Question

A rod of length $2m$ is at a temperature of ${20}^{o}C$. Find the free expansion of the rod, if the temperature is increased to ${50}^{o}C$, then find stress produced when the rod isFully prevented to expand

• A. $5\times {10}^{7}N/m^2$
• B. $18\times {10}^{7}N/m^2$
• C. $9\times {10}^{8}N/m^2$
• D. $18\times {10}^{8}N/m^2$

Answer 1

The correct option is A $5\times {10}^{7}N/m^2$

Free expansion of the rod $=\alpha L\Delta \theta$

$=15\times {10}^{-60}C\times 2m\times (50-20{)}^{0}C$

$=9\times {10}^{-}4m=0.9mm$

If the expansion is fully prevented, then

$Strain=\frac{9\times {10}^{-4}}{2}=4.5\times {10}^{-4}$

Temperature Stress $=$ Strain $\times Y$

$=4.5\times {10}^{-4}\times 2\times {10}^{11}=9\times {10}^{7}N/m^2$

If $0.4$ mm expansion is allowed, then length restricted to expand

$=0.9-0.4=0.5mm$

$Strain=\frac{5\times {10}^{-4}}{2}=2.5\times {10}^{-4}$

Temperature stress $=$ Strain $\times Y=2.5\times {10}^{-4}\times 2\times {10}^{11}=5\times {10}^{7}N/m^2$