Please check the answers given in the book for the following questions.

11. A metre scale of steel is caliberated at 20^oC to give correct reading. Find the distance between 50cm and 51cm mark if the scale is used at 10^oC.

Coefficient of linear expansion of steel = 1.1 x 10 ^-5.

Answer given : 1.00011cm. ( I think it should be 0.99989 cm. Shouldn't it decrease ?? )

27. A steel rod is clamped at its two ends and rests on a fixed horizontal base. The rod is unstretched at 20^oC. Find the longitudinal strain developed in the rod if the temperature rises to 50^oC.

Coefficient of linear expansion of steel = 1.2 x 10 ^-5.

Answer given : - 3.6 x 10 ^-4 . ( Why minus ?? )

Let's break down the questions you've posed regarding thermal expansion and clarify the concepts involved. Understanding these principles will help you grasp why the answers provided in your book are correct.

Question 11: Distance Between 50 cm and 51 cm Marks

To determine the distance between the 50 cm and 51 cm marks on a steel scale when the temperature changes from 20 °C to 10 °C, we need to apply the formula for linear expansion:

ΔL = L₀ × α × ΔT

Where:

• ΔL = change in length
• L₀ = original length (1 cm in this case, as we are looking at the distance between two marks)
• α = coefficient of linear expansion (1.1 x 10^-5 °C^-1 for steel)
• ΔT = change in temperature (final temperature - initial temperature = 10 °C - 20 °C = -10 °C)

Now, substituting the values into the formula:

ΔL = 1 cm × (1.1 x 10^-5) × (-10) = -1.1 x 10^-4 cm

This means that the scale contracts by 0.00011 cm. Therefore, the new distance between the 50 cm and 51 cm marks is:

New distance = Original distance + ΔL = 1 cm - 0.00011 cm = 0.99989 cm

Your calculation is indeed correct; the distance should decrease to approximately 0.99989 cm, not 1.00011 cm. The scale contracts as the temperature decreases, which is why you observed a decrease in the distance.

Question 27: Longitudinal Strain in a Steel Rod

For the second question regarding the longitudinal strain in a steel rod clamped at both ends, we can use the formula for strain due to thermal expansion:

Strain = α × ΔT

Where:

• α = coefficient of linear expansion (1.2 x 10^-5 °C^-1)
• ΔT = change in temperature (50 °C - 20 °C = 30 °C)

Substituting the values into the formula gives:

Strain = (1.2 x 10^-5) × (30) = 3.6 x 10^-4

Now, regarding the negative sign in the answer provided in your book, it indicates that the rod is experiencing compressive strain. Since the rod is clamped at both ends, it cannot expand freely when the temperature increases. This restriction leads to a situation where the material is effectively being compressed, hence the negative sign. In essence, the rod wants to expand, but the clamps prevent it from doing so, resulting in a compressive strain.

In summary, the answers in your book are correct based on the principles of thermal expansion and the constraints placed on the materials. Understanding these concepts helps clarify why the values are what they are.