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Chapter 10: Problem 13

What is the maximum load that could be suspended from a copper wire of length \(1.0 \mathrm{m}\) and radius \(1.0 \mathrm{mm}\) without permanently deforming the wire? Copper has an elastic limit of \(2.0 \times 10^{8} \mathrm{Pa}\) and a tensile strength of \(4.0 \times 10^{8} \mathrm{Pa}\).

Short Answer

Expert verified
Answer: The maximum load that can be suspended from the copper wire without permanently deforming it is approximately 127.9 N.

Step by step solution

01

Calculate cross-sectional area of the wire

To determine the cross-sectional area of the wire, we can use the formula for the area of a circle, which is \(A = \pi r^2\). In this case, the radius of the wire is given as \(1.0\,\text{mm} = 1.0 \times 10^{-3}\,\text{m}\). So the area is: \(A = \pi (1.0 \times 10^{-3}\,\text{m})^2\)
02

Use the elastic limit and cross-sectional area to determine maximum force

The elastic limit of the material is given as \(2.0 \times 10^{8}\,\text{Pa}\). The formula to calculate the force required to reach this stress level can be obtained using the equation: \(Stress = \frac{Force}{Area}\) We can rearrange the equation to find the maximum force: \(Force = Stress \times Area\) Substitute the given values for stress and area: \(Force = (2.0 \times 10^{8}\,\text{Pa}) \times \pi (1.0 \times 10^{-3}\,\text{m})^2\)
03

Calculate the maximum load

To find the maximum load that could be suspended from the wire without permanently deforming it, we can divide the maximum force by the acceleration due to gravity (\(9.81\,\text{m/s}^2\)): \(Load = \frac{Force}{g}\) Substitute the expression for the maximum force and the value of g: \(Load = \frac{(2.0 \times 10^{8}\,\text{Pa}) \times \pi (1.0 \times 10^{-3}\,\text{m})^2}{9.81\,\text{m/s}^2}\) Finally, calculate the maximum load: \(Load \approx 127.9\,\text{N}\) So, the maximum load that could be suspended from a copper wire of length \(1.0\,\text{m}\) and radius \(1.0\,\text{mm}\) without permanently deforming the wire is approximately \(127.9\,\text{N}\).

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Most popular questions from this chapter

An anchor, made of cast iron of bulk modulus \(60.0 \times 10^{9} \mathrm{Pa}\) and of volume \(0.230 \mathrm{m}^{3},\) is lowered over the side of the ship to the bottom of the harbor where the pressure is greater than sea level pressure by \(1.75 \times 10^{6} \mathrm{Pa} .\) Find the change in the volume of the anchor.
A \(230.0-\mathrm{g}\) object on a spring oscillates left to right on a frictionless surface with a frequency of \(2.00 \mathrm{Hz}\). Its position as a function of time is given by \(x=(8.00 \mathrm{cm})\) sin \(\omega t\) (a) Sketch a graph of the elastic potential energy as a function of time. (b) The object's velocity is given by \(v_{x}=\omega(8.00 \mathrm{cm}) \cos \omega t .\) Graph the system's kinetic energy as a function of time. (c) Graph the sum of the kinetic energy and the potential energy as a function of time. (d) Describe qualitatively how your answers would change if the surface weren't frictionless.
The maximum strain of a steel wire with Young's modulus $2.0 \times 10^{11} \mathrm{N} / \mathrm{m}^{2},\( just before breaking, is \)0.20 \%$ What is the stress at its breaking point, assuming that strain is proportional to stress up to the breaking point?
Luke is trying to catch a pesky animal that keeps eating vegetables from his garden. He is building a trap and needs to use a spring to close the door to his trap. He has a spring in his garage and he wants to determine the spring constant of the spring. To do this, he hangs the spring from the ceiling and measures that it is \(20.0 \mathrm{cm}\) long. Then he hangs a \(1.10-\mathrm{kg}\) brick on the end of the spring and it stretches to $31.0 \mathrm{cm} .$ (a) What is the spring constant of the spring? (b) Luke now pulls the brick \(5.00 \mathrm{cm}\) from the equilibrium position to watch it oscillate. What is the maximum speed of the brick? (c) When the displacement is \(2.50 \mathrm{cm}\) from the equilibrium position, what is the speed of the brick? (d) How long will it take for the brick to oscillate five times?
How would the volume of \(1.00 \mathrm{cm}^{3}\) of aluminum on Earth change if it were placed in a vacuum chamber and the pressure changed to that of the Moon (less than \(10^{-9} \mathrm{Pa}\) )?
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