---
title: "A uniform rigid rod of mass \\( M \\) and length \\( L \\) is suspended from a horizontal pivot point \\( P \\) that is located a distance \\( x \\) from the center of mass of the rod, where \\( 0 < x < L/2 \\). The rod is free to swing in a vertical plane without friction. The acceleration due to gravity is \\( g \\)."
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url: "https://nerd-notes.com/ubq/117746/"
date_modified: "2026-08-04T07:55:53+00:00"
---

# A uniform rigid rod of mass \( M \) and length \( L \) is suspended from a horizontal pivot point \( P \) that is located a distance \( x \) from the center of mass of the rod, where \( 0 < x < L/2 \). The rod is free to swing in a vertical plane without friction. The acceleration due to gravity is \( g \).

A uniform rigid rod of mass \( M \) and length \( L \) is suspended from a horizontal pivot point \( P \) that is located a distance \( x \) from the center of mass of the rod, where \( 0 < x < L/2 \). The rod is free to swing in a vertical plane without friction. The acceleration due to gravity is \( g \).

![A tall, narrow vertical rectangle representing the uniform rod. A horizontal dashed line passes through the exact center of the rectangle, labeled 'Center of Mass'. Above this center line, a small circular dot marks the pivot point on the rod, labeled 'P'. An arrow labeled 'x' points vertically from the Center of Mass dashed line up to the pivot point P. A vertical bracket along the entire left side of the rod indicates its total length is L. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830152-vMsHab.jpg)

**Part a)** **Use** integral calculus to **derive** the rotational inertia \( I_{cm} \) of the rod about an axis perpendicular to the rod and passing through its center of mass. Express your answer in terms of \( M \) and \( L \). *(2 points)*

**Part b)** **Derive** an expression for the rotational inertia \( I_P \) of the rod about an axis perpendicular to the rod and passing through the pivot point \( P \). Express your answer in terms of \( M \), \( L \), and \( x \). *(1 points)*

**Part c)** The rod is displaced by a small angle \( \theta \) from the vertical and released from rest. **Write**, but do NOT solve, a differential equation that can be used to determine the angular displacement \( \theta \) of the rod as a function of time \( t \). Express your answer in terms of \( M \), \( L \), \( x \), \( \theta \), and physical constants, as appropriate. *(2 points)*

**Part d)** **Derive** an expression for the period \( T \) of the small-angle oscillations of the rod. Express your answer in terms of \( L \), \( x \), and physical constants, as appropriate. *(2 points)*

**Part e)** **Calculate** the distance \( x \) from the center of mass to the pivot that minimizes the period \( T \) of the pendulum. Express your answer in terms of \( L \). *(3 points)*

**Part f)** **Predict** whether the period \( T \) approaches zero, approaches infinity, or approaches a non-zero finite constant as the pivot point \( P \) is moved very close to the center of mass (as \( x \) approaches zero). **Justify** your answer using the expression derived in part (d). *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117746/*