Question

Two cylindrical rods have the same mass. One is made of silver (density = 10 500 kg/m³), and one is made of iron (density = 7860 kg/m³). Both rods conduct the same amount of heat per second when the same temperature difference is maintained across their ends. What is the ratio (silver-to-iron) of (a) the lengths and (b) the radii of these rods? Give your answers as numbers with no units.

Answer 1

Mass density*volume. Volume area *length.
d is density, this is
m1 = m2
d1 a1 l1 = d2 a2 l2

Thermal conductivity = conductivity coefficient area / length;
let w = watts per degree
Now;
w1 = w2
c1 a1/l1 = c2 a2/l2

Taking the appropriate ratios, we have
l1/l2 = sqrt((c1 d2)/(c2 d1)) = sqrt((420*7860)/(10500*79)) ≈ 1.995 ----- answer (a)

a1/a2 = sqrt((c2 d2)/(c1 d1)) = sqrt((79*7860)/(420*10500)) ≈ .37523689
area ratio = the square of the radius ratio, so that ratio is
r1/r2 ≈ sqrt(.37523689) ≈ .6126 --------- answer (b)