Q: A steel wire of length $$l_0 = 2.0$$\ \mathrm{m}$$ and cross-sectional area A = 1.0$$ \times $$10^{-6}$$\ \mathrm{m}^2$$ is stretched by $$\Delta l = 1.0\ \mathrm{mm}$$ under a load. If Young's modulus for steel is Y = 2.0$$ \times $$10^{11}$$\ \mathrm{N}\cdot\mathrm{m}^{-2}$$, what is the magnitude of the force applied?

$$F = Y \frac{A \Delta l}{l_0$} = 2.0 \times 10$$^{11}$$ \times \frac{1.0 \times 10$$^{-6}$$ \times 1.0 \times 10$$^{-3}$$}{2.0} = 100\ \mathrm{N}$$

Q: A cube of metal with edge length $$l_0 = 0.10$$\ \mathrm{m}$$ is subjected to a shear force of F_T = 1.5$$ \times $$10^6$$\ \mathrm{N}$$, causing one face to be displaced by a = 0.1$$\ \mathrm{mm}$$. What is the shear modulus G$ of the material?

$$G = \frac{F_T l_0$}{A a} = \frac{1.5 \times 10^6$ \times 0.10}{0.01 \times 0.0001} = 1.5 \times 10$$^{11}$$\ \mathrm{N}\cdot\mathrm{m}^{-2}$$

Q: A submarine hull is subjected to an increase in hydrostatic pressure $$\Delta P = 5.0 \times 10^7$\ \mathrm{Pa}. If $$the bulk modulus of the hull material is $$B = 1.0 \times 10$$^{11}$$\ \mathrm{N}\cdot\mathrm{m}^{-2}$$, what is the fractional change in volume $$\epsilon$$?

$$\epsilon = -\frac{\Delta P}{B} = -\frac{5.0 \times 10^7$}{1.0 \times 10$$^{11}$$} = -5.0 \times 10$$^{-4}$$

Q: A cylindrical rod of radius $$r = 2.0\ \mathrm{mm}$$ and length $$l_0 = 1.0$$\ \mathrm{m}$$ is compressed by a force of F = 1000$$\ \mathrm{N}$$. What is the compressive stress in the rod?

$$\delta = \frac{F}{A} = \frac{1000}{\pi (2.0 \times 10$$^{-3}$$)^2$$} = 7.96 \times $$10^7$$\ \mathrm{N}\cdot\mathrm{m}^{-2}$

Question 1

Which of the following best describes the difference between tensile stress and compressive stress in a solid rod?

Accepted Answer

Tensile stress stretches the rod, increasing its length, while compressive stress shortens the rod, decreasing its length.

Question 2

A steel wire of length $$l_0 = 2.0$$\ \mathrm{m}$$ and cross-sectional area A = 1.0$$ 	imes $$10^{-6}$$\ \mathrm{m}^2$$ is stretched by $$\Delta l = 1.0\ \mathrm{mm}$$ under a load. If Young's modulus for steel is Y = 2.0$$ 	imes $$10^{11}$$\ \mathrm{N}\cdot\mathrm{m}^{-2}$$, what is the magnitude of the force applied?

Accepted Answer

$$F = Y \frac{A \Delta l}{l_0$} = 2.0 	imes 10$$^{11}$$ 	imes \frac{1.0 	imes 10$$^{-6}$$ 	imes 1.0 	imes 10$$^{-3}$$}{2.0} = 100\ \mathrm{N}$$

Question 3

Which of the following statements about the stress-strain curve for a ductile metal is correct?

Accepted Answer

The slope of the initial linear region (Oa) of the curve is equal to the Young's modulus.

Question 4

A cube of metal with edge length $$l_0 = 0.10$$\ \mathrm{m}$$ is subjected to a shear force of F_T = 1.5$$ 	imes $$10^6$$\ \mathrm{N}$$, causing one face to be displaced by a = 0.1$$\ \mathrm{mm}$$. What is the shear modulus G$ of the material?

Accepted Answer

$$G = \frac{F_T l_0$}{A a} = \frac{1.5 	imes 10^6$ 	imes 0.10}{0.01 	imes 0.0001} = 1.5 	imes 10$$^{11}$$\ \mathrm{N}\cdot\mathrm{m}^{-2}$$

Question 5

Which of the following best explains why the stress-strain curve for an elastomer (such as rubber) forms a hysteresis loop?

Accepted Answer

Because the energy lost during stretching and relaxing is converted into heat, and the loading and unloading paths are different.

Question 6

A submarine hull is subjected to an increase in hydrostatic pressure $$\Delta P = 5.0 	imes 10^7$\ \mathrm{Pa}. If $$the bulk modulus of the hull material is $$B = 1.0 	imes 10$$^{11}$$\ \mathrm{N}\cdot\mathrm{m}^{-2}$$, what is the fractional change in volume $$\epsilon$$?

Accepted Answer

$$\epsilon = -\frac{\Delta P}{B} = -\frac{5.0 	imes 10^7$}{1.0 	imes 10$$^{11}$$} = -5.0 	imes 10$$^{-4}$$

Question 7

A cylindrical rod of radius $$r = 2.0\ \mathrm{mm}$$ and length $$l_0 = 1.0$$\ \mathrm{m}$$ is compressed by a force of F = 1000$$\ \mathrm{N}$$. What is the compressive stress in the rod?

