A uniform metal bar of length (10 ~m ) with a crack at its midpoint is clamped between two rigid supports. The bar buckles upward due to temperature rise of (40^ C ). If the coefficient of linear expansion of the metal is (2.5 10⁻⁶ ^ C ⁻¹ ), the maximum displacement of the mid-point of the bar is

Step-by-step solution

Given, length of bar (L=10 ~m ), rise in temperature, ( T=40^ C ) and coefficient of linear expansion, ( =25 10⁻⁶ ^ C ⁻¹ ) As, linear expansion, ( aligned L & =L t=10 2.5 10⁻⁶ 40 L & =0.1 ~cm aligned ) In the figure below, let the displacement of mid point of bar is (x ), Now, from ( O B A ), ( aligned x & = (A B)^2-(O A)^2 x & = ( 10+0.001 2 )^2-(5)^2 x & =0.223 ~m aligned ) ( Hence, x=22.3 ~cm ) Hence, the correct option is (b).