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Chapter 11: Problem 25

A metal rod that is 4.00 \(\mathrm{m}\) long and 0.50 \(\mathrm{cm}^{2}\) in cross- sectional area is found to stretch 0.20 \(\mathrm{cm}\) under a tension of 5000 \(\mathrm{N}\) . What is Young's modulus for this metal?

Short Answer

Expert verified
Young's modulus is \(2 \times 10^{11}\,\mathrm{N/m}^2\).

Step by step solution

01

Understanding the Problem

We need to find the Young's modulus of a metal rod given its original length, cross-sectional area, amount it stretches under a specific tension, and the tension force applied. Young's modulus is a measure of the stiffness of a material and is defined by the formula:\[ E = \frac{F \cdot L}{A \cdot \Delta L} \]where \(E\) is Young's modulus, \(F\) is the force applied, \(L\) is the original length, \(A\) is the cross-sectional area, and \(\Delta L\) is the change in length.
02

Calculating Given Values

Before substituting into the formula, ensure all units are consistent. The original length \(L\) is given as \(4.00\,\mathrm{m}\), which is \(400\,\mathrm{cm}\). The cross-sectional area \(A\) is \(0.50\,\mathrm{cm}^2\). The change in length \(\Delta L\) is \(0.20\,\mathrm{cm}\). The force \(F\) is \(5000\,\mathrm{N}\).
03

Substitute and Solve for Young's Modulus

Substitute the given values into the formula for Young's modulus:\[E = \frac{5000\,\mathrm{N} \times 400\,\mathrm{cm}}{0.50\,\mathrm{cm}^2 \times 0.20\,\mathrm{cm}}\]Calculate the numerator and the denominator:- Numerator: \(5000 \times 400 = 2000000\,\mathrm{N\cdot cm}\)- Denominator: \(0.50 \times 0.20 = 0.10\,\mathrm{cm}^3\)Now, divide to find \(E\):\[ E = \frac{2000000}{0.10} = 20000000\,\mathrm{N/cm}^2 \]Convert \(\mathrm{N/cm}^2\) to \(\mathrm{N/m}^2\) (or pascals) by multiplying by \(10^4\), since \(1\,\mathrm{m} = 100\,\mathrm{cm}\):\[ E = 20000000 \times 10000 = 2 \times 10^{11}\,\mathrm{N/m}^2 \]
04

Conclusion

After performing the calculations, Young's modulus for the metal is determined to be \(2 \times 10^{11}\,\mathrm{N/m}^2\). This signifies the stiffness of the metal rod under the given tension.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Material Stiffness
Material stiffness refers to how much a material resists deformation when a force is applied. It's an essential property in determining how suitable a material is for specific applications. Stiff materials, like steel, don't bend or stretch easily. They hold their shape much better when pushed or pulled.
This quality is crucial in areas like construction or manufacturing, where materials are frequently under stress. By understanding a material's stiffness, engineers can predict how structures will behave under various conditions.
Young's modulus, which quantifies stiffness, helps in comparing different materials and choosing the right one for specific purposes. A higher Young's modulus value signifies a stiffer material.
Tension
Tension is the force that stretches something. Think of pulling on a rope; this pulling action generates a tension force within the rope. In the context of the metal rod problem, tension refers to the force trying to elongate the rod.
Materials can behave differently under tension. Some may stretch slightly and return to their original size once the tension is released. Others might not return to their original length if stretched beyond a certain point. To design safe systems, understanding how materials respond to tension is fundamental.
Materials with high resistance to tension are less prone to snapping or deforming permanently when a force is applied.
Mechanics of Materials
Mechanics of materials is a branch of engineering that studies how different materials react under various forces. This area encompasses many factors, including tension, compression, bending, and twisting.
The primary goal is to understand the behavior of materials and predict their performance when used in constructions like bridges or airplanes. Since each material has unique properties, engineers use concepts from mechanics of materials to decide which material will work best for a particular application.
This knowledge allows for the optimization of material use, ensuring safety and efficiency in design.
Calculating Young's Modulus
Computing Young's modulus helps determine a material's stiffness quantitatively. The calculation involves several known variables, including force (\(F\)), original length (\(L\)), cross-sectional area (\(A\)), and the change in length (\(\Delta L\)). The formula used is: \[E = \frac{F \cdot L}{A \cdot \Delta L}\]To apply this formula correctly:
  • Ensure all measurements are in consistent units for accurate results.
  • Substitute the known values into the formula.
  • Solve for \(E\).
This exercise provided an example with a metal rod where basic arithmetic gives Young's modulus, aiding in selecting materials for engineering projects.
Young's modulus is expressed in pascals (\(\mathrm{N/m}^2\)), clarifying the stiffness metric.

