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Chapter 13: Problem 16

Given two springs of identical size and shape, one made of steel and the other made of aluminum, which has the higher spring constant? Why? Does the difference depend more on the shear modulus or the bulk modulus of the material?

Short Answer

Expert verified
Answer: The shear modulus has a more significant impact on the spring constant because it determines the material's resistance to shearing deformation (twisting), which is a primary concern in spring systems.

Step by step solution

01

Understanding the spring constant

The spring constant (k) measures a spring's stiffness, determining how much force is needed to cause a certain displacement. It can be found using Hooke's Law: F = kx, where F is the force applied to the spring, and x is the displacement from the spring's resting position.
02

The relationship between spring constant and material properties

The spring constant depends on the material's properties and the spring geometry. Generally, the spring constant can be written as k = (d^4 * G) / (8 * D^3 * N), where d is the wire diameter, G is the shear modulus, D is the mean coil diameter, and N is the number of active coils. In our case, since the springs are of identical size and shape, the factors related to geometry (d, D, and N) are equal. Therefore, we can conclude that k is directly proportional to G (shear modulus).
03

Comparing the shear modulus of steel and aluminum

The shear modulus of steel is around 80 GPa, while the shear modulus of aluminum is around 26 GPa. Since the spring constant is directly proportional to the shear modulus, the spring made of steel has a higher spring constant than the one made of aluminum.
04

Explaining the importance of shear modulus and bulk modulus

In this context, the shear modulus (G) is the dominant factor determining the spring constant because it describes how resistant a material is to shearing deformation (twisting). Bulk modulus (B) measures a material's resistance to uniform compression, which is not a primary concern in spring systems. Therefore, the difference in spring constants depends more on the shear modulus of the material than the bulk modulus.

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Most popular questions from this chapter

You are in a boat filled with large rocks in the middle of a small pond. You begin to drop the rocks into the water. What happens to the water level of the pond? a) It rises. d) It rises momentarily and then b) It falls. falls when the rocks hit bottom. c) It doesn't change. e) There is not enough information to say.

A basketball of circumference \(75.5 \mathrm{~cm}\) and mass \(598 \mathrm{~g}\) is forced to the bottom of a swimming pool and then released. After initially accelerating upward, it rises at a constant velocity, a) Calculate the buoyant force on the basketball. b) Calculate the drag force the basketball experiences while it is moving upward at constant velocity.

A water-powered backup sump pump uses tap water at a pressure of \(3.00 \mathrm{~atm}\left(p_{1}=3 p_{\mathrm{atm}}=\right.\) \(3.03 \cdot 10^{5} \mathrm{~Pa}\) ) to pump water out of a well, as shown in the figure \(\left(p_{\text {well }}=p_{\text {ttm }}\right)\). This system allows water to be pumped out of a basement sump well when the electric pump stops working during an electrical power outage. Using water to pump water may sound strange at first, but these pumps are quite efficient, typically pumping out \(2.00 \mathrm{~L}\) of well water for every \(1.00 \mathrm{~L}\) of pressurized tap water. The supply water moves to the right in a large pipe with cross-sectional area \(A_{1}\) at a speed \(v_{1}=2.05 \mathrm{~m} / \mathrm{s}\). The water then flows into a pipe of smaller diameter with a cross-sectional area that is ten times smaller \(\left(A_{2}=A_{1} / 10\right)\). a) What is the speed \(v_{2}\) of the water in the smaller pipe, with area \(A_{2} ?\) b) What is the pressure \(p_{2}\) of the water in the smaller pipe, with area \(A_{2} ?\) c) The pump is designed so that the vertical pipe, with cross-sectional area \(A_{3}\), that leads to the well water also has a pressure of \(p_{2}\) at its top. What is the maximum height, \(h,\) of the column of water that the pump can support (and therefore act on ) in the vertical pipe?

The Hindenburg, the German zeppelin that caught fire in 1937 while docking in Lakehurst, New Jersey, was a rigid duralumin-frame balloon filled with \(2.000 \cdot 10^{5} \mathrm{~m}^{3}\) of hydrogen. The Hindenburg's useful lift (beyond the weight of the zeppelin structure itself) is reported to have been \(1.099 \cdot 10^{6} \mathrm{~N}(\) or \(247,000 \mathrm{lb}) .\) Use \(\rho_{\text {air }}=1.205 \mathrm{~kg} / \mathrm{m}^{3}, \rho_{\mathrm{H}}=\) \(0.08988 \mathrm{~kg} / \mathrm{m}^{3}\) and \(\rho_{\mathrm{He}}=0.1786 \mathrm{~kg} / \mathrm{m}^{3}\) a) Calculate the weight of the zeppelin structure (without the hydrogen gas). b) Compare the useful lift of the (highly flammable) hydrogen-filled Hindenburg with the useful lift the Hindenburg would have had had it been filled with (nonflammable) helium, as originally planned.

A fountain sends water to a height of \(100 . \mathrm{m}\). What is the difference between the pressure of the water just before it is released upward and the atmospheric pressure?
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