Transmission Of Heat Question 57
Question: Two spheres of different materials one with double the radius and one-fourth wall thickness of the other, are filled with ice. If the time taken for complete melting ice in the large radius one is 25 minutes and that for smaller one is 16 minutes, the ratio of thermal conductivities of the materials of larger sphere to the smaller sphere is
[EAMCET 1991]
Options:
A) 4 : 5
B) 5 : 4
C) 25 : 1
D) 1 : 25
Show Answer
Answer:
Correct Answer: D
Solution:
$ Q=\frac{KA,(\Delta \theta )t}{l} $
$ \because $ Q and $ \Delta \theta $ are same for both spheres hence $ K\propto \frac{l}{At}\propto \frac{l}{r^{2}t} $
Therefore $ \frac{{K _{\text{larger}}}}{{K _{\text{smaller}}}}=\frac{l _{l}}{l _{s}}\times {{( \frac{r _{s}}{r _{l}} )}^{2}}\times \frac{t _{s}}{t _{l}} $ . It is given that $ r _{l}=2r _{s}, $
$ l _{l}=\frac{1}{4}l _{s} $ and $ t _{1}=25 $ min, $ t _{s}=16 $ min.
Therefore $ \frac{{k _{\text{larger}}}}{{k _{\text{smaller}}}}=( \frac{1}{4} ){{( \frac{1}{2} )}^{2}}\times \frac{16}{25}=\frac{1}{25} $
BETA
