   Chapter 10, Problem 54AP

Chapter
Section
Textbook Problem

Consider an object with any one of the shapes displayed in Table 8.1. What is the percentage increase in the moment of inertia of the object when it is warmed from 0°C to 100.°C if it is composed of (a) copper or (b) aluminum? Assume the average linear expansion coefficients shown in Table 10.1 do not vary between 0°C and 100.°C. (c) Why are the answers for parts (a) and (b) the same for all the shapes?

(a)

To determine
The percentage increase in the moment of inertia for copper.

Explanation

Given info: The temperature changes from T1=0οC to T2=100οC . The co-efficient of linear expansion of copper ( αc ) is 17×106/οC . The co-efficient of linear expansion of aluminum ( αa ) is 24×106/οC .

Formula to calculate the percentage increase in the moment of inertia is,

%increase=(II0I0)×100% (I)

• I is the moment of inertia after the change in temperature.
• I0 is the initial moment of inertia.

The moment of inertia is proportional to the square of length.

Il2I0l02 (II)

• l0 and l are the initial and final lengths.

Formula to calculate l is,

l=l0[1+αc(T2T1)] (III)

From Equations (I), (II) and (III),

%increase=(l02

(b)

To determine
The percentage increase in the moment of inertia for aluminum.

(c)

To determine
Why the answers of (a) and (b) is the same for all shapes.

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