---
title: "A simple pendulum consists of a small bob of mass \\(m\\) attached to the end of a light string of length \\(L\\). The pendulum is released from rest at an initial angular displacement \\(\\theta_0\\) from the vertical.  Which of the following statements correctly compares the actual period of oscillation \\(T\\) to the theoretical small-angle period \\(T_0 = 2\\pi\\sqrt{\\dfrac{L}{g}}\\) when \\(\\theta_0\\) is large, and provides the correct physical justification?"
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url: "https://nerd-notes.com/ubq/120976/"
date_modified: "2026-08-23T04:44:54+00:00"
---

# A simple pendulum consists of a small bob of mass \(m\) attached to the end of a light string of length \(L\). The pendulum is released from rest at an initial angular displacement \(\theta_0\) from the vertical.

Which of the following statements correctly compares the actual period of oscillation \(T\) to the theoretical small-angle period \(T_0 = 2\pi\sqrt{\dfrac{L}{g}}\) when \(\theta_0\) is large, and provides the correct physical justification?

A simple pendulum consists of a small bob of mass \(m\) attached to the end of a light string of length \(L\). The pendulum is released from rest at an initial angular displacement \(\theta_0\) from the vertical.

Which of the following statements correctly compares the actual period of oscillation \(T\) to the theoretical small-angle period \(T_0 = 2\pi\sqrt{\dfrac{L}{g}}\) when \(\theta_0\) is large, and provides the correct physical justification?

![A schematic diagram showing a simple pendulum suspended from a horizontal fixed ceiling line. A vertical dashed line extends downward from the pivot point on the ceiling to indicate the vertical reference. A solid line representing a light string of length \(L\) is displaced to the right of the dashed vertical line by an angle labeled \(\theta\). At the lower end of the string, a solid filled circle represents a pendulum bob of mass \(m\). An arc with an arrow shows the angle \(\theta\) between the dashed vertical line and the string. A downward vertical arrow originating at the center of the bob is labeled \(mg\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/pendulum-stem-diagram-1787460294-8rwC3w.jpg)

- **A.** The actual period is less than \(T_0\) because the tangential component of gravity increases at larger displacements, producing higher speeds that shorten the total cycle time.
- **B.** The actual period is equal to \(T_0\) because the period of an ideal pendulum depends exclusively on \(L\) and \(g\), making the period strictly independent of amplitude.
- **C.** The actual period is greater than \(T_0\) because the restoring torque magnitude \(mgL\sin\theta\) is less than the linear approximation \(mgL\theta\), resulting in smaller angular accelerations throughout the swing.
- **D.** The actual period is greater than \(T_0\) because the rotational inertia of the pendulum about the pivot increases with the angular displacement \(\theta\), reducing the angular acceleration.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/120976/*