Physics
Fluid-Mechanics
Easy

Question

Consider a vessel filled with a liquid upto height H. The bottom of the vessel lies in the X-Y plane passing through the or origin. The density of the liquid varies with Z-axis as
rho left parenthesis z right parenthesis equals rho subscript 0 left square bracket 2 minus left parenthesis z over H right parenthesis squared right square bracket. text If  end text P subscript 1 space text and  end text P subscript 2 are the pressures at the bottom surface and top surface of the liquid, the magnitude of left parenthesis P subscript 1 minus P subscript 2 right parenthesis is:

  1. rho subscript 0 g H
  2. 8 over 5 rho subscript 0 g H
  3. 3 over 2 rho subscript 0 g H
  4. 5 over 3 rho subscript 0 g H

The correct answer is: 5 over 3 rho subscript 0 g H


    table attributes columnalign left columnspacing 1em rowspacing 4 pt end attributes row cell text Pressure of bottom end text equals g cross times integral subscript z equals 0 end subscript fraction numerator 1 over denominator pi r squared end fraction cross times d z cross times rho subscript 0 left parenthesis 2 minus z squared over H squared right parenthesis end cell row cell equals g cross times integral subscript z equals 0 end subscript superscript z equals H end superscript rho subscript 0 left parenthesis 2 minus z squared over H squared right parenthesis d z equals g cross times left square bracket rho left parenthesis 2 z minus fraction numerator z squared over denominator 3 H squared end fraction right parenthesis right square bracket subscript z equals 0 end subscript superscript z equals H end superscript end cell row cell equals g rho subscript 0 left parenthesis 2 H minus fraction numerator H cubed over denominator 3 H squared end fraction right parenthesis equals g rho subscript 0 left parenthesis 2 H minus H over 3 right parenthesis end cell row cell equals g rho subscript 0 left parenthesis fraction numerator 5 H over denominator 3 end fraction right parenthesis equals 5 over 3 rho subscript 0 g h end cell end table

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