Chapter 18, Problem 1Q

The initial length L, change in temperature ΔT, and change in length ΔL of four rods are given in the following table. Rank the rods according to their coefficients of thermal expansion, greatest first.

Rod	L(m)	ΔT(C°)	ΔL (m)
a	2	10	4 × 10−4
b	1	20	4 × 10−4
c	2	10	8 × 10−4
d	4	5	4 × 10−4

Expert Solution & Answer

To determine

To rank

the four rods according to their coefficients of thermal expansion.

Answer to Problem 1Q

Solution:

The ranking of the rods according to their coefficients of thermal expansion is

${\alpha }_c>{\alpha }_a={\alpha }_b={\alpha }_d$

Explanation of Solution

1) Concept:

Using the Equation 18-9 for the change in the length of rod $∆L$, we can find the coefficients of thermal expansion $\alpha$ for each rod, and by comparing these values of $\alpha$ for each rod, we can rank the rods according to their coefficient of thermal expansion.

2) Formula:

$∆L=L\alpha ∆T$

Where $\alpha$ is the coefficient of linear expansion

3) Given:

i) The length of rod a is $L=2 m$.

ii) The length of rod b is $L=1 m$.

iii) The length of rod c is $L=2 m$.

iv) The length of rod d is $L=4 m$.

v) The change in temperature of rod a is $∆T=10 ℃$.

vi) The change in temperature of rod b is $∆T=20 ℃$.

vii) The change in temperature of rod c is $∆T=10 ℃$.

viii) The change in temperature of rod d is $∆T=5 ℃$

ix) The change in the length of rod a is $∆L=4\times {10}^{-4} m$.

x) The change in the length of rod b is $∆L=4\times {10}^{-4} m$.

xi) The change in the length of rod c is $∆L=8\times {10}^{-4} m$.

xii) The change in the length of rod d is $∆L=4\times {10}^{-4} m$.

4) Calculations:

Using Equation 18-9- If the temperature of the metal rod of length L is released by an amount $∆T$, its length is found to increase by an amount

$∆L=L\alpha ∆T$

Where $\alpha$ is the coefficient of linear expansion.

Therefore, the coefficient $\alpha$ of linear expansion is given by

$\alpha =\frac{∆L}{L∆T}$

Let ${\alpha }_a$, ${\alpha }_b$, ${\alpha }_c$ and ${\alpha }_d$ be the coefficients of linear expansion of  rods a, b, c, and d respectively.

So, for rod a; $L=2 m$, $∆T=10 ℃$ and $∆L=4\times {10}^{-4} m$, adding the values, we get ${\alpha }_a=\frac{4\times {10}^{-4}}{2\times 10}$

${\alpha }_a=2\times {10}^{-5} per ℃$

Similarly, for rod b; $L=1 m$, $∆T=20 ℃$ and $∆L=4\times {10}^{-4} m$, adding  the values, we get ${\alpha }_b=\frac{4\times {10}^{-4}}{1\times 20}$

${\alpha }_b=2\times {10}^{-5} per ℃$

For rod c; $L=2 m$, $∆T=10 ℃$ and $∆L=8\times {10}^{-4} m$, adding the values we get ${\alpha }_c=\frac{8\times {10}^{-4}}{2\times 10}$

${\alpha }_c=4\times {10}^{-5} per ℃$

And for rod d; $L=4 m$, $∆T=5 ℃$ and $∆L=4\times {10}^{-4} m$, adding the values we get ${\alpha }_d=\frac{4\times {10}^{-4}}{4\times 5}$

${\alpha }_d=2\times {10}^{-5} per ℃$

Therefore, we get

${\alpha }_c>{\alpha }_a={\alpha }_b={\alpha }_d$

Conclusion:

Using the formula for change in length due to linear expansion, we can rank the rods according to their coefficient of thermal expansion.

To determine

To rank

the four rods according to their coefficients of thermal expansion.

Answer to Problem 1Q

Solution:

The ranking of the rods according to their coefficients of thermal expansion is

${\alpha }_c>{\alpha }_a={\alpha }_b={\alpha }_d$

Explanation of Solution

1) Concept:

Using the Equation 18-9 for the change in the length of rod $∆L$, we can find the coefficients of thermal expansion $\alpha$ for each rod, and by comparing these values of $\alpha$ for each rod, we can rank the rods according to their coefficient of thermal expansion.

2) Formula:

$∆L=L\alpha ∆T$

Where $\alpha$ is the coefficient of linear expansion

3) Given:

i) The length of rod a is $L=2 m$.

ii) The length of rod b is $L=1 m$.

iii) The length of rod c is $L=2 m$.

iv) The length of rod d is $L=4 m$.

v) The change in temperature of rod a is $∆T=10 ℃$.

vi) The change in temperature of rod b is $∆T=20 ℃$.

vii) The change in temperature of rod c is $∆T=10 ℃$.

viii) The change in temperature of rod d is $∆T=5 ℃$

ix) The change in the length of rod a is $∆L=4\times {10}^{-4} m$.

x) The change in the length of rod b is $∆L=4\times {10}^{-4} m$.

xi) The change in the length of rod c is $∆L=8\times {10}^{-4} m$.

xii) The change in the length of rod d is $∆L=4\times {10}^{-4} m$.

4) Calculations:

Using Equation 18-9- If the temperature of the metal rod of length L is released by an amount $∆T$, its length is found to increase by an amount

$∆L=L\alpha ∆T$

Where $\alpha$ is the coefficient of linear expansion.

Therefore, the coefficient $\alpha$ of linear expansion is given by

$\alpha =\frac{∆L}{L∆T}$

Let ${\alpha }_a$, ${\alpha }_b$, ${\alpha }_c$ and ${\alpha }_d$ be the coefficients of linear expansion of  rods a, b, c, and d respectively.

So, for rod a; $L=2 m$, $∆T=10 ℃$ and $∆L=4\times {10}^{-4} m$, adding the values, we get ${\alpha }_a=\frac{4\times {10}^{-4}}{2\times 10}$

${\alpha }_a=2\times {10}^{-5} per ℃$

Similarly, for rod b; $L=1 m$, $∆T=20 ℃$ and $∆L=4\times {10}^{-4} m$, adding  the values, we get ${\alpha }_b=\frac{4\times {10}^{-4}}{1\times 20}$

${\alpha }_b=2\times {10}^{-5} per ℃$

For rod c; $L=2 m$, $∆T=10 ℃$ and $∆L=8\times {10}^{-4} m$, adding the values we get ${\alpha }_c=\frac{8\times {10}^{-4}}{2\times 10}$

${\alpha }_c=4\times {10}^{-5} per ℃$

And for rod d; $L=4 m$, $∆T=5 ℃$ and $∆L=4\times {10}^{-4} m$, adding the values we get ${\alpha }_d=\frac{4\times {10}^{-4}}{4\times 5}$

${\alpha }_d=2\times {10}^{-5} per ℃$

Therefore, we get

${\alpha }_c>{\alpha }_a={\alpha }_b={\alpha }_d$

Conclusion:

Using the formula for change in length due to linear expansion, we can rank the rods according to their coefficient of thermal expansion.