Question

A ruler is accurate when the temperature is 26.7 °C. When the temperature drops to -12.3 °C, the ruler shrinks and no longer measures distances accurately. However, the ruler can be made to read correctly if a force of magnitude 1.72 x 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 1.39 x 10^-5 m², and it is made from a material whose coefficient of linear expansion is 2.5 x 10^-5 (C°)^-1. What is Young's modulus for the material from which the ruler is made?

Answer 1

Change in temperature = T = 26.7^o C - ( - 12.3)^o C = 39^o C.

Coeffitient of linear expansion = = 2.5 x 10^-5oC^-1.

Hence, change in length L, per unit original length L = L / L = T = 2.5 x 10^-5oC^-1 x 39^o C ~ 9.75 x 10^-4.

Force applied to retuen the length of the ruler to its original value = F = 1.72 x 10³ N.

Cross - sectional area of the ruler = A = 1.39 x 10^-5 m².

Hence, Young's modulus for the material from which the ruler is made = Y = ( F / A ) / ( L / L )

or, Y = ( 1.72 x 10³ N ) / ( 1.39 x 10^-5 m² x 9.75 x 10^-4 ) ~ 127 x 10⁹ N / m²

or, Y = 127 GPa.