---
title: "Students are investigating the periodic motion of a physical pendulum to determine the acceleration due to gravity \\(g\\). They are provided with a uniform rigid rod of length \\(L = 1.20 \\text{ m}\\) and mass \\(M\\). The rod has several small holes drilled along its length, allowing it to be pivoted about a frictionless axle. The distance from the center of mass of the rod to the pivot is \\(x\\).  When displaced by a small angle \\(\\theta\\) and released, the rod oscillates. The theoretical period of oscillation \\(T\\) is given by the equation:  \\[ T = 2\\pi \\sqrt{ \\dfrac{\\dfrac{1}{12}L^2 + x^2}{gx} } \\]"
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url: "https://nerd-notes.com/ubq/113152/"
date_modified: "2026-05-05T04:28:34+00:00"
---

# Students are investigating the periodic motion of a physical pendulum to determine the acceleration due to gravity \(g\). They are provided with a uniform rigid rod of length \(L = 1.20 \text{ m}\) and mass \(M\). The rod has several small holes drilled along its length, allowing it to be pivoted about a frictionless axle. The distance from the center of mass of the rod to the pivot is \(x\).

When displaced by a small angle \(\theta\) and released, the rod oscillates. The theoretical period of oscillation \(T\) is given by the equation:

\[ T = 2\pi \sqrt{ \dfrac{\dfrac{1}{12}L^2 + x^2}{gx} } \]

Students are investigating the periodic motion of a physical pendulum to determine the acceleration due to gravity \(g\). They are provided with a uniform rigid rod of length \(L = 1.20 \text{ m}\) and mass \(M\). The rod has several small holes drilled along its length, allowing it to be pivoted about a frictionless axle. The distance from the center of mass of the rod to the pivot is \(x\).

When displaced by a small angle \(\theta\) and released, the rod oscillates. The theoretical period of oscillation \(T\) is given by the equation:

\[ T = 2\pi \sqrt{ \dfrac{\dfrac{1}{12}L^2 + x^2}{gx} } \]

![A tall, uniform rectangular rod hanging vertically. A pivot pin passes through a hole near the top of the rod. The center of mass is clearly marked with a black dot labeled 'C.M.' in the exact geometric center of the rod. A dimension line labeled '\(x\)' indicates the distance from the pivot pin down to the C.M. Another dimension line to the side labeled '\(L\)' indicates the total length of the rod. A dashed line shows the vertical equilibrium axis, and the rod is slightly tilted by a small angle \(\theta\) from the dashed vertical line.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1777955313-mY3Wml.jpg)

**Part a)** The students have access to the rod (with pre-drilled holes), a pivot stand, a meterstick, a stopwatch, a protractor, and other standard laboratory equipment. **Describe** an experimental procedure the students could use to collect the data needed to determine \(g\). In your description, state the quantities that would be measured and the equipment used to measure them. Describe the steps in sufficient detail so that another student could replicate the procedure. Include steps taken to reduce experimental uncertainty. *(3 points)*

**Part b)** The students want to create a linear graph from their data to determine \(g\). **Indicate** the quantities that could be plotted on the vertical and horizontal axes to yield a straight line whose slope could be used to calculate \(g\). Vertical axis: Horizontal axis: *(1 points)*

**Part c)** The students perform the experiment and record the following data. | \(x \text{ (m)}\) | \(T \text{ (s)}\) | | | |---|---|---|---| | 0.10 | 2.29 | | | | 0.20 | 1.80 | | | | 0.30 | 1.68 | | | | 0.40 | 1.68 | | | | 0.50 | 1.73 | | | *(6 points)*

**Part d)** A student performs the experiment but consistently releases the rod from an initial angle of \(45^\circ\) instead of a small angle (e.g., \(5^\circ\)). **Indicate** whether the measured period will be greater than, less than, or equal to the period predicted by the theoretical equation provided. - [ ] Greater than - [ ] Less than - [ ] Equal to **Justify** your reasoning using physical principles. *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/113152/*