Figure shows a very delicate balancing act where on one side

Figure shows a very delicate balancing act, where on one side of a weightless wedge there is fluid of density rho_1, and on the other side is fluid of density rho_2. If the wedge is just balanced, find the ratio rho_2/rho_1 in terms of H1, H2 and theta. Assume constant width W.

Solution

From the given diagram,

       ₁= Flat side fluid density

       ₂= Inclined side fluid density

       H1= Flat side fluid Head

       H2= Inclined side fluid Head

       = Inclined side angle

W=width of the wedge

Since the wedge is weightless,

For Balancing of the given Wedge the moments about the resting point should be equal

I.e.

   Clockwise moments    = Counter Clockwise moments

   Force on flat side* 0.5H1 = Force on wedge side * (0.5H1)/sin

   ₁*g*H1*Area1*0.5H1     = ₂*g*H2*Area2*(0.5H1)/sin

   ₁*g*H1*(H1*W)*0.5H1 = ₂*g*H2*(H2*W)*(0.5H1)/sin

   ₁*H1*H1*H1 = ₂*H2*H2*H1/sin

  

       ₂/₁= (H1)³*Sin/(H2)³