# Buoyant Force

When an object is floating on or immersed in a fluid, such as an iceberg in water or a whale in the ocean, the fluid pushes on the object from many directions. This situation might seem too complicated to analyze mathematically, but it turns out that it is not. We can summarize the effect of the fluid in one force, the buoyant force. The following principle is called*Archimedes*'

*principle*, after its discoverer:

F_{B} = pVg |
...(7) |

_{B}is the buoyant force, p is the density of the fluid, and V is volume of the displaced fluid.

*m*and

*F*. (Use

*m*=

*ρ*

*V*,

*F*=

*PA*, and of course

*F*

_{B}=

*ρ*

*Vg*.)

A bathtub duck floats in water with one third of its volume above the water line. What is its specific gravity?

First, we DRAW A DIAGRAM. (See Figure 10-1.)

**Figure 10-1**

Second, we write a force equation, which is a force balance equation, since the duck is not accelerating:

0 = *F*_{B} – *mg*

Next we replace *F*_{B }with *ρ*_{H2O}*V*_{disp}*g* , and we replace *m* with *ρ**V*, where *ρ* is the density of the duck, and *V* is its volume:

Since one third of the duck is shown above the water, the displaced volume is 2*V*/3, and we write

Here we canceled the factors *V* and *g*. The answer is 2/3

A crown, apparently made of gold, is weighed in air, and the weight is 50 N. The crown is weighed again by hanging it from a string and submerging it in water. (See Figure 10-2.) What reading will the force meter give if the crown is true gold? (Use for the density of gold 19.3 g/cm^{3}; and for water, 1.0 g/cm^{3}. Also, use *g* = 10 m/s^{2}.)

**Figure 10-2**

First, we DRAW A DIAGRAM with all the forces on the crown. (See Figure 10-3.)

**Figure 10-3**

We draw the force of gravity first. Next, what is touching the crown? The fluid and the string are, so we know to draw the buoyant force and the force of tension. It is the force of tension that the meter reads. (That is what a force meter does: it provides a force and then tells you what that force is.)

We have force balance because there is no acceleration:

0 = *F*_{m} + *F*_{B} – *mg*

In this equation *m* is the mass of the crown, and *mg* is 50 N. Let's first solve for *F*_{m} (which we want), then replace *F*_{B} and *m*. We obtain