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Chapter 10: Problem 19

A sphere of copper is subjected to 100 MPa of pressure. The copper has a bulk modulus of 130 GPa. By what fraction does the volume of the sphere change? By what fraction does the radius of the sphere change?

Short Answer

Expert verified
Answer: The change in volume fraction is approximately -0.000769, which means the volume decreases by about 0.0769%. The change in radius fraction is approximately -0.00059, which means the radius decreases by about 0.059%.

Step by step solution

01

Understand Bulk Modulus formula

Bulk modulus (B) is the ratio of the change in pressure (∆P) to the fractional change in volume (∆V/V). The formula for bulk modulus is as follows: B = -∆P / (∆V/V) We have been given the bulk modulus (B = 130 GPa) and the pressure (∆P = 100 MPa). We now need to find the change in volume fraction (∆V/V).
02

Convert units

Before moving further, we need to convert the given pressure and bulk modulus values to the same unit. Since the bulk modulus is given in GPa, let's convert the pressure to GPa as well: ∆P = 100 MPa = 100 × 10^{-3} GPa = 0.1 GPa
03

Calculate the change in volume fraction

Now, we can plug in the given values into the bulk modulus formula and solve for the change in volume fraction (∆V/V): 130 GPa = -(0.1 GPa)/(∆V/V) Rearranging the equation to find (∆V/V): ∆V/V = -(0.1 GPa)/(130 GPa) = -0.000769
04

Relate change in volume fraction to change in radius fraction

In order to relate the change in volume to the change in radius, we need to consider the formula for the volume of a sphere: V = (4/3)πr^3 The fractional change in volume is given by: ∆V/V = [(4/3)π(r+∆r)^3 - (4/3)πr^3] / (4/3)πr^3 Simplifying and canceling out common terms: 1 + (∆V/V) = (r+∆r)^3/r^3
05

Calculate the change in radius fraction

Now, we can plug in the change in volume fraction ∆V/V = -0.000769 and solve for the change in radius fraction (∆r/r): 1 - 0.000769 = (r+∆r)^3 / r^3 Taking the cube root of both sides: (1 - 0.000769)^{1/3} = r+∆r / r Let x = (1 - 0.000769)^{1/3} x = r+∆r / r Now calculating the change in radius fraction (∆r/r): ∆r/r = x - 1 Using a calculator, we get: x ≈ 0.99941 ∆r/r = 0.99941 - 1 = -0.00059
06

Interpret the results

The change in volume fraction is -0.000769. This means the volume of the sphere decreases by approximately 0.0769% when subjected to a pressure of 100 MPa. The change in radius fraction is -0.00059. This means the radius of the sphere decreases by approximately 0.059% under the same pressure.

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Most popular questions from this chapter

A sewing machine needle moves with a rapid vibratory motion, rather like SHM, as it sews a seam. Suppose the needle moves \(8.4 \mathrm{mm}\) from its highest to its lowest position and it makes 24 stitches in 9.0 s. What is the maximum needle speed?
A mass-and-spring system oscillates with amplitude \(A\) and angular frequency \(\omega\) (a) What is the average speed during one complete cycle of oscillation? (b) What is the maximum speed? (c) Find the ratio of the average speed to the maximum speed. (d) Sketch a graph of \(v_{x}(t),\) and refer to it to explain why this ratio is greater than \(\frac{1}{2}\).
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A hedge trimmer has a blade that moves back and forth with a frequency of $28 \mathrm{Hz}$. The blade motion is converted from the rotation provided by the electric motor to an oscillatory motion by means of a Scotch yoke (see Conceptual Question 7 ). The blade moves \(2.4 \mathrm{cm}\) during each stroke. Assuming that the blade moves with SHM, what are the maximum speed and maximum acceleration of the blade?
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)
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