1. oil water Geometric Proof of Archimedes’ principal---From Pressure on a submerged surface Consider a submerged body in an oil-water layered system (shown in cross section in Fig 1) Fig 1 To net upward Buoyancy force F B acting on this body is Net Vertical pressure force Acting UP on bottom (blue) surface Net Vertical pressure Force Acting Down On top (red) surface Pressures always act towards a surface Go to next slide to see how to calculate the vertical pressure forces

2. oil water Vertical Pressure Force acting on top surface of a submerged solid body Step 1: Imagine that you can separate the top surface from the body and move it horizontally in The fluid oil water Top surface without body Then Since the points A and A’ are at the same depth And we can conclude that the pressure acting On the top of the separated red surface is IDENTICAL to the pressure acting on the Top surface of our body HENCE oil water The Net vertical pressure force acting DOWN on top surface of the SUBMERGED BODY is the Weight of the fluidS ABOVRE the separated red Surface (Shown as a shaded region in Fig 2) Fig 2 F red

3. Vertical Pressure Force acting UP on BOTTOM surface of a submerged solid body To get the vertical pressure force acting up on the Bottom surface of the submerged body imagine Separating and moving horizontally the bottom (dark blue) surface We know that (since at a point pressure is the same in all directions) if our separated surface is thin, at a point A’-- the value of the pressure on the upper part of the surface MUST match the pressure acting on the lower part of the surface. FURTHER this pressure is equal to the pressure Acting on the matching point A on the lower surface Of the submerged body HENCE the vertical pressure force acting UP on the bottom surface of the submerged body Is the weight of the fluids ABOVE the separated Bottom (Dark Blue) surface (shaded in Fig 3) Fig 3 F blue

4. oil waterFig 4 Which will be the weight of fluids Contained in in the shaded region Of Fig 4— Which is identical to the weight of fluids displaced by our submerged body Archimedes oil waterFig 2 F red With the vertical pressure forces On our submerged body identified The net upward acting buoyancy force F B is found by Subtracting the weight above the Red surface of Fig 2 Fig 3 F blue From the weight above the dark blue surface in Fig 3 Archimedes