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?
Set the (negative) thermal expansion equal per unit length to the strain due to tension.
alpha*(delta T) = (F/A)/Y
Y = (F/A)/[(delta T)*alpha]
Y is Young's modulus
A is the cross sectional area, 1.5*10^-5 m^2
delta T = 41 C
F = 1200 N
alpha = 2.1*10^-5 C^-1 is the coefficient of thermal expansion
Solve for Y
I also get 9.29*10^10 N/m^2. That is a typical value for a metal.
To find Young's modulus for the material, we need to use the equation:
Young's modulus (Y) = Stress (σ) / Strain (ε)
First, we need to calculate the stress experienced by the ruler. Stress is defined as the force applied divided by the cross-sectional area:
Stress (σ) = Force (F) / Area (A)
Using the given values:
Force (F) = 1.2 * 10^3 N (force applied to each end)
Area (A) = 1.5 * 10^-5 m^2 (cross-sectional area of the ruler)
Calculating the stress, we get:
Stress (σ) = 1.2 * 10^3 N / 1.5 * 10^-5 m^2
Now, let's calculate the strain experienced by the ruler. Strain is defined as the change in length divided by the original length:
Strain (ε) = ΔL / L
To find the change in length, we need to consider the change in temperature. The temperature change is given by:
ΔT = T2 - T1
where T2 is the final temperature (-16°C) and T1 is the initial temperature (25°C).
Next, we calculate ΔL using the coefficient of linear expansion and the change in temperature:
ΔL = α * L * ΔT
where α is the coefficient of linear expansion (2.10 * 10^-5 (C°)^-1) and L is the original length.
Now, we substitute the values into the equation:
ΔL = 2.10 * 10^-5 (C°)^-1 * L * (−16°C − 25°C)
Finally, we substitute the values of stress and strain into the equation for Young's modulus:
Young's modulus (Y) = Stress (σ) / Strain (ε)