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Chapter 10: Problem 14

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 breaking 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 the copper wire can hold without breaking is approximately \(1,256 \,\text{N}\).

Step by step solution

01

Find the cross-sectional area of the wire

First, we need to find the cross-sectional area (A) of the copper wire. This can be calculated using the formula for the area of a circle: $$ A = \pi r^{2} $$ Where A is the area, and r is the radius of the circle. The radius of the copper wire is given as \(1.0 \,\text{mm}\) which can be converted to meters: $$ 1.0 \,\text{mm} = 0.001 \,\text{m} $$
02

Calculate the cross-sectional area of the wire

Now, we can substitute the converted radius value to find the cross-sectional area of the wire: $$ A = \pi (0.001)^{2} $$ $$ A \approx 3.14 \times 10^{-6} \, \text{m}^2 $$ We have now calculated the cross-sectional area of the copper wire to be approximately \(3.14 \times 10^{-6} \, \text{m}^2\).
03

Calculate the maximum force supported by the wire

We can now determine the maximum force (F) that the copper wire can support by using the tensile strength of copper (\(4.0 \times 10^{8} \, \text{Pa}\)). Stress is defined as force (F) divided by the cross-sectional area (A), so we can rearrange this equation to find the force: $$ \text{Tensile Strength} = \frac{F}{A} \,\Rightarrow\, F = \text{Tensile Strength} \times A $$ Substitute the given tensile strength and the calculated cross-sectional area into the equation: $$ F = (4.0 \times 10^8 \, \text{Pa}) \times (3.14 \times 10^{-6} \, \text{m}^2) $$
04

Calculate the maximum load that can be suspended from the wire

Now, we can perform the calculations to find the maximum force that the copper wire can support without breaking: $$ F \approx (4.0 \times 10^8) \times (3.14 \times 10^{-6}) $$ $$ F \approx 1.256 \times 10^3 \, \text{N} $$ So, the maximum load that can be suspended from the copper wire without breaking is approximately \(1,256 \,\text{N}\).

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Most popular questions from this chapter

Two steel plates are fastened together using four bolts. The bolts each have a shear modulus of \(8.0 \times 10^{10} \mathrm{Pa}\) and a shear strength of $6.0 \times 10^{8} \mathrm{Pa} .\( The radius of each bolt is \)1.0 \mathrm{cm} .$ Normally, the bolts clamp the two plates together and the frictional forces between the plates keep them from sliding. If the bolts are loose, then the frictional forces are small and the bolts themselves would be subject to a large shear stress. What is the maximum shearing force \(F\) on the plates that the four bolts can withstand? (IMAGE NOT COPY)
A pendulum of length \(L_{1}\) has a period \(T_{1}=0.950 \mathrm{s}\). The length of the pendulum is adjusted to a new value \(L_{2}\) such that $T_{2}=1.00 \mathrm{s} .\( What is the ratio \)L_{2} / L_{1} ?$
Spider silk has a Young's modulus of $4.0 \times 10^{9} \mathrm{N} / \mathrm{m}^{2}\( and can withstand stresses up to \)1.4 \times 10^{9} \mathrm{N} / \mathrm{m}^{2} . \mathrm{A}$ single webstrand has across-sectional area of \(1.0 \times 10^{-11} \mathrm{m}^{2}\) and a web is made up of 50 radial strands. A bug lands in the center of a horizontal web so that the web stretches downward. (a) If the maximum stress is exerted on each strand, what angle \(\theta\) does the web make with the horizontal? (b) What does the mass of a bug have to be in order to exert this maximum stress on the web? (c) If the web is \(0.10 \mathrm{m}\) in radius, how far down does the web extend? (IMAGE NOT COPY)
A \(0.50-\mathrm{kg}\) object, suspended from an ideal spring of spring constant \(25 \mathrm{N} / \mathrm{m},\) is oscillating vertically. How much change of kinetic energy occurs while the object moves from the equilibrium position to a point \(5.0 \mathrm{cm}\) lower?
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.
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