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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 51
Chapter 12, Problem 51

Assume the supports of the uniform cantilever shown in Fig. 12–79 (m = 2900 kg) are made of wood. Calculate the minimum cross-sectional area required of each, assuming a safety factor of 9.0.
Diagram of a uniform cantilever with forces and dimensions labeled, illustrating support reactions and load distribution.

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1
Determine the total weight of the cantilever by using the formula for weight: \( W = m \cdot g \), where \( m \) is the mass of the cantilever (2900 kg) and \( g \) is the acceleration due to gravity (9.8 m/s²).
Calculate the total force that each support must bear. Since the cantilever is uniform, the weight is evenly distributed, so each support will bear half of the total weight: \( F_{support} = \frac{W}{2} \).
Apply the safety factor to the force. The safety factor ensures the supports can handle more than the expected load. Multiply the force on each support by the safety factor: \( F_{safe} = F_{support} \cdot 9.0 \).
Use the formula for stress to relate the force to the cross-sectional area: \( \sigma = \frac{F}{A} \), where \( \sigma \) is the maximum allowable stress for wood (a material property that must be provided or looked up). Rearrange the formula to solve for the cross-sectional area: \( A = \frac{F_{safe}}{\sigma} \).
Substitute the values for \( F_{safe} \) and \( \sigma \) into the equation to calculate the minimum cross-sectional area required for each support. Ensure the units are consistent throughout the calculation.

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Key Concepts

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

Cantilever Beam

A cantilever beam is a structural element that is fixed at one end and free at the other. It supports loads applied along its length, creating bending moments and shear forces. Understanding the behavior of cantilever beams is crucial for analyzing their strength and stability under various loading conditions.
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Safety Factor

The safety factor is a design criterion that provides a margin of safety in engineering structures. It is defined as the ratio of the maximum load a structure can withstand to the intended load. A higher safety factor indicates greater reliability and is essential for ensuring that structures can handle unexpected loads or material defects.
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Cross-Sectional Area

The cross-sectional area of a structural element is the area of its section perpendicular to its length. It is a critical factor in determining the strength and stiffness of materials under load. In the context of cantilevers, calculating the minimum cross-sectional area ensures that the supports can safely carry the weight and forces acting on the beam.
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