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

Giancoli Douglas 5th edition

Ch. 12 - Static Equilibrium; Elasticity and Fracture

Problem 40c

Chapter 12, Problem 40c

A marble column of cross-sectional area 1.4m² supports a mass of 22,000 kg. By how much is the column shortened if it is 8.6 m high?

Verified step by step guidance

Determine the force exerted on the marble column due to the mass it supports. Use the formula for weight:

F = m g

, where

m

is the mass (22,000 kg) and

g

is the acceleration due to gravity (9.8 m/s²).

Calculate the stress on the column using the formula:

σ = \frac{F}{A}

, where

F

is the force calculated in step 1 and

A

is the cross-sectional area (1.4 m²).

Use the relationship between stress and strain to find the strain:

ε = \frac{σ}{Y}

, where

σ

is the stress from step 2 and

Y

is the Young's modulus for marble (a known material property, typically around 5 × 10¹⁰ Pa).

Relate the strain to the change in length of the column using the formula:

ε = \frac{ΔL}{L}

, where

ΔL

is the change in length, and

L

is the original length of the column (8.6 m). Rearrange to solve for

ΔL

ΔL = ε L

Substitute the values for strain (from step 3) and the original length (8.6 m) into the formula from step 4 to calculate the amount by which the column is shortened.

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Key Concepts

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

Stress

Stress is defined as the force applied per unit area within materials. In this context, it is calculated by dividing the weight of the mass supported by the column by the cross-sectional area of the column. Understanding stress is crucial for determining how materials deform under load.

Strain

Strain is the measure of deformation representing the displacement between particles in a material body. It is defined as the change in length divided by the original length. In this problem, strain will help us understand how much the marble column shortens when subjected to the load.

Young's Modulus

Young's Modulus is a material property that measures the stiffness of a solid material. It is defined as the ratio of stress to strain in the linear elastic region of the material's stress-strain curve. This concept is essential for calculating the amount of shortening of the column under the applied load.

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