# Find height in cylinder

1 vote
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A piston of mass M = 3kg and radius R = 4cm has a hole into which a thin pipe
of radius r = 1cm is inserted. The piston can enter a cylinder tightly and without
friction, and initially it is at the bottom of the cylinder. 750gm of water is now
poured into the pipe so that the piston & pipe are lifted up as shown. Find the
height H of water in the cylinder and height h of water in the pipe.
![][1]

By blacing force on piston
$PA=mg$
$(1)(16-1)\pi h =30$
$h=2/\pi$
But now how can we find H
[1]: http://i.imgur.com/kpp2g3G.jpg

1 vote

The air pressure on the top of the cylinder is $P_0$. The pressure at the bottom of the pipe is $P=P_0+\rho g h$. Balancing forces on the cylinder gives
$(P-P_0)A=(\rho g h)\pi(R^2-r^2) = mg$
$(1gm/cm^3)h\pi(4-1)(4+1)cm^2=3000gm$
$h\pi=200cm=2.0m$.

The total mass of the water is $750gm$, so the total volume of water in the cylinder + pipe is $750cm^3$.

Volume in pipe is $(\pi h)r^2=200cm^3$. Volume in cylinder is $750-200=550cm^3$ This is also equal to $\pi R^2H=16\pi H$ so
$H=550/16\pi \approx 11 cm$.
This agrees with the given answer of $11/32\pi$, which is in metres.

I wonder if the question-setter intended that the mass of the hole should be removed from the 3kg mass of the cylinder. However, this would not make much difference to the answer.

answered Feb 13, 2017 by (28,876 points)
selected Feb 13, 2017 by koolman
Do you mean H (not h)
Answer is given as H=11/32$\pi$