Question

Chapter 18, Problem 014 When the temperature of a copper coin is raised by 150 C°, its diameter increases by 0.26%. 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.

Answer 1

Part A.

Given that when temperature of copper coin is raised, then it's Diameter increases by 0.26%

We know that:

A = Area of coin = pi*R^2 = pi*D^2/4

D = diameter of coin

Now differentiating both sides w.r.t. D

dA/dD = d(pi*D^2/4)/dD = 2*pi*D/4

dA/dD = pi*D/2

dA = pi*D*dD/2

Also, A = pi*D^2/4

divide both equations:

dA/A = 2*(dD/D)

Now since diameter increases by 0.26%, So dD/D = 0.26%

So, percent Increase in Area of the face will be:

dA/A = 2*0.26 = 0.52%

Part B.

Thickness is a one dimensional quantity, So

dh/h = dD/D

dh/h = 0.26%

Part C

Similarly, Volume is a three dimension quantity, So

dV/V = 3*(dD/D) = 3*0.26

dV/V = 0.78%

Part D.

Since there is no mass loss, So mass of coin will remain constant, So

percentage increase in mass = 0%

Part E.

We know that relation between change in diameter and temperature is given by:

dD = D**dT

= (dD/D)*(1/dT)

dD/D = 0.26% = 0.0026

dT = change in temperature = 150 C

So,

= 0.0026/150

= 1.733*10^-5 /C

In two significant figures

= 1.7*10^-5 /C = coefficient of linear expansion

Let me know if you've any query.