Question

Chapter 18, Problem 014 When the temperature of a copper coin is raised by 140 Co, its diameter increases by 0.24%. To two significant figures, give the percent increase in (a) the area of a face, (b) the thickness, (c) the volume, and (d) the mass of the coin. (e) Calculate the coefficient of linear expansion of the coin. (a) Number (b) Number (c) Number (d) Number (e) Number Units Units Units Units Units

Answer 1

Area of the face of coin - A = pi r^2 A = pi D^2/4 (because r = D/2) ln(A) = ln(pi) + 2 ln(D) - ln(4) (where ln is the natural logarithm) Taking differentiation on both sides - dA/A = 0 + 2dD/D - 0 (Differentiation of constant is zero) dA/A times 100 = 2 times (dD/D times 100) rightarrow (1) But given that increase in diameter is 0.24% i.e. dD/D times 100 = 0.24% so from eq^n (1) change in area - dA/A times 100 = 2 times 0.24 = 0.48% \r\n Since linear expansion will be same along all direction so if percentage increase along diameter is 0.24% so - Percentage increase along thickness (dx/x times 100) = 0.24% We know that - volume of coin = Area of face times thickness v = A middot x Taking natural logarithm on both sides - ln(v) = ln(A) + ln(x) Taking differentiation on both sides - dv/v = dA/A + dx/x