Will the ice crack?

Classical Mechanics Level 5

Ice has formed on a shallow pond and a steady state has been reached, with the air (above the ice) at $\displaystyle -5^o C$ , and the bottom of the pond at $\displaystyle 4^o C$ .

If the total depth of $\displaystyle \text{ice + water}$ is $\displaystyle 1.4m$ , then find the thickness of the ice (in m ) layer on the pond.

Details and Assumptions:
$\bullet$ $\displaystyle K_{ice} = 0.4 \text{cal}/m-^oC$
$\bullet$ $\displaystyle K_{water} = 0.125 \text{cal}/m-^oC$ , where $\displaystyle K$ denotes the thermal conductivity.

Note: This problem appeared in our AITS (All India Test Series)

The answer is 1.12.

3 solutions

Pratik Shastri
May 29, 2014

At the point where the ice meets the water, $T=0^oC$

Now, let $d$ be the thickness of the ice formed.

In the equilibrium condition, the $\frac{dQ}{dt}$ from ice to water equals that from water to ice.

So,

$\frac {K_{ice}*A*4}{d}= \frac{K_{water}*A*5}{1.4-d}$

Plugging the values and solving for d, we get

$\boxed {d=1.12m}$

Maybe the equation should be K(ice) A 5/d=K(water)*4/d Plaese reply Like this answer is 0.4

Aishwary Omkar - 6 years, 8 months ago

can know which AITS is this from??

Abhranil Das - 4 years, 1 month ago

Chew-Seong Cheong
May 27, 2014

Let the thickness of ice be x. It should be noted that the temperature at the bottom of the ice is 0 degree Celcius. At equilibrium, the heat Q conducted through the (1.4 - x) m thick layer of water is same as that through the x m thick layer of ice.

Therefore, Q = 0.125 (4-0)/(1.4 - x) = 0.4 (0 -(-5))/x

And this gives: 2.5x = 2.8 and x = 1.12

Murlidhar Sharma
May 9, 2014

Equating KA\Delta T/L for ice and water and putting the temperature of junction of ice and water as 0 we get the answer.

Isn't the thickness of the ice 0.28 m (=1.4 - 1.12)?

Sungil Cho - 7 years ago

No.... why?

Md Zuhair - 3 years, 4 months ago

it was 1.12 i think u made a small mistake

Murlidhar Sharma - 7 years ago

From which AITS this problem was?

Riya chauahan - 7 years ago

Will the ice crack?

The answer is 1.12.

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