Question

A 3-ft-diameter vertical cylindrical tank open to the atmosphere contains 1-ft-high water. The tank is now rotated about the centerline, and the water level drops at the center while it rises at the edges.

Determine the angular velocity at which the bottom of the tank will first be exposed.


Also determine the maximum water height at the moment.

Answer 1

The angular velocity is 7.56 rad/s

the maximum water height is 2 ft

Step-by-step explanation:

The z-position as a function of r is equal to

`z_(s(r)) =h_(0) -(w^(2)(R^(2)-2r^(2) )/(4g)` (eq. 1)

where

h0 = initial height = 1 ft

w = angular velocity

R = radius of the cylinder = 1.5 ft

zs(r) = 0 when the free surface is lowest at the centre

Replacing and clearing w

`w=\sqrt{(4gh_(0) )/(R^(2) ) } =\sqrt{(4*32.17*1)/(1.5^(2) ) } =7.56rad/s`

if you consider the equation 1 for the free surface at the edge is equal to

`z_(s(R)) =h_(0) +(w^(2)R^(2) )/(4g) =1+((7.56^(2))*(1.5^(2) ) )/(4*32.17) =1.99ft=2ft`