A metallic bar of Young’s modulus, 0.5 × 10^{11} N m^{–2} and coefficient of linear thermal expansion 10^{–5} °C^{–1}, length

Question

A metallic bar of Young’s modulus, \(0.5 \times10^{11}\ N\ m^{-2}\) and coefficient of linear thermal expansion \(10^{–5} \ {}^\circ C^{–1}\) , length 1 m and area of cross-section \(10^{–3}\) \(m^2\) is heated from 0°C to 100°C without expansion or bending. The compressive force developed in it is : 

(1) 5 × \(10^3\) N 

(2) 50 × \(10^3\) N 

(3) 100 × \(10^3\) N 

(4) 2 × \(10^3\) N

Answer 1

Correct option is : (2) \( 50 \times 10^3 \ N\) 

Thermal strain = Longitudinal strain = \(\alpha\ \Delta\ T \)

\(\Rightarrow\ \text{Longitudinal} \ \text{strain},\ \delta = \ 10^{-5}\times10^{2} = 10^{-3} \) 

\(\Rightarrow\) Compressive stress \( = \delta\times \text{Young's}\ \text{modulus}\) 

\(= 10^{-3}\times 0.5 \times 10^{11}\) 

\( = 0.5 \times 10^8\) 

\(\Rightarrow\) Compressive force \(=0.5\times10^8 \times10^{-3} = 0.5 \times10^{5}\)

\(=5\times10^{4}\times\frac{10}{10}\) 

\(=0.5\times10^3\ N\)