A vessel cylindrical in shape and closed at the top and the bottom of the radius R and height H.The vessel is completely filled with water?
If it is rotated about its vertical axis with a speed of w Rad/sec, what is the total pressure force exerted by water on the top and bottom of the vessel?
If it is rotated about its vertical axis with a speed of w Rad/sec, what is the total pressure force exerted by water on the top and bottom of the vessel?
1 Answer
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Let us consider a drop of water of mass
The
P_"top"=pi/4ρomega^2R^4
whereomega is angular speed of cylinder about its central vertical axis,rho density of water andR radius of cylinder.
We know Newton's Third Law of Motion that for every action there is there is equal and opposite reaction. As such this pressure generated on the top has a reaction. The pressure on the bottom thus is
P_"bottom"=P_"top"+"Pressure due to weight of water"
P_"bottom"=pi/4ρomega^2R^4+piR^2rhoHg
whereH is height of water in the cylinder
.-.-.-.-.-.-.-.-.-.
slideplayer.com/slide/7232137/
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Ncostheta=mg .....(1)
Nsintheta=mromega^2 ....(2)
Dividing (2) with (1) we get
tantheta=(romega^2)/g
Taking
tantheta=(dz)/(dr)
wherer andz are cylindrical coordinates.
Above expression becomes
(romega^2)/g=(dz)/(dr)
This separates and integrates into the parabolic surface profile
z=(omega^2r^2)/(2g)+C
whereC is constant of integration.
- The cylinder is closed at the top. As such water in the cylinder does not spill over.
- For mass of water below the center of paraboloid formed by free surface, reference
z=0 , normal reactionR=mg . Therefore, mass of water contributing to the upwards pressure is only which is contained within the paraboloid surface and the walls of the cylinder.
At
z=(omega^2r^2)/(2g) ........(3)
Mass of rotating water creating pressure towards top
M=int_0^R\ rhoz(2pir)dr
Using (3)
M=int_0^R\ rho(omega^2r^2)/(2g)(2pir)dr
=>M=|(rhopiomega^2r^4)/(4g)|_0^R
=>M=(rhopiomega^2R^4)/(4g)
Pressure on the top
P_"top"=(rhopiomega^2R^4)/(4g)g
=>P_"top"=pi/4rhoomega^2R^4
Note that as expected it is independent of