---
title: "A uniform rod of length \\(L\\) and mass \\(M\\) is launched into the air. At time \\(t = 0\\), the rod’s center of mass is located at the origin \\((0,0)\\) of an \\(xy\\)-coordinate system. The rod is perfectly vertical, with a labeled point \\(P\\) at the top end of the rod. At this instant, the center of mass has an initial horizontal velocity \\(v_{0x}\\) to the right and an initial vertical velocity \\(v_{0y}\\) upward. The rod is rotating counterclockwise in the \\(xy\\)-plane with a constant angular velocity \\(\\omega_0\\). Air resistance is negligible."
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url: "https://nerd-notes.com/ubq/110395/"
date_modified: "2026-04-02T07:24:36+00:00"
---

# A uniform rod of length \(L\) and mass \(M\) is launched into the air. At time \(t = 0\), the rod’s center of mass is located at the origin \((0,0)\) of an \(xy\)-coordinate system. The rod is perfectly vertical, with a labeled point \(P\) at the top end of the rod. At this instant, the center of mass has an initial horizontal velocity \(v_{0x}\) to the right and an initial vertical velocity \(v_{0y}\) upward. The rod is rotating counterclockwise in the \(xy\)-plane with a constant angular velocity \(\omega_0\). Air resistance is negligible.

A uniform rod of length \(L\) and mass \(M\) is launched into the air. At time \(t = 0\), the rod's center of mass is located at the origin \((0,0)\) of an \(xy\)-coordinate system. The rod is perfectly vertical, with a labeled point \(P\) at the top end of the rod. At this instant, the center of mass has an initial horizontal velocity \(v_{0x}\) to the right and an initial vertical velocity \(v_{0y}\) upward. The rod is rotating counterclockwise in the \(xy\)-plane with a constant angular velocity \(\omega_0\). Air resistance is negligible.

![A Cartesian coordinate system with the origin (0,0) marked. A solid vertical line segment representing the rod is centered on the origin. The total length of the rod is L. The top end of the rod is labeled 'P'. A dashed arrow points horizontally to the right from the origin, labeled 'v_0x'. Another dashed arrow points vertically upward from the origin, labeled 'v_0y'. A curved arrow forms a counterclockwise circle around the origin, labeled '\omega_0'.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1775107327-2rjlZ4.jpg)

**Part a)** The rod is in the air until it returns to the height \(y = 0\). The rod completes exactly one-half of a full rotation (\(\pi\) radians) between \(t = 0\) and the instant its center of mass reaches its maximum height. On the axes provided, **sketch** the trajectory of the rod's center of mass. On your sketch, **draw** the rod at the instant it reaches its maximum height, clearly indicating its orientation and the location of point \(P\). *(3 points)*

**Part b)** **Sketch** a graph of the rod's angular velocity \(\omega\) as a function of time \(t\) and the vertical velocity \(v_y\) of the rod's center of mass as a function of time \(t\) from \(t = 0\) until the rod returns to \(y = 0\). Let counterclockwise and upward be the positive directions. *(3 points)*

**Part c)** At the instant the rod's center of mass reaches its maximum height, **determine** whether the absolute speed of the top end of the rod is greater than, less than, or equal to the absolute speed of the bottom end of the rod. **Justify** your answer using physical principles. *(3 points)*

**Part d)** A student proposes the following equation for the vertical position \(y_P\) of point \(P\) as a function of time: \[ y_P = v_{0y}t - \dfrac{1}{2}gt^2 + \dfrac{L}{2}\cos(\omega_0 t) \] **Explain** how the student's equation correctly models the motion of point \(P\) in the limiting case where the rod has no initial angular velocity (\(\omega_0 = 0\)). *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/110395/*