Accepted Answer

$$\delta = \frac{F}{A} = \frac{1000}{\pi (2.0 	imes 10$$^{-3}$$)^2$$} = 7.96 	imes $$10^7$$\ \mathrm{N}\cdot\mathrm{m}^{-2}$

Question 8

A steel bar with Young's modulus $$Y = 2.0 	imes 10$$^{11}$$\ \mathrm{N}\cdot\mathrm{m}^{-2}$$ and cross-sectional area $$A = 2.0 	imes 10$$^{-4}$$\ \mathrm{m}^2 is $$used to pull a truck of mass $$m = 10^4$\ \mathrm{kg} up a $$slope. If the bar stretches by $$\Delta l = 0.11\ \mathrm{mm}$$, what is the force in the bar?

Accepted Answer

$$F = Y \frac{A \Delta l}{l_0$}$$; if $$l_0 = 0.5$$\ \mathrm{m}$$, F = 2.0$$ 	imes $$10^{11}$$ 	imes \frac{2.0 	imes $$10^{-4}$$ 	imes 1.1 	imes $$10^{-4}$$}{0.5} = 8.8 	imes $$10^3$$\ \mathrm{N}$$

Question 9

Which of the following is true regarding the elastic limit of a material?

Accepted Answer

It is the maximum stress a material can withstand without permanent deformation.

Question 10

A rubber band is stretched and released repeatedly. After several cycles, it feels warm to the touch. What is the best explanation for this observation?

Accepted Answer

Energy is lost as heat due to hysteresis in the stress-strain cycle of the elastomer.

Question 11

A steel wire of length $$l_0 = 10.0$$\ \mathrm{m}$$ stretches by $$\Delta l = 5.0\ \mathrm{mm}$$ under a load of F = 100.0$$\ \mathrm{N}$$. If Young's modulus is Y = 2$$ 	imes $$10^{11}$$\ \mathrm{N}\cdot\mathrm{m}^{-2}$$, what is the diameter d$ of the wire?

Accepted Answer

$$d = \sqrt{\frac{4 F l_0$}{\pi Y \Delta l}} = \sqrt{\frac{4 	imes 100.0 	imes 10.0}{\pi 	imes 2 	imes 10$$^{11}$$ 	imes 5.0 	imes 10$$^{-3}$$}} = 1.12\ \mathrm{mm}$$

Question 12

Which of the following best describes the physical meaning of Young's modulus?

Accepted Answer

It is a measure of the stiffness of a material, defined as the ratio of stress to strain within the elastic limit.

Question 13

A rugby player of mass $$m = 100\ \mathrm{kg}$$ running at $$v = 4.0\ \mathrm{m}\cdot\mathrm{s}^{-1} is $$brought to rest in $$\Delta t = 0.2\ \mathrm{s} by a $$tackle. If the cross-sectional area of the bone in his leg is $$A = 3.0 	imes 10$$^{-4}$$\ \mathrm{m}^2$$ and the shear modulus is $$G = 1.0 	imes 10$$^{10}$$\ \mathrm{N}\cdot\mathrm{m}^{-2}$$, what is the maximum shear stress experienced by the bone?

Accepted Answer

$$\delta = \frac{F}{A} = \frac{m v / \Delta t}{A} = \frac{100 	imes 4.0 / 0.2}{3.0 	imes 10$$^{-4}$$} = 6.67 	imes 10^6$\ \mathrm{N}\cdot\mathrm{m}^{-2}$$

Question 14

Which of the following statements about the geometry of hollow versus solid cylinders is correct, according to the study materials?

Accepted Answer

A hollow cylinder can have nearly the same strength as a solid cylinder but with much less mass.

Question 15

A cube of alcohol with edge length $$l_0 = 0.25$$\ \mathrm{m}$$ is subjected to an absolute pressure of P = 5.0$$ 	imes $$10^7$$\ \mathrm{Pa}$$. If the compressibility of alcohol is k = 3$$ 	imes $$10^{-8}$$\ \mathrm{m}^2\cdot\mathrm{N}^{-1}$$, what is the new volume of the cube?

Accepted Answer

The fractional change in volume is $$\epsilon = -k P = -3 	imes 10$$^{-8}$$ 	imes 5.0 	imes 10^7$ = -1.5. $$The new volume is $$V = V_0$ (1 + \epsilon) = 0.0156 	imes (1 - 1.5) = 0.0078\ \mathrm{m}^3$$

Question 16

Which of the following is the correct expression for the work done in stretching a rod of length $$l_0$$, cross-sectional area $$A$$, and Young's modulus $$Y by an $$extension $$x$$ within the elastic limit?

Accepted Answer

$$W = \frac{1}{2} Y A \frac{x^2$}{l_0$}$$

Question 17

A steel wire is used to swing a ball of mass $$m = 5.0\ \mathrm{kg} in a $$vertical circle of radius $$r. If $$the tension in the wire at the lowest point is $$T = 6mg$$, what is the strain in the wire at that point if its cross-sectional area is $$A = 2.0 	imes 10$$^{-6}$$\ \mathrm{m}^2$$ and Young's modulus is $$Y = 2.0 	imes 10$$^{11}$$\ \mathrm{N}\cdot\mathrm{m}^{-2}$$?

Accepted Answer

$$\epsilon = \frac{T}{A Y} = \frac{6 	imes 5.0 	imes 9.8}{2.0 	imes 10$$^{-6}$$ 	imes 2.0 	imes 10$$^{11}$$} = 7.4 	imes 10$$^{-4}$$

Elasticity: Stress, Strain, and Material Properties

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