A unifrom steel rod of cross section `5mm^(2)` is heated from ` 0^(@)C` to`25^(@)C`. Find the force which must be exerted to prevent it from expanding. Also, find the energy stored per unit volume of the rod.

Explanation

Data : ` A = 5 mm^(2) = 5 xx 10^(-6) m^(2) , Delta theta = 25^(@) C - 0^(@) C = 25^(@)C`
` prop_("steel") = 12 xx 10^(-6).^(@) C^(-1) , Y_("Steel") = 2xx 10^(11) N//m^(2)`
The thermal strain in the rod is
` (DeltaL)/(L_(0)) = prop_("steel") Deltatheta = (12 xx 10^(-6)) (25) = 3 xx 10^(-4)`
To prevent the rod from expanding , it must be subjected to a compressive stress, ` F/A = Y xx ` strain.
` = (2 xx 10^(11)) (3 xx 10^(-4)) = 6 xx 10^(7) N//m^(2)`
The compressive force that must be exerted on the rod is
F = compressive stress ` xx A = ( 6 xx 10^(7))( 5 xx 10^(-6))= 300 N`
The energy stored per unit volume of the rod
` 1/2 xx " stress" xx "strain" = 1/2(6xx 10^(7)) (3 xx 10^(-4)) = 9 xx 10^(3) J//m^(3)`