---
title: "A uniform rigid beam of mass \\(M\\) and length \\(L\\) is attached to a vertical wall by a frictionless hinge. The beam is held in a horizontal position by a lightweight cable that is attached to the right end of the beam and to the wall. The cable makes an angle \\(\\theta\\) with the beam. A small block of mass \\(m\\) is placed on the beam at a distance \\(x\\) from the hinge. The entire system is initially in static equilibrium."
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url: "https://nerd-notes.com/ubq/111554/"
date_modified: "2026-04-12T03:11:39+00:00"
---

# A uniform rigid beam of mass \(M\) and length \(L\) is attached to a vertical wall by a frictionless hinge. The beam is held in a horizontal position by a lightweight cable that is attached to the right end of the beam and to the wall. The cable makes an angle \(\theta\) with the beam. A small block of mass \(m\) is placed on the beam at a distance \(x\) from the hinge. The entire system is initially in static equilibrium.

A uniform rigid beam of mass \(M\) and length \(L\) is attached to a vertical wall by a frictionless hinge. The beam is held in a horizontal position by a lightweight cable that is attached to the right end of the beam and to the wall. The cable makes an angle \(\theta\) with the beam. A small block of mass \(m\) is placed on the beam at a distance \(x\) from the hinge. The entire system is initially in static equilibrium.

![A vertical line on the left represents a wall. A horizontal rectangle represents a uniform beam, extending to the right from a hinge attached to the wall. A diagonal line represents a cable connecting the right end of the beam to a higher point on the wall, forming an angle \(\theta\) between the beam and the cable. A small square block labeled 'm' rests on top of the beam at a distance \(x\) from the hinge. The total length of the beam is indicated as \(L\) with a dimension line. The beam is labeled 'Mass \(M\)'.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1775963499-1ZyBvy.jpg)

**Part a)** **Derive** an expression for the tension \(F_T\) in the cable. Express your answer in terms of \(M\), \(m\), \(L\), \(x\), \(\theta\), and fundamental constants, as appropriate. *(4 points)*

**Part b)** The cable suddenly snaps. The block has a sticky coating so that it sticks to the beam and does not slip as the beam falls. **Derive** an expression for the initial angular acceleration \(\alpha\) of the beam-block system immediately after the cable snaps. Express your answer in terms of \(M\), \(m\), \(L\), \(x\), and fundamental constants, as appropriate. *(Note: The rotational inertia of a uniform rod of mass \(M\) and length \(L\) about one end is \(\dfrac{1}{3}ML^2\))* *(3 points)*

**Part c)** **Indicate** whether the magnitude of the angular acceleration of the beam increases, decreases, or stays the same as the beam rotates downward from the horizontal position. - [ ] Increases - [ ] Decreases - [ ] Stays the same **Justify** your answer. *(2 points)*

**Part d)** In a new trial, the original cable is replaced, but the block of mass \(m\) is moved to a new distance \(x_2\) from the hinge, where \(x_2 > x\). The system is once again held in horizontal static equilibrium. **Indicate** whether the new tension in the cable is greater than, less than, or equal to the original tension \(F_T\). - [ ] Greater than - [ ] Less than - [ ] Equal to **Justify** your answer 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/111554/*