The relationship between upthrust (or buoyant force) and the volume of a body immersed in a fluid is explained by Archimedes’ Principle. This principle is a cornerstone of fluid mechanics and describes how objects behave when partially or fully submerged in a liquid or gas.
What is Upthrust?
Upthrust, also known as buoyant force, is the upward force exerted by a fluid on a body that is submerged in it. This force opposes the weight of the object and is responsible for making objects float or feel lighter in water.
Archimedes’ Principle
Archimedes’ Principle states that:
The upthrust acting on a body immersed in a fluid is equal to the weight of the fluid displaced by the body.
Mathematically:
Fb=ρf⋅Vd⋅gF_b = \rho_f \cdot V_d \cdot g
Where:
- FbF_b = Buoyant force (upthrust)
- ρf\rho_f = Density of the fluid
- VdV_d = Volume of the fluid displaced (equivalent to the volume of the submerged part of the body)
- gg = Acceleration due to gravity
How Upthrust is Related to Volume
The upthrust depends directly on the volume of the fluid displaced, which in turn is determined by the volume of the body that is submerged in the fluid. The relationship can be explained as follows:
- Partially Submerged Object:
For an object that is only partially submerged, the upthrust is proportional to the submerged volume. For instance, if an object is half submerged, the upthrust corresponds to the weight of the fluid displaced by the submerged half of the object. - Fully Submerged Object:
When an object is completely submerged, the upthrust is proportional to the total volume of the object, provided the fluid is incompressible. A larger volume will displace more fluid, leading to a greater upthrust. - Floating Objects:
For objects that float, the upthrust equals the object’s weight. In this case, only a part of the object is submerged, and the submerged volume adjusts itself such that the displaced fluid’s weight matches the object’s weight.
Key Observations
- Direct Proportionality:
The upthrust is directly proportional to the volume of the displaced fluid, and consequently, to the volume of the submerged part of the body. This means:- Larger submerged volumes result in greater upthrust.
- Smaller submerged volumes result in lesser upthrust.
- Fluid Density:
The upthrust also depends on the density of the fluid. For the same submerged volume:- Denser fluids (like saltwater) provide more upthrust.
- Less dense fluids (like oil) provide less upthrust.
Practical Applications
- Floating of Ships:
Ships float because their design allows them to displace a volume of water equal in weight to their own weight. This ensures the upthrust balances the ship’s weight. - Submarines:
By controlling their volume (through ballast tanks), submarines can adjust their submerged volume and, hence, the upthrust to either float, sink, or remain at a certain depth. - Hydrometers:
Hydrometers are devices that measure fluid density based on how much of their volume is submerged. The upthrust changes with the fluid’s density, allowing for the measurement.
The upthrust experienced by an object in a fluid is directly proportional to the volume of the object that is submerged. This relationship, governed by Archimedes’ Principle, plays a crucial role in understanding phenomena like floating, sinking, and buoyancy. By considering the volume of the submerged part of a body and the density of the fluid, one can predict and explain the behavior of objects in different fluid environments.