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Ch. 12 - Static Equilibrium; Elasticity and Fracture
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 12 - Static Equilibrium; Elasticity and FractureProblem 25a
Chapter 12, Problem 25a

A uniform rod AB of length 4.5 m and mass M = 3.8 kg is hinged at A and held in equilibrium by a light cord, as shown in Fig. 12–69. A load W = 22 N hangs from the rod at a distance d so that the tension in the cord is 85 N. Draw a free-body diagram for the rod.
A uniform rod hinged at A with a load W = 22 N hanging vertically and a cord at an angle of 37° at B.

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Start by identifying all the forces acting on the rod. These include: (1) the weight of the rod, which acts downward at its center of gravity (the midpoint of the rod), (2) the load W = 22 N, which acts downward at a distance d from point A, (3) the tension T = 85 N in the cord, which acts at an angle θ to the horizontal, and (4) the reaction force at the hinge A, which has both horizontal and vertical components.
Draw the free-body diagram for the rod. Represent the rod as a straight horizontal line. Mark the hinge at point A on the left end. Indicate the weight of the rod (Mg) acting downward at the midpoint of the rod. Show the load W acting downward at a distance d from A. Draw the tension T acting at an angle θ to the horizontal at the right end of the rod. Finally, include the reaction forces at the hinge: a horizontal force (Ax) and a vertical force (Ay).
Apply the condition for rotational equilibrium: the sum of the torques about any point must be zero. Choose point A as the pivot to eliminate the hinge forces from the torque equation. Write the torque contributions: (1) the torque due to the rod's weight, (2) the torque due to the load W, and (3) the torque due to the tension T. Use the equation: Στ = 0.
Apply the condition for translational equilibrium: the sum of forces in both the horizontal and vertical directions must be zero. Write two equations: (1) ΣFx = 0 for the horizontal forces, which includes the horizontal component of the tension (Tcosθ) and the horizontal reaction force (Ax), and (2) ΣFy = 0 for the vertical forces, which includes the vertical component of the tension (Tsinθ), the weight of the rod (Mg), the load W, and the vertical reaction force (Ay).
Solve the system of equations step by step. Use the torque equation to find the distance d. Then use the force equilibrium equations to find the hinge reaction forces (Ax and Ay). Ensure all trigonometric functions (sinθ and cosθ) are correctly applied based on the geometry of the problem.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Free-Body Diagram

A free-body diagram (FBD) is a graphical representation used to visualize the forces acting on an object. In the context of the rod, it helps identify all forces, including gravitational forces, tension, and any applied forces. By isolating the rod and illustrating these forces, one can analyze the equilibrium conditions and solve for unknowns.
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Equilibrium Conditions

An object is in equilibrium when the net force and net torque acting on it are both zero. For the rod in this problem, this means that the upward forces (tension in the cord) must balance the downward forces (weight of the rod and the load). Understanding these conditions is crucial for solving problems involving static structures.
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Torque & Equilibrium

Torque

Torque is a measure of the rotational force applied to an object and is calculated as the product of the force and the distance from the pivot point (hinge). In this scenario, the torques created by the weight of the rod and the load must be balanced by the torque from the tension in the cord to maintain equilibrium. Analyzing torque is essential for understanding how forces cause rotation.
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