The boom is supported by the winch cable that has an...

The boom is supported by the winch cable that has an allowable normal stress of $\sigma_{\text {allow }}=24 \mathrm{ksi}$. If it supports the 5000 lb crate when $\phi=20^{\circ},$ determine the smallest diameter of the cable to the nearest $\frac{1}{16}$ in.

Key Concepts

Cross-sectional Area of a Circular Cable

The cross-sectional area of a circular cable is given by the formula A = (?d²)/4, where d is the diameter. This relationship is key to linking the material's allowable stress and the cable's diameter, allowing for the determination of the minimum necessary diameter to safely support a given load.

Normal Stress

Normal stress is the force acting perpendicularly on a cross-sectional area. In materials like cables, it is defined as the ratio of the applied tensile force to the cable's cross-sectional area. This concept is crucial for determining whether the material can safely support a given load without failure.

Allowable Stress

Allowable stress is the maximum stress that a material is permitted to withstand under service conditions without experiencing failure or excessive deformation. In engineering design, ensuring that the induced stress from an applied load does not exceed the allowable stress is fundamental for safe and reliable structures.

Resolving Forces for Angled Loads

When a load is applied at an angle, the effective force that a structural element, such as a cable, must resist is higher than the component of the load in the direction of support. Trigonometric functions are used to resolve the applied load into its components, ensuring that the component causing tension in the cable is properly accounted for.

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