1.6. Problem on buoyant force

The buoyant force exerted on a body by a static fluid, which is fully or partially submerged, is always equal to the weight of the fluid displaced by the body. This force acts in the opposite direction of gravity.

The buoyant force is the upward force exerted by a fluid on an object that is immersed in it. When an object is submerged, it displaces a volume of fluid. The weight of this displaced fluid equates to the buoyant force acting on the object. It’s important to note that this force acts upward, counteracting the downward force of gravity. Thus, if the buoyant force equals the weight of the object, it will float; if it is less, the object will sink.

Imagine a beach ball submerged underwater. As you push it down, it pushes up against your hand with equal force. This upward force is the buoyant force, and it’s trying to return the ball to the surface. It’s like if you try to hold a ball underwater; the water pushes it back up to the surface to where it can float.

In equilibrium, the weight of the body supports the weight of the displaced liquids. For a block of wood, density ρ1 floating at the interface of two liquids, ρ2 and ρ3, the relationship can be stated as: ρ1 V1 g = ρ2 V2 g + ρ3 V3 g.

When an object like a block of wood is floating at the interface of two liquids, we can analyze the forces acting on it. The weight of the wood (determined by its density and total volume) must balance the weights of the volumes of the two displaced liquids. In mathematical terms, this is expressed as the weight of the wood (ρ1 V1 g) being equal to the sum of the weights of the liquids above (ρ2 V2 g) and below (ρ3 V3 g) the block. By understanding these relationships, you can solve problems involving floating objects in different fluid environments.

Think of a sponge sitting on water. The sponge is soaked with water (density ρ1) and it floats because the weight of the water it displaces (which is still much lighter than itself) pushes it upwards. If you try adding too many heavy things on the sponge, it will sink because it can’t displace enough weight in water to remain afloat.

A block of wood with density ρ1 is floating completely submerged. We can calculate the weights and volumes through the relationship of their densities, resulting in a series of equations to balance the forces.

To solve problems involving buoyancy, first, visualize the situation, then identify the forces acting on the body. By establishing equilibrium (where the total upward buoyant force equals the total downward weight), you can derive relationships between the volumes and densities of the different fluids and the object. The calculations will often involve substituting known physics values and then simplifying equations to find unknowns, such as volume or density of the block.

Consider a rubber duck in a bathtub. It displaces a certain amount of water equal to its own weight. If you add a weight to the rubber duck – say a paperclip – it still floats, but if you add too many paperclips, it will sink. The rubber duck's buoyant force can only support a certain amount of weight; beyond that, it can no longer equal the downward gravitational forces.

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