When an object is immersed in a fluid, the upward force on the bottom of an object is greater than the downward force on the top of the object. The result is a net upward force (a **buoyant** force) on any object in any fluid.

### Buoyant Force: Cause and Calculation

Pressure increases with depth. When an object is immersed in a fluid, the upward force on the bottom of an object is greater than the downward force on the top of the object. The result is a net upward force (a buoyant force) on any object in any fluid. If the buoyant force is greater than the object’s weight, the object will rise to the surface and float. If the buoyant force is less than the object’s weight, the object will sink. If the buoyant force equals the object’s weight, the object will remain suspended at that depth. The buoyant force is always present in a fluid, whether an object floats, sinks or remains suspended.

The buoyant force is a result of pressure exerted by the fluid. The fluid pushes on all sides of an immersed object, but as pressure increases with depth, the push is stronger on the bottom surface of the object than in the top (as seen in ).

You can calculate the buoyant force on an object by adding up the forces exerted on all of an object’s sides. For example, consider the object shown in.

The top surface has area AA and is at depth h1h1; the pressure at that depth is:

P1=h1ρg

where ρρ is the density of the fluid and g≈9.81ms2g≈9.81ms2 is the gravitational acceleration. The magnitude of the force on the top surface is:

F1=P1A=h1ρgA

This force points downwards. Similarly, the force on the bottom surface is:

F2=P2A=h2ρgA

and points upwards. Because it is cylindrical, the net force on the object’s sides is zero—the forces on different parts of the surface oppose each other and cancel exactly. Thus, the net upward force on the cylinder due to the fluid is:

FB=F2−F1=ρgA(h2−h1)

### The Archimedes Principle

Although calculating the buoyant force in this way is always possible it is often very difficult. A simpler method follows from the Archimedes principle, which states that the buoyant force exerted on a body immersed in a fluid is equal to the weight of the fluid the body displaces. In other words, to calculate the buoyant force on an object we assume that the submersed part of the object is made of water and then calculate the weight of that water (as seen in ).

**Archimedes principle**: The buoyant force on the ship (a) is equal to the weight of the water displaced by the ship—shown as the dashed region in (b).

The principle can be stated as a formula:

FB=wfl

The reasoning behind the Archimedes principle is that the buoyancy force on an object depends on the pressure exerted by the fluid on its submerged surface. Imagine that we replace the submerged part of the object with the fluid in which it is contained, as in (b). The buoyancy force on this amount of fluid must be the same as on the original object (the ship). However, we also know that the buoyancy force on the fluid must be equal to its weight, as the fluid does not sink in itself. Therefore, the buoyancy force on the original object is equal to the weight of the “displaced fluid” (in this case, the water inside the dashed region (b)).

## Complete Submersion

The buoyancy force on a completely submerged object of volume is FB=VρgFB=Vρg.

learning objectivies

- Identify factors determining the buoyancy force on a completely submerged object

The Archimedes principle is easiest to understand and apply in the case of entirely submersed objects. In this section we discuss a few relevant examples. In general, the buoyancy force on a completely submerged object is given by the formula:

FB=Vρg

where VV is the volume of the object, ρρ is the density of the fluid, and gg is gravitational acceleration. This follows immediately from the Archimedes’ principle, and the fact that the object is completely submerged (and so the volume of the fluid displaced is just the volume of the object).

### Cylinder

In the previous section, we calculated the buoyancy force on a cylinder (shown in ) by considering the force exerted on each of the cylinder’s sides. Now, we’ll calculate this force using Archimedes’ principle. The buoyancy force on the cylinder is equal to the weight of the displaced fluid. This weight is equal to the mass of the displaced fluid multiplied by the gravitational acceleration:

**Buoyant force**: The fluid pushes on all sides of a submerged object. However, because pressure increases with depth, the upward push on the bottom surface (F2) is greater than the downward push on the top surface (F1). Therefore, the net buoyant force is always upwards.

FB=wfl=mflg

The mass of the displaced fluid is equal to its volume multiplied by its density:

mfl=Vflρ

However (*and this is the crucial point*), the cylinder is entirely submerged, so the volume of the displaced fluid is just the volume of the cylinder (see ), and:

**Archimedes principle**: The volume of the fluid displaced (b) is the same as the volume of the original cylinder (a).

mfl=Vflρ=Vcylinderρ

The volume of a cylinder is the area of its base multiplied by its height, or in our case:

Vcylinder=A(h2−h1).

Therefore, the buoyancy force on the cylinder is:

FB=mflg=Vcylinderρg=(h1−h2)ρgA.

This is the same result obtained in the previous section by considering the force due to the pressure exerted by the fluid.

Bouyancy related experiment:-