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Blazing goIITian

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24 Jun 2008 16:55:54 IST

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Fluid mechanics--rates assured

None

Q. A cube of ice of edge 4 cm is placed in an empty cylindrical glass of inner diameter 6 cm. Assume that the ice melts uniformly from each side so that it always retains its cubical shape. Remembering that the ice is lighter than water, find the length of the edge of the ice cube at the instant it just leaves contact with the bottom of the glass.

P.S. this HCV , vol I , page 275 q.no 25.

Rates assured.

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Chirag Sangani

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24 Jun 2008 17:18:56 IST

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Initial mass = final mass

Hence,

Which gives

So side of cube = 2x = 2.26 cm

ramyani chakrabarty

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24 Jun 2008 17:21:40 IST

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owa ! Thanks a ton friend !

A liitle explanation plz!

ramyani chakrabarty

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24 Jun 2008 17:23:31 IST

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why 4 cube into half ?

Chirag Sangani

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24 Jun 2008 17:25:28 IST

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Hmm...

initial mass is volume of cube into relative density

Final mass is Volume of water into density + remaining ice

Volume of water is Area of cylindrical base into x minus the volume of the cube submerged which is 4x³

ramyani chakrabarty

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24 Jun 2008 17:29:04 IST

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is nt the density of ice 0.9 ?

ramyani chakrabarty

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24 Jun 2008 17:32:57 IST

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look u hav done wonder in secs.

But why don't u be a little less cryptic ? May be silly. But explain the steps. why 8x cube etc? I don't mind to be called dull.

But i want to undertstand.

harsh _27

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24 Jun 2008 17:33:03 IST

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But the relative density of ice must be 0.9g/cc right.Can you explain why did you write 0.5 there?

Chirag Sangani

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24 Jun 2008 17:40:49 IST

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lol.. i mistook density for specific heat

no problem...

ramyani chakrabarty

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24 Jun 2008 17:41:24 IST

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where r my other frnds ?

Akansha

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24 Jun 2008 17:41:37 IST

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concept is that ice floats with 9/10th of its volume inside water so if that
much water(9/10 of the vol of remaining ice cube) is formed it will just leave contact!!

ramyani chakrabarty

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24 Jun 2008 17:45:12 IST

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in that case a cubic equn comes.

Chirag Sangani

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24 Jun 2008 17:47:17 IST

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Ok, initial mass of total H2O will be equal to the final mass

Initially, volume of ice was 64 cu. cm. Mass of H2O initiall will be 64 x 0.9 gm

Finally, as per the diagram, mass of H2O will be Mass of ice plus mass of water

Mass of ice = (10X)³ x 0.9

Mass of water = where is the volume of ice submerged in the water

Equating, the value of 10x comes out to be 2.26cm again!!

EDIT: solved cubic eqn by scientific calc

Tarin Bansal

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24 Jun 2008 17:52:22 IST

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I am posting my solution. I feel its correct but I am not getting the right answer.

Mass of ice melted = mass of water in container

(4³ - a³)0.9 = pi.3².h (where, a = edge of cube after melting of ice, h=height of water in container)

So, h=(4³ - a³)/10pi

Now, as it leaves contact with the floor, means mg=buoyant force

0.9 a³ g = 1 a² h g

Finally u get this eq-

a³ + 9.pi.a - 64=0

Now, I didnt have enuf courage to solve it . So I used this site-

http://www.hostsrv.com/webmab/app1/MSP/quickmath/02/pageGenerate?site=quickmath&s1=equations&s2=solve&s3=basic

And according to it, a = 1.98656 cm.

PLZ correct me if I am wrong.

Chirag Sangani

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24 Jun 2008 17:57:17 IST

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@tarinbansal

your mistake is that you have taken the mass of water in container as pi.3²h, whithout removing that part of the volume of ice which is submerged. It should be pi.3²h - a²h

Tarin Bansal

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24 Jun 2008 20:12:38 IST

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Ohh, ya.
Silly me. x-(

abhishek sinha

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25 Jun 2008 17:56:34 IST

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Is the answer 2.515 ?

abhishek sinha

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25 Jun 2008 18:01:18 IST

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Suppose that the feight of water in the container =hcm and the side of the cube is x cm .

Now from conservation of mass

pi 3^2 h + 0.9x^3 -x^2h = 4^3.............(1)

And from archemedes' law

x^3 *0.9 = x^2h ......................(2)

from (2) h=0.9x

from (1) h= 64/9pi

x = 64/(9pi*0.9)= 2.515 cm

Tarin Bansal

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25 Jun 2008 20:46:01 IST

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Hmm, the answer given in HCV is 2.26 cm.

newton

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29 May 2010 14:52:31 IST

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is anyone's answer matching with hcvs ans cause i am getting ans as 2.45 after solving the above equations on a scientific calculator!!!

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