Question

Three rods of copper, brass and steel are welded together to form an inverted Y-shaped structure. End of copper rod is maintained at $100^{\circ }C$ whereas ends of brass and steel are kept at $0^{\circ }C.$ All the three rods have same length of $9cm$ and cross-section of $4.5c{m}^2.$ If the rate of heat flow through copper rod is $xcal/s$ . FInd $10x$ .
Given that the rods are thermally insulated from surroundings except at ends.
(Take thermal conductivities of copper, brass and steel to be $0.9,0.24$ and $0.11CGS$ units respectively)

Answer 1

Answer: 126

$Q=Q_1+Q_2....\left(i\right)$
But $Q=KA\left(\frac{\Delta \theta }{\Delta x}\right)t$
Here, for copper rod, $\frac{Q}{t}=\frac{0.9\times 4.5\times (100-T)}{9}$
For brass and steel rods,
$\frac{Q_1+Q_2}{t}=\frac{[0.24\times 4.5\times (T-0)]+[0.11\times 4.5\times (T-0)]}{9}$
From equation (i),
$0.9\times 4.5\times (100-T)=[(0.24\times T)+\left(0.11\times T\right)\times 4.5$
$\therefore 0.9\times (100-T)=0.24\times T+0.11\times T$
$\therefore 90-0.9T=0.35T$
$\therefore 90=1.25T$
$\therefore T=72^{\circ }C$
$\therefore \frac{Q}{t}=\frac{0.9\times 4.5\times 28}{9}=12.6cal/s$
$10x=126$