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title: "A physical pendulum consists of a rigid body of mass \\(M\\) and rotational inertia \\(I\\) pivoted about a frictionless horizontal axis located a distance \\(d\\) from its center of mass. The pendulum is released from rest at an initial angular displacement \\(\\theta_0\\), where \\(0 < \\theta_0 < \\pi\\), and oscillates with mechanical energy \\(E\\). Which of the following best describes the qualitative shape of the graph of angular velocity \\(\\omega\\) versus angular displacement \\(\\theta\\) over one complete cycle, and how that shape deforms as \\(\\theta_0\\) approaches \\(\\pi\\)?"
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url: "https://nerd-notes.com/ubq/124556/"
date_modified: "2026-09-28T14:08:37+00:00"
---

# A physical pendulum consists of a rigid body of mass \(M\) and rotational inertia \(I\) pivoted about a frictionless horizontal axis located a distance \(d\) from its center of mass. The pendulum is released from rest at an initial angular displacement \(\theta_0\), where \(0 < \theta_0 < \pi\), and oscillates with mechanical energy \(E\). Which of the following best describes the qualitative shape of the graph of angular velocity \(\omega\) versus angular displacement \(\theta\) over one complete cycle, and how that shape deforms as \(\theta_0\) approaches \(\pi\)?

A physical pendulum consists of a rigid body of mass \(M\) and rotational inertia \(I\) pivoted about a frictionless horizontal axis located a distance \(d\) from its center of mass. The pendulum is released from rest at an initial angular displacement \(\theta_0\), where \(0 < \theta_0 < \pi\), and oscillates with mechanical energy \(E\). Which of the following best describes the qualitative shape of the graph of angular velocity \(\omega\) versus angular displacement \(\theta\) over one complete cycle, and how that shape deforms as \(\theta_0\) approaches \(\pi\)?

![A grayscale schematic of a physical pendulum. A fixed horizontal ceiling is shown at the top with diagonal hatching. A frictionless pivot point is marked with a small solid circle attached to the ceiling. Hanging from the pivot is a smooth, irregular rigid body extending downward. A dashed vertical line extends downward from the pivot to represent the equilibrium position. A straight line connects the pivot to a crosshair marked CM inside the body, located at a distance labeled d from the pivot. The line of the pendulum is displaced to the right of the vertical by an angle labeled \theta_0. A curved directional arrow labeled \omega indicates angular velocity about the pivot. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604517-oN0idH.jpg)

- **A.** A closed, symmetric loop that is nearly elliptical for small \(\theta_0\), but develops pointed tips at \(\pm \theta_0\) as \(\theta_0 \to \pi\) because the restoring torque approaches zero, causing the slope \(\dfrac{d\omega}{d\theta}\) to approach a finite value at the turning points.
- **B.** A closed, symmetric loop that is nearly elliptical for small \(\theta_0\), but becomes flattened along the \(\omega\)-axis and rectangular as \(\theta_0 \to \pi\) because the pendulum spends most of its period moving at constant maximum angular velocity.
- **C.** A pair of intersecting parabolas meeting at \(\pm \theta_0\) for small \(\theta_0\), which deform into a circle of radius \(\theta_0\) as \(\theta_0 \to \pi\) because the kinetic and potential energies become equal at all angular positions.
- **D.** A closed, symmetric ellipse whose eccentricity is strictly independent of \(\theta_0\), because the ratio of maximum angular velocity to maximum angular displacement remains constant for all oscillation amplitudes.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124556/*