Using a sketch and the relevant equations, show that the buoyancy or upthrust force on a floating body, F' = rho*g*V_{t}

Sure, here is the step-by-step explanation:

1. Sketch: Imagine a body of volume V submerged in a fluid. The body displaces a volume V_t of the fluid. The fluid exerts a force on every surface of the body. The force is perpendicular to the surface and its magnitude is given by the pressure of the fluid at the depth of the surface times the area of the surface.

2. Pressure: The pressure at a depth h in a fluid is given by P = P_0 + rhogh, where P_0 is the pressure at the surface of the fluid, rho is the density of the fluid, g is the acceleration due to gravity, and h is the depth.

3. Buoyancy Force: The buoyancy force or upthrust force on the body is the net force exerted by the fluid on the body. It is the difference between the force exerted on the bottom surface of the body and the force exerted on the top surface of the body.

4. Calculation: The force on the bottom surface is P_bottomA = (P_0 + rhogh_bottom)A and the force on the top surface is P_topA = (P_0 + rhogh_top)A. The buoyancy force is F' = P_bottomA - P_topA = rhog(h_bottom - h_top)A = rhog*V_t, where V_t = (h_bottom - h_top)*A is the volume of the fluid displaced by the body.

So, the buoyancy force on a floating body is F' = rhogV_t. This is known as Archimedes' principle.