Question

Determine a formula for the change in surface area of a uniform solid sphere of radius r if its coefficient of linear expansion is alpha A (assume constant) and its temperature is changed by Delta T.Assume the temperature chan ge is not large, Delta T^2 ~ 0.

Answer 1

Area A = 4 pi R^2
lnA = ln(4 pi) + 2 ln R
dA/A = 2 dR/R = 2*(alpha)*dT
delta A = 2A*alpha*(delta T)

OR

A at T + deltaT
delta A = 4 pi (R + deltaR)^2 - A
= 4 pi R(1 + alpha*deltaT)^2 -A
= 4 pi R^2 *[1 + 2*alpha*deltaT +
(alpha*deltaT)^2] - A
= 2*A*alpha*deltaT + A*(alpha*deltaT)^2

(Ignore the second term in deltaT^2)