Question

The lengths of steel and copper rods are so that the length of the steel rod is $5cm$ longer than that of the copper rod at all temperatures. What will be the length of each rod?

Answer 1

Step 1: Given data

The length of the steel rod is $5cm$ longer than the copper rod at all temperatures

Step 2: Formula used

Change in length with temperature $∆l=l\alpha ∆T$

Where $l$ is the original length

$\alpha$ coefficient of linear expansion

$∆T$ is the change in temperature

Step 3: Assumption

Let the length of the copper rod be $xcm$

Then the length of the steel rod will be $(x+5)cm$

$∆l$of both rods has to be the same in order for their length difference to remain constant

Step 4: Calculation

Change in length of steel $∆l_s=l_s{\alpha }_s∆T$

Change in length of copper $∆l_c=l_c{\alpha }_c∆T$

⇒$∆l_s=∆l_c$

⇒$l_s{\alpha }_s∆T=l_c{\alpha }_c∆T$

⇒$(x+5){\alpha }_s=x{\alpha }_c$

The experimental values of ${\alpha }_s$and ${\alpha }_c$ are $1.1\times {10}^{-5}$ and $1.7\times {10}^{-5}$

⇒ $(x+5)1.1=1.7x$

⇒ $0.6x=5.5$

⇒ $x=9.17cm$

⇒ $x+5=14.17cm$

The length of the copper rod is $9.17cm$ and that of the steel rod is $14.17cm$