What is upthrust?

Upthrust is the upward force that a fluid exerts on an object that is either floating or submerged in the fluid. It's the force that allows you to float on water when you swim.

This force is caused by the pressure of the fluid pushing upwards on the object. Pressure arises from the motion of the fluid molecules colliding with the object, and the net effect of all these collisions is an upward force. 

Pressure in a fluid
Upthrust is caused by the pressure exerted by the fluid. Before we derive U=pgh, let's take a look at pressure in a fluid.
  • The pressure in a fluid increases with depth. This is because the weight of the fluid above exerts a force on the fluid below.
  • This pressure acts in all directions, including upwards on any object immersed in the fluid.
  • The equation for pressure in a fluid is given by the equation P=ρgh, where ρ is the density of the fluid, g is the acceleration due to gravity and h is the depth.

Deriving U= ρgV

  • Consider a submerged object with a top surface at a depth h₁ and a bottom surface at a depth h₂. The pressure at the top surface, p₁, is ρgh₁, and the pressure at the bottom surface, p₂, is ρgh₂.
  • The force due to pressure is given by F = pA, where A is the area.
  • The upward force on the bottom of the object (F₂) is p₂A = ρgh₂A, while the downward force on the top of the object (F₁) is p₁A = ρgh₁A.
  • The net upward force, which is the upthrust (U), is given by the difference between these forces:
  • U = F₂ - F₁
  • U = ρgh₂A - ρgh₁A
  • The height of the object is the difference in depth, or h₂ - h₁. If the height of the object is h, then h = h₂ - h₁.
  • Thus, the upthrust can be written as:
  • U = ρg(h₂ - h₁)A
  • U = ρghA
  • Since V = Ah, U = ρgV
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