Pressure Variation – Rigid Body Motion

 

Example:   The U-tube shown below contains mercury and rotates about the off-center axis a – a.

At rest the depth of mercury in each leg is 150 mm.  Find the angular velocity for which the

difference in heights, h, between the two legs is 75 mm.  d = 220 mm  and  e = 90 mm

                                      

 

Strategy:  Apply the equation of motion for constant angular velocity about the a-a axis.

                 

                                                ∂P/∂r = ρrω2

 

              Apply the equation of motion in the z-direction.

 

                                               -∂P/∂z = ρg

 

Now            dP  =  (∂P/∂r) dr  +  (∂P/∂z) dz  =  ρrω2 dr  -  ρg dz

 

On the free surface  dP = 0.         0  =  ρrω2 dr  -  ρg dz

 

   dz/dr  =  rω2 / g        Integrate to obtain equation of free surface.    z  =  r2ω2 / 2g  +  C    

 

So on the free surface:          h  =  [ω2 / 2g] [ d2 – e2 ]  and   ω2 =  2gh / (d2 – e2)

 

    ω = √ [ 2gh / (d2 – e2) ] =  √ [ 2(9.8)m/sec2(0.075m) / [( 0.220)2 – (0.090)2 ] m2

 

                        ω =  6.04 rad/sec             (result)

 

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