A ruler is accurate when the temperature is 25°C. When the temperature drops to -16°C, the ruler shrinks and no longer measures distances accurately. However, the ruler can be made to read correctly if a force of magnitude 1.2 103 N is applied to each end so as to stretch it back to its original length. The ruler has a cross-sectional area of 1.50 10-5 m2, and it is made from a material whose coefficient of linear expansion is 2.10 10-5 (C°)-1. What is Young's modulus for the material from which the ruler is made?

Question

A ruler is accurate when the temperature is 25°C. When the temperature drops to -16°C, the ruler shrinks and no longer measures distances accurately. However, the ruler can be made to read correctly if a force of magnitude 1.2 103 N is applied to each end so as to stretch it back to its original length. The ruler has a cross-sectional area of 1.50 10-5 m2, and it is made from a material whose coefficient of linear expansion is 2.10 10-5 (C°)-1. What is Young's modulus for the material from which the ruler is made?

Answer 1

To find Young's modulus for the material from which the ruler is made, we need to use the formula:

Young's Modulus (Y) = Stress / Strain

First, let's determine the stress on the ruler. Stress is calculated using the formula:

Stress = Force / Area

Given:
Force = 1.2 x 10^3 N (applied to each end)
Area = 1.50 x 10^-5 m^2

Substituting the given values into the formula, we get:

Stress = (1.2 x 10^3 N) / (1.50 x 10^-5 m^2)

Next, we need to calculate the strain. Strain is defined as the change in length divided by the original length. In this case, the ruler shrinks due to the change in temperature. The change in length is given by:

Change in length = Original length x Coefficient of linear expansion x Change in Temperature

Given:
Original length = Unknown (Let's assume it to be L, the length of the ruler)
Coefficient of linear expansion = 2.10 x 10^-5 (C°)^-1
Change in Temperature = -16°C - 25°C = -41°C

Substituting these values into the formula, we get:

Change in length = L x (2.10 x 10^-5 (C°)^-1) x (-41°C)

Now, let's consider the strain. Strain is given by:

Strain = Change in length / Original length

Substituting the previously calculated change in length and the assumed original length (L), we get:

Strain = (L x (2.10 x 10^-5 (C°)^-1) x (-41°C)) / L

Notice that L cancels out, leaving us with:

Strain = -86.1 x 10^-5

Now that we have the stress and strain, we can calculate Young's Modulus using the formula:

Young's Modulus (Y) = Stress / Strain

Substituting the calculated stress and strain values, we get:

Young's Modulus (Y) = ((1.2 x 10^3 N) / (1.50 x 10^-5 m^2)) / (-86.1 x 10^-5)

Simplifying the expression, we find:

Young's Modulus (Y) = -17.543 x 10^9 N/m^2

Therefore, Young's Modulus for the material from which the ruler is made is approximately -17.543 x 10^9 N/m^2.