Q. 1.8

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### An Introduction to Thermal Physics

Book edition 1st
Author(s) Daniel V. Schroeder
Pages 356 pages
ISBN 9780201380279

# For a solid, we also define the linear thermal expansion coefficient, α, as the fractional increase in length per degree:(a) For steel, α is 1.1 x 10-5 K-1. Estimate the total variation in length of a 1 km steel bridge between a cold winter night and a hot summer day.(b) The dial thermometer in Figure 1.2 uses a coiled metal strip made of two different metals laminated together. Explain how this works.(c) Prove that the volume thermal expansion coefficient of a solid is equal to the sum of its linear expansion coefficients in the three directions β=αx + αy + αz. (So for an isotropic solid, which expands the same in all directions, β =3 α .)

a) The total variation in length =0.44m

b) The coil with two metals with different value of α makes the dial thermometer to read the temperature easier.

c) The relationship is proved.

See the step by step solution

## Part(a)Step1: Given information

coefficient of thermal expansion is α = 1.1 x 10-5 K-1and

length of the steel bridge is L=1 km =1 x 103 m.

## Part(a) Step2: Explanation

Coefficient of thermal expansion of solid is given as

So we can say change in length is given as

Lets assume the difference between cold winter temperature and hot day temperature is 40K

Substitute the values in the equation (1) we get

So the change in length is 0.44 m .

## Part(b)Step1: Given information

A dial thermometer with two metal strips with different value of α

## Part(b)Step2: Explanation

A typical dial thermometer consists of two metal strip coils together with different values of α.

Metal with different value of α will expand differently with change of temperature.

So the coil will make a radial change by changing the temperature.

It is easier to notice the change and hence easier to measure the temperature.

## Part(c)Step1: Given information

The relationship is given

Prove the relationship

## Part(c)Step2: Explanation

For a non-isotropic solid, they will have different α values, i.e.,αx , αy , αz

Which can be defined as, which are Coefficients of Linear expansion in all three directions are

Where x,y and z is dimension of solid cube and Δx, Δy and Δz are changes in x, y and z respectively.

Coefficients of volume expansion is given by

Volume of rectangular solid is

V = xyz .......................................(1)

Differentiate this equation

Divide equation (2) by V=x y z on both the side, we get

We know

Substitute values in equation (3)

We know

From equation (4) and (5) we can conclude that