The boom has a uniform weight of 600 lb and is hoisted into...

The boom has a uniform weight of 600 lb and is hoisted into position using the cable $B C$. If the cable has a diameter of 0.5 in., plot the average normal stress in the cable as a function of the boom position $\theta$ for $0^{\circ} \leq \theta \leq 90^{\circ}$

Key Concepts

Normal Stress

Normal stress is defined as the force acting perpendicular to a given cross-sectional area divided by that area. In the context of a cable, it is the tensile force distributed uniformly over the cable's cross-sectional area, quantifying the internal stresses the cable experiences when subjected to external loads.

Cross-Sectional Area Calculation

For a cable with a circular cross-section, the area is calculated using the formula A = (?/4)×d², where d is the diameter. This calculation is essential for converting the applied tensile force into an average normal stress by dividing the force by the computed area.

Static Equilibrium

Static equilibrium is a fundamental concept in mechanics stating that the sum of forces and moments acting on a body must be zero if the body is at rest or moving at constant velocity. In this problem, it is used to analyze how the weight of the boom is balanced by the tension in the cable as the boom’s position changes.

Force Resolution and Trigonometry

Force resolution involves breaking down a force into its components along specified directions, typically using trigonometric functions. As the boom rotates, the orientation changes the effective component of the weight that is resisted by the cable, necessitating the use of trigonometric relationships to express the cable tension as a function of the boom’s angle.

Function Plotting

Plotting a function entails representing the relationship between variables graphically. In this problem, the average normal stress in the cable is expressed as a function of the boom position angle. Analyzing and plotting this function helps visualize how the internal stress varies with changes in the boom’s orientation, which is critical for understanding the cable’s performance under different load conditions.

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