---
title: "A uniform rigid beam of mass \\(M\\) and length \\(L\\) is attached to a vertical wall by a hinge. The beam is held horizontally by a light cable attached to the right end of the beam. The cable makes an angle \\(\\theta\\) with the beam and is attached to the wall above the hinge. A small block of mass \\(m_0\\) rests on the beam at a horizontal distance \\(x\\) from the hinge. The system is initially at rest in static equilibrium."
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url: "https://nerd-notes.com/ubq/110288/"
date_modified: "2026-04-01T04:47:21+00:00"
---

# A uniform rigid beam of mass \(M\) and length \(L\) is attached to a vertical wall by a hinge. The beam is held horizontally by a light cable attached to the right end of the beam. The cable makes an angle \(\theta\) with the beam and is attached to the wall above the hinge. A small block of mass \(m_0\) rests on the beam at a horizontal distance \(x\) from the hinge. The system is initially at rest in static equilibrium.

A uniform rigid beam of mass \(M\) and length \(L\) is attached to a vertical wall by a hinge. The beam is held horizontally by a light cable attached to the right end of the beam. The cable makes an angle \(\theta\) with the beam and is attached to the wall above the hinge. A small block of mass \(m_0\) rests on the beam at a horizontal distance \(x\) from the hinge. The system is initially at rest in static equilibrium.

![A vertical wall is on the left side of the image. A horizontal uniform rectangular beam extends to the right, attached to the wall by a circular hinge at its left end. A small square block labeled \(m_0\) sits on top of the beam. A horizontal dimension line below the beam indicates the distance from the wall to the center of the block is \(x\). Another dimension line indicates the full length of the beam is \(L\). A taut cable connects the right end of the beam to a point higher up on the vertical wall. An arc indicates the angle between the horizontal beam and the cable, labeled \(\theta\).](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1775018840-pnHEZi.jpg)

**Part a)** **On the diagram below, draw and label** the forces (not components) that are exerted on the beam. **Represent** the force exerted by the hinge as two separate perpendicular components (horizontal and vertical). Draw each force as a distinct arrow starting on, and pointing away from, the point where the force is applied. *(3 points)*

**Part b)** **Derive** an expression for the tension \(T\) in the cable. Express your answer in terms of \(M\), \(L\), \(m_0\), \(x\), \(\theta\), and physical constants, as appropriate.

**Part c)** **Derive** an expression for the vertical component of the force exerted by the hinge on the beam, \(F_{Hy}\). Express your answer in terms of \(M\), \(L\), \(m_0\), \(x\), \(\theta\), and physical constants, as appropriate.

**Part d)** The block is now moved to a new position further to the right, increasing \(x\). **Indicate** whether the vertical component of the hinge force, \(F_{Hy}\), increases, decreases, or stays the same. - [ ] Increases - [ ] Decreases - [ ] Stays the same **Justify** your answer using physical principles or your derived equations.

**Part e)** The block is now placed at the far right end of the beam (\(x = L\)). The cable suddenly snaps. **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 \(M\), \(L\), \(m_0\), and physical constants, as appropriate. (The rotational inertia of a uniform beam of mass \(M\) and length \(L\) rotated about one end is \(I = \dfrac{1}{3}ML^2\)).


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/110288/*