/124618/"
date_modified: "2026-09-28T14:08:51+00:00"
---

# A physical pendulum consists of a uniform thin rod of mass \(M\) and length \(L\) pivoted frictionlessly at one end, and an adjustable point mass \(m\) clamped at a distance \(x\) from the pivot, where \(0 \le x \le L\).

For small-angle oscillations, the period of the system as a function of the clamp position is given by
\[ T(x) = 2\pi \sqrt{\dfrac{\dfrac{1}{3}ML^2 + mx^2}{g\left(\dfrac{1}{2}ML + mx\right)}} \]

Which of the following correctly identifies the period of oscillation in the limit \(x \to 0\) and explains whether the period initially increases or decreases as the point mass is moved slightly away from the pivot?

A physical pendulum consists of a uniform thin rod of mass \(M\) and length \(L\) pivoted frictionlessly at one end, and an adjustable point mass \(m\) clamped at a distance \(x\) from the pivot, where \(0 \le x \le L\).

For small-angle oscillations, the period of the system as a function of the clamp position is given by
\[ T(x) = 2\pi \sqrt{\dfrac{\dfrac{1}{3}ML^2 + mx^2}{g\left(\dfrac{1}{2}ML + mx\right)}} \]

Which of the following correctly identifies the period of oscillation in the limit \(x \to 0\) and explains whether the period initially increases or decreases as the point mass is moved slightly away from the pivot?

![A vertical uniform rod of length \(L\) and mass \(M\) is suspended from a small circular pivot at its top end. A small solid circular clamp representing a point mass \(m\) is attached to the rod at a variable distance \(x\) below the pivot. A vertical dashed reference line extends downward from the pivot parallel to the rod. A double-headed dimension arrow labeled \(x\) spans from the center of the pivot to the center of mass \(m\). A second double-headed dimension arrow labeled \(L\) spans the full length of the rod from the pivot to its bottom free end. The rod is shown hanging vertically at rest. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604531-Ix5B9l.jpg)

- **A.** \(T \to 2\pi\sqrt{\dfrac{2L}{3g}}\), and the period decreases because the restoring torque increases linearly with \(x\) while the added rotational inertia increases quadratically with \(x\).
- **B.** \(T \to 2\pi\sqrt{\dfrac{2L}{3g}}\), and the period increases because moving the point mass farther from the pivot increases the rotational inertia of the system.
- **C.** \(T \to 2\pi\sqrt{\dfrac{2ML}{3(M+m)g}}\), and the period decreases because the added mass increases the total gravitational force while contributing zero rotational inertia at the pivot.
- **D.** \(T \to 0\), and the period increases because the effective pendulum length is proportional to the distance \(x\) between the pivot and the point mass.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124618/*