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Most popular questions from this chapter

Bulk Modulus of an Ideal Gas. The equation of state (the equation relating pressure, volume, and temperature) for an ideal gas is \(p V=n R T\) , where \(n\) and \(R\) are constants. (a) Show that if the gas is compressed while the temperature \(T\) is held constant, the bulk modulus is equal to the pressure. (b) When an ideal gas is compressed without the transfer of any heat into or out of it, the pressure and volume are related by \(p V^{\gamma}=\) constant, where \(\gamma\) is a constant having different values for different gases. Show that, in this case, the bulk modulus is given by \(B=\gamma p\) .

A door 1.00 \(\mathrm{m}\) wide and 2.00 \(\mathrm{m}\) high weighs 280 \(\mathrm{N}\) and is supported by two hinges, one 0.50 \(\mathrm{m}\) from the top and the other 0.50 \(\mathrm{m}\) m from the bottom.Each hinge supports half the total weight of the door. Assuming that the door's center of gravity is at its center, find the horizontal components of force exerted on the door by each hinge.

A metal wire 3.50 \(\mathrm{m}\) long and 0.70 \(\mathrm{mm}\) in diameter was given the following test. A load weighing 20 \(\mathrm{N}\) was originally hung from the wire to keep it taut. The position of the lower end of the wire was read on a scale as load was added. $$ \begin{array}{cc}{\text { Added Load (N) }} & {\text { Scale Reading (cm) }} \\\ \hline 0 & {3.02} \\ {10} & {3.07} \\ {20} & {3.12} \\ {30} & {3.17} \\\ {40} & {3.22} \\ {50} & {3.27} \\ {60} & {3.32} \\ {70} & {4.27}\end{array} $$ (a) Graph these values, plotting the increase in length horizontally and the added load vertically. \((b)\) Calculate the value of Young's modulus. (c) The proportional limit occurred at a scale reading of 3.34 \(\mathrm{cm} .\) What was the stress at this point?

A \(15,000-N\) crane pivots around a friction-free axle at its base and is supported by a cable making a \(25^{\circ}\) angle with the crane (Fig. 11.29 ). The crane is 16 \(\mathrm{m}\) long and is not uniform, its center of gravity being 7.0 \(\mathrm{m}\) from the axle as measured along the crane. The cable is attached 3.0 \(\mathrm{m}\) from the upper end of the crane. When the crane is raised to \(55^{\circ}\) above the horizontal holding an \(11,000-N\) pallet of bricks by a 2.2 -in very light cord, find (a) the tension in the cable, and \((b)\) the horizontal and vertical components of the force that the axle exerts on the crane. Start with a free-body diagram of the crane.

A nonuniform beam 4.50 \(\mathrm{m}\) long and weighing 1.00 \(\mathrm{kN}\) makes an angle of \(25.0^{\circ}\) below the horizontal. It is held in position by a frictionless pivot at its upper right end and by a cable 3.00 \(\mathrm{m}\) farther down the beam and perpendicular to it (Fig. 11.31\()\) . The center of gravity of the beam is 2.00 \(\mathrm{m}\) down the beam from the pivot. Lighting equipment exerts a \(5.00-\mathrm{kN}\) downward force on the lower left endof the beam. Find the tension \(T\) in the cable and the horizontal and vertical components of the force exerted on the beam by the pivot. Start by sketching a free-body diagram of the beam.
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