Question

At ${20^ \circ }C$ a steel ruler $20\;cm$ long is graduated to give correct reading but when it is used at a temperature of ${40^ \circ }C$ what will be the actual length of the steel ruler? [${\alpha _{steel}} = 1.2 \times {10^{ - 5}}{\left( {{}^ \circ C} \right)^{ - 1}}$]
A. $22.02\;cm$
B. $19.6\;cm$
C. $20.0048\;cm$
D. $18.0002\;cm$

Answer 1

All solids, liquids and gases have a property of expanding when heated. The concept of thermal expansion is applied wherein metals like steel tend to expand by a certain amount changing its length by a slight amount. The formula for linear expansion is applied in order to find the actual length of the ruler.

Formula used:
The linear expansion of the material is given by:
$l' = {l_0}(1 + \alpha \Delta T)$
Where, $l'$ is the change in length, ${l_0}$ is the original length, $\alpha$ is the coefficient of linear expansion and $\Delta T$ is the change in temperature.

Complete step by step answer:
The above problem revolves around the concept of thermal expansion. First, let us look into the definition of thermal expansion. Thermal expansion is said to be the increase in size of the body or the material at hand when it is heated. When a solid like a metal is heated it has a tendency to increase in its length and hence the type of expansion considered here is the linear expansion.

Linear expansion is defined as the change or the increase in length when a metal is heated. Here, the metal used is given as steel. In the above question we are to find out the actual length, that is, the original length of the steel ruler before it was heated or in other words before there was a change in its length.

Now, let us discuss the cause of this thermal expansion. We are all aware that metal is a type of solid and by the properties of solids we know that the molecules and atoms which make up the solid are tightly packed. This means that they are bonded to each other or are connected to each other by intermolecular forces that keep the molecules of the solid bound to each other.

As per the concepts of chemistry, when these molecules are provided heat they tend to vibrate about their positions and gain kinetic energy in order to move more violently with more energy leading to the breakage of the intermolecular bonds between them. Hence due to this, the distance of separation between these molecules will tend to increase resulting in expansion.

Metals tend to undergo linear expansion, that is, a change in its length. The change in length is due to the increase in temperature in order to provide more heat. Hence the linear expansion of the metal is given by a formula:
$l' = {l_0}(1 + \alpha \Delta T)$ -------($1$)
Here the coefficient of linear expansion refers to a measure or the extent to which there is a change in length that is observed in the material. It is defined as the increase in length per unit length per degree rise in the temperature. Hence, we can say that the length increases linearly with temperature.

Let us now extract all the data given in the question. It is given that before the ruler is graduated, that is, before heating the length is $20\;cm$ which is said to be the original length. The change in temperature from ${20^ \circ }C$ to $40{}^ \circ C$ is given. The coefficient of linear expansion and the original length is also given. We are asked to find the actual length that is the changed length.

Given, ${\alpha _{steel}} = 1.2 \times {10^{ - 5}}{\left( {{}^ \circ C} \right)^{ - 1}}$ and ${l_0} = 20\;cm$.Let us now determine the change in temperature. The change in temperature refers to the difference between them and hence we get:
$\Delta T = (40 - 20){}^ \circ C$
$\Rightarrow \Delta T = {20^ \circ }C$
Now we substitute all these given values into equation ($1$) to get the value of actual length.
$l' = 20(1 + (1.2 \times {10^{ - 5}} \times 20))$
We further solve the equation in order to get the length:
$\Rightarrow l' = 20(1 + 24 \times {10^{ - 5}})$
$\Rightarrow l' = 20 \times 1.00024$
$\therefore l' = 20.0048\;cm$
Hence the actual length is said to be $20.0048\;cm$.

Hence the correct answer is option C.

Note: An alternative method which we can use is to first find the change in length due to linear expansion and then add the original length to it to get the actual length of the ruler. The common error observed in this question is that the given original length is considered as the changed length which would give the wrong answer.