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Home| Questions|Q 76099
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Varun Kumar
Varun Kumar
Asked: September 5, 20222022-09-05T15:40:00+05:30 2022-09-05T15:40:00+05:30In: Fluid Mechanics

What is Pascal’s Law? Explain its utilities

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    1. Priyanka Kashyap
      2022-09-05T16:36:53+05:30Added an answer on September 5, 2022 at 4:36 pm

      Pascal’s Law : It states that intensity of pressure for a fluid at rest is equal in all directions. This is proved as :

      The fluid element is of very small dimensions i.e. dx, dy, ds.
      Consider a arbitrary fluid element of the wedge shape in a fluid mass at rest. Let the width of the element perpendicular to the plane of paper is unity and
      px, py , pz are the pressure intensity acting on the face AB, AC, BC respectively.
      Then the forces acting on the element are :
      i. Pressure forces normal to the surfaces ,
      ii. Weight of element in the vertical direction ,
      The forces on the faces are :
      Forces on the face AB = px × Area of face AB = px × dy × 1
      Forces on the face AC = py × area of face AC = py × dx × 1
      Forces on the face BC = pz × area of face BC = pz × ds × 1
      Weight of element = ( Mass of element ) × g = ( Volume × rho ) × g
      = [ ( AB × AC ) / 2 × 1 ] × rho × g
      where, rho = density of fluid
      Resolving the forces in x – direction, we have
      px × dy × 1 – pz.ds × 1 cos ( thita ) = 0
      px × dy × 1 – pz × dy × 1 = 0
      px = pz ……(i)
      { ds cos ( thita ) = AB = dy }
      Similarly, Resolving the forces in y – direction, we get
      py × dx × 1 – pz × ds × 1 cos ( 90 ° – thita ) – ( dx × dy ) / 2 × 1 × rho × g = 0
      py × dx – pz ds sin ( thita ) – ( dx . dy ) / 2 × rho × g = 0
      But [ ds . Sin ( thita ) ] = dx and also the element is very small and hence the weight is negligible.
      Therefore, py.dx – pz × dx = 0
      py = pz ……(ii)
      From (i) and (ii) ; we have
      px = py = pz
      This equation shows that the pressure at any point in x, y and z directions is equal.

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