# Attack point of buoyancy force

I have seen free body diagrams of submerged objects that show the buoyancy force acting on the centre of mass of the object. However, the buoyancy force arises due to pressure differences between the upper and lower surfaces of the object. Therefore I have been thinking that the buoyancy force should be sketched in the free body diagram as a vector with an "attack point" on the bottom of the object.

Which free body diagram would be correct?

## 3 Answers

The buoyancy force acts at the "center of pressure".

• Varying pressure is exerted across the entire surface of the object, but when you add it up, it is as if a buoyant force is acting at a single point, the center of buoyancy. This is similar to how gravity on every part of an object acts as if gravity were acting only on the center of mass. – G. Smith Nov 24 '18 at 18:47
• Where would the centre of pressure be for a cube, for example, submerged in a resting fluid, e.g. water? – FizzleDizzle Nov 24 '18 at 18:55

It is in fact quite rare for an applied force to be applied right at the centre of mass. Think of the normal reaction force, for example: that is a force acting at the surface, and furthermore it acts throughout the part of the surface that is in contact with another surface. Next consider the gravitational force: its acts throughout the body, like lots of little force vectors attached to each atom in the body. You are perfectly correct to say that pressure acts at the surface of the body, not at the centre of mass.

In each case, a good question to ask, in the case of a rigid body, is: does this spread-out force (normal reaction, gravity, pressure) produce an overall torque on the body? If not, then it is safe to say that the net effect of the force on the body's motion is just as if you had a single force vector running through the centre of mass. So the practice of drawing a force vector through the centre of mass amounts to a claim: it is not saying 'this is exactly what is happening' it is saying 'this force through the centre of mass is equivalent to the ones really acting, in its effects on the overall motion of the body'.

The center of gravity and center of mass coincide as long as the gravitational field is uniform (negligible changes from point to point on the object). Under these conditions, for analyzing motion, stability, etc., it is correct to model the gravitational force as acting through the center of mass.

The center of buoyancy and center of volume of a submerged object coincide if, again, the gravitational field doesn't change over the object, and additionally, the density of the fluid does not change appreciably in the region surrounding the object. It is correct under these conditions to model the buoyancy force as acting through the center of volume (the "average" of all the points contained in the object). This center of volume, in general, only coincides with the center of mass if the object is of uniform density. If it is not--has dense areas and light areas--the center of mass and center of buoyancy are not the same, and if you pretend they are, you will reach incorrect conclusions.

As has been pointed out in other answers, neither gravity nor buoyancy actually act at a concentrated point; but for a rigid body all that matters is the integrated force over the entire body.