---
title: "A non-uniform beam of mass \\(M\\) and length \\(L\\) is attached to a vertical wall by a frictionless hinge at point P. The mass of the beam is distributed such that its center of mass is located at a distance \\(3L/4\\) from the hinge. A light cable is attached to the right end of the beam and to the wall, making an angle \\(\\theta\\) with the beam. The cable keeps the beam perfectly horizontal. A small block of mass \\(m\\) is placed on the beam at a distance \\(x\\) from the hinge, where \\(0 \\le x \\le L\\)."
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url: "https://nerd-notes.com/ubq/112633/"
date_modified: "2026-05-01T04:42:29+00:00"
---

# A non-uniform beam of mass \(M\) and length \(L\) is attached to a vertical wall by a frictionless hinge at point P. The mass of the beam is distributed such that its center of mass is located at a distance \(3L/4\) from the hinge. A light cable is attached to the right end of the beam and to the wall, making an angle \(\theta\) with the beam. The cable keeps the beam perfectly horizontal. A small block of mass \(m\) is placed on the beam at a distance \(x\) from the hinge, where \(0 \le x \le L\).

A non-uniform beam of mass \(M\) and length \(L\) is attached to a vertical wall by a frictionless hinge at point P. The mass of the beam is distributed such that its center of mass is located at a distance \(3L/4\) from the hinge. A light cable is attached to the right end of the beam and to the wall, making an angle \(\theta\) with the beam. The cable keeps the beam perfectly horizontal. A small block of mass \(m\) is placed on the beam at a distance \(x\) from the hinge, where \(0 \le x \le L\).

![A 2D physical diagram showing a vertical wall on the left. A horizontal rectangular beam extends to the right from the wall. The left end of the beam connects to the wall at a circular hinge labeled 'P'. A straight line representing a cable extends from the far right end of the beam, angling upward and leftward to attach to the vertical wall. The angle between the cable and the horizontal beam is marked with an arc and the symbol \(\theta\). A bracket below the beam indicates its total length is \(L\). A small square block labeled \(m\) rests on top of the beam. A bracket indicates the horizontal distance from the hinge P to the center of the block is \(x\). A prominent dot is drawn on the beam at exactly three-quarters of its length from the hinge, labeled 'Center of Mass'.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1777499068-aV4egs.jpg)

**Part a)** On the diagram below, **draw** and **label** the forces (not components) that act on the beam when the block is at distance \(x\). Each force must be represented by a distinct arrow starting on, and pointing away from, the point at which the force is exerted. *(3 points)*

**Part b)** **Derive** an expression for the tension \(T\) in the cable when the block is at a distance \(x\) from the hinge. Express your answer in terms of \(M\), \(m\), \(L\), \(x\), \(\theta\), and fundamental constants, as appropriate. *(3 points)*

**Part c)** **Sketch** a graph of the tension \(T\) in the cable as a function of the block's distance \(x\) from the hinge, from \(x = 0\) to \(x = L\). *(2 points)*

**Part d)** The block is now removed from the beam. The cable is suddenly cut, and the beam begins to swing downward around the hinge. Let \(\phi\) be the angle the beam makes with the horizontal. **Sketch** a graph of the magnitude of the beam's angular acceleration \(\alpha\) as a function of \(\phi\), from \(\phi = 0^{\circ}\) (horizontal) to \(\phi = 90^{\circ}\) (vertical). **Justify** the shape of your graph using physical principles. *(4 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/112633/*