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Vertical motion

Buoyant force

The area S of the top surface of the cylinder is equal to the area of the bottom surface of the cylinder.
The downward force due to the air pressure at the top surface of the cylinder equals

F_top = pressure × area = -p_air S

whereas the upward force due to the total pressure from air and liquid at the bottom surface of the cylinder equals

F_bottom = pressure × area = (p_air+p_liquid) S

So, there are in total three forces acting on the cylinder: the gravitational attraction from the Earth which is given by the weight -mg, the force of air pressure on its top surface -p_air S and the force of the total pressure at the bottom (p_air+p_liquid) S. Since the cylinder is in equilibrium while floating, the sum of those forces must vanish, i.e.

- mg - p_air S + (p_air+p_liquid) S = 0     hence     mg = p_liquid S

The pressure inside the liquid depends on the depth d below the surface and on the density of the liquid ρ_liquid, according to

p_liquid = dgρ_liquid

Consequently

mg = p_liquid S = dgρ_liquid  S

Furthermore is the volume V_imm of that part of the cylinder which sticks in the liquid equal to V_imm=dS. On substituting this fact in the above formula, we obtain

mg = p_liquid S = gV_imm ρ_liquid

which is clearly equal to the weight of the liquid which is displaced by the cylinder.

More general and also for gases this is known as Archimedes' principle which states that the buoyant force on an object equals the weight of the fluid displaced by the object, or the density of the fluid multiplied by the submerged volume times the gravitational acceleration g.

Some videos you might like to watch:

• Float or Sink - Cool Science Experiment (3:25 min)
  
• Why Do Things Float? An Easy Lesson (3:34 min)
  
• Archimedes' Principle (7:46 min)
  
• Buoyancy and Density (12:09 min)

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