A walkway suspended across a hotel lobby is supported at...

A walkway suspended across a hotel lobby is supported at numerous points along its edges by a vertical cable above each point and a vertical column underneath. The steel cable is 1.27 $\mathrm{cm}$ in diameter and is 5.75 $\mathrm{m}$ long before loading. The aluminum column is a hollow cylinder with an inside diameter of $16.14 \mathrm{cm},$ an outside diameter of $16.24 \mathrm{cm},$ and unloaded length of $3.25 \mathrm{m} .$ When the walk-way exerts a load force of 8500 $\mathrm{N}$ on one of the support points, how much does the point move down?

Key Concepts

Hooke's Law

Hooke's Law is the foundational principle in elasticity that states that, within the elastic limit of a material, the deformation (strain) is directly proportional to the applied load (stress). This principle is used to calculate how much a structural component will stretch or compress under an axial load, assuming the material returns to its original shape once the load is removed.

Young's Modulus

Young's Modulus is a material property that quantifies its stiffness by relating stress to strain. It is defined as the ratio of the applied stress to the resulting strain in the linear elastic region. A higher Young's Modulus indicates a stiffer material that deforms less under a given load, and it is crucial for determining how much a component will elongate or compress under an axial force.

Axial Deformation Calculation

Axial deformation refers to the change in length of a component when subjected to an axial load. It can be calculated using the formula ?L = FL/(AE), where F is the applied force, L is the original length, A is the cross-sectional area, and E is Young’s Modulus. This calculation is essential for predicting how much a structural member will extend or compress in response to loading.

Cross?Sectional Area Calculation

The cross?sectional area is a key geometric property that directly influences the magnitude of deformation under load. For circular components, it is calculated using the formula A = ?r², where r is the radius of the cross section. Knowing the cross-sectional area is necessary for determining the stress experienced by a material and directly affects the deformation predicted by Hooke's Law and axial deformation formulas.

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