A U.S. penny has a diameter of 1.9000 cm at 20.0o C. The coin is made of a metal alloy (mostly zinc) for which the coefficient of linear expansion is 2.6 X 10-5 K-1. What would its diameter be on a hot day in Death Valley (48.0°C)? On a cold night in the mountains of Greenland (– 53o C)?

Solution 13E Step1 : Coefficient of linear expansion of zinc is = 2.6 × 10 K 5 1 Diameter of penny L = 1.9000 cm 0 Temperature T = 20.0 C Change in temperature on hot day T = ( 20 C 48 C) = 28 C 0 0 1 0 0 0 Change in temperature on cool day T = ( 20 2 ( 53 C)) = 73 C After contraction, the final length is given by L= L(1 T) Case (1) On hot day L = L(1 T ) 1 5 1 0 Substitute L = 1.9000 cm = 2.6 × 10 K and T = 1 28 C 1.9000cm = L(1 2.6 × 10 K 5 1 × 28 C) 1.9000cm L = (12.6×10 K ×28 C) L = 1.9000cm 1.000728 L = 1.8986 cm 0 Thus we have diameter at temperature 48 C will be 1.8986 cm Case (2) On hot day L L(1 T ) 2 5 1 0 Substitute L = 1.9000 cm = 2.6 × 10 K and T = 23 C 1.9000cm = L(1 2.6 × 10 K 5 1 × 73 C) 1.9000cm L = 5 1 0 (12.6×10 K ×73 C) L = 1.9000cm 0.998102 L = 1.9036 cm 0 Thus we have diameter at temperature 53 C will be 1.9036 cm