Q16PE - College Physics (Urone)

Question

Question: (a) Suppose a meter stick made of steel and one made of invar (an alloy of iron and nickel) are the same length at 0ºC . What is their difference in length at 22.0ºC? (b) Repeat the calculation for two 30.0-m-long surveyor’s tapes.

Step-by-Step Solution

Answer

Answer

(a) The difference in length of meter stick made of steel and invar $\text{2.44}\times {\text{10}}^{-4} m$ is  

for 1 meter of length, and

(b) 7.33mm for 30 meters of length

Step 1: Introduction

We calculate the change in length of the steel and invar stick by the formula for linear expansion in solids and find the difference in lengths.

Step 2: Given parameters and formula for linear expansion

Coefficient of linear expansion of steel =$1.2\times {10}^5{/}^{O}C$

The formula for length expansion of solids$∆L=\alpha L∆T$

Here,is the coefficient of linear expansion, $∆T$is original length, andis the temperature change. 

Step 3: Increase in length for steel & invar of length 1 meter (a)

For steel

$$\begin{gathered} ∆L=\alpha L∆T \\ =(1.5\times {10}^5{/}^{O}C)\times 1m\times \left({22}^{0}C\right) \\ =\text{2.64}\times {\text{10}}^{-4} m \end{gathered}$$

For invar

$$\begin{gathered} ∆L=\alpha L∆T \\ =(9.0\times {10}^6{/}^{O}C)\times 1m\times \left({22}^{0}C\right) \\ =1.98\times {\text{10}}^{-4} m \end{gathered}$$

$$\begin{gathered} Differenceinlength=\left(\text{2.64}\times {\text{10}}^{-4} m\right)-\left(\text{1.98}\times {\text{10}}^{-5} m\right) \\ =\text{2.44}\times {\text{10}}^{-4} m \end{gathered}$$

Step 4: Increase in length for steel & invar of length 30 meters (b)

For steel

$$\begin{gathered} ∆L=\alpha L∆T \\ =(1.2\times {10}^5{/}^{O}C)\times 30m\times \left({22}^{0}C\right) \\ =7.92\times {\text{10}}^{-4} m \end{gathered}$$

For invar

$$\begin{gathered} ∆L=\alpha L∆T \\ =(3.0\times {10}^{-6}{/}^{O}C)\times 30m\times \left({22}^{0}C\right) \\ =5.94\times {\text{10}}^{-4} m \end{gathered}$$

$$\begin{gathered} Differenceinlength=\left(\text{7.92}\times {\text{10}}^{-3} m\right)-\left(\text{5.94}\times {\text{10}}^{-4} m\right) \\ =\text{7.33}\times {\text{10}}^{-3} m\left(\frac{1 mm}{{10}^{-3} m}\right) \\ =\text{7.33} mm \end{gathered}$$

The difference in length of meter stick made of steel and invar is for 1 meter of length and for 30 meters of length.