Upthrust
- Buoyancy is experienced by a body partially or totally immersed in a fluid, such as a water
- The size of the force that produces this is equal to the weight of the fluid displacement (sometimes referred to as Archimedes principle)
- This force, called upthrust, keeps boats afloat and allows balloons to rise through the air
- When a body travels through a fluid, it also experiences a buoyancy force (upthrust) due to the displacement of the fluid
- Where:
- Fb = buoyancy force (N)
- ρ = density of the fluid (kg m–3)
- Vg = volume of the fluid displaced (m3)
- g = gravitational field strength (m s–2)
- If you were to take a hollow ball and submerge it into a bucket of water, you would feel some resistance
- Some water will flow out of the bucket as it is displaced by the ball
- The buoyancy force, Fb will act upwards on the ball to bring it to the surface
- The ball will remain stationary floating when its weight acting downwards, Fg equals the buoyancy force acting upwards, Fb
The ball floats when it is balanced by the buoyancy force and its weight
- Notice that
which is the weight of the object submerged in the fluid
- Where:
- m = mass of the ball (kg)
- ρ = density of the ball (kg m–3)
- V = volume of the ball (m3)
Drag Force at Terminal Velocity
- Terminal velocity is useful when working with Stoke’s Law since at terminal velocity the forces in each direction are balanced
(equation 1)
- Where:
- Ws = weight of the sphere
- Fd = the drag force (N)
- Fb = the buoyancy force / upthrust (N)
At terminal velocity, the forces on the sphere are balanced
- The weight of the sphere is found using volume, density and gravitational field strength
(equation 2)
- Where
- Vs = volume of the sphere (m3)
- ρs = density of the sphere (kg m–3)
- r = radius of the sphere (m)
- g = gravitational field strength (N kg−1)
- Recall Stoke’s Law
(equation 3)
- Where v is the terminal velocity
- The buoyancy force equals the weight of the displaced fluid
- The volume of displaced fluid is the same as the volume of the sphere
- The weight of the fluid is found from volume, density and gravitational
(equation 4)
- Substitute equations 2, 3 and 4 into equation 1
- Rearrange to make terminal velocity the subject of the equation
- Finally, cancel out r from the top and bottom to find an expression for terminal velocity in terms of the radius of the sphere and the coefficient of viscosity
- This final equation shows that terminal velocity is:
- directly proportional to the square of the radius of the sphere
- inversely proportional to the viscosity of the fluid
Worked example
Icebergs typically float with a large volume of ice beneath the water. Ice has a density of 917 kg m-3 and a volume of Vi.
The density of seawater is 1020 kg m-3.
What fraction of the iceberg is above the water?
A. 0.10 Vi B. 0.90 Vi C. 0.97 Vi D. 0.20 Vi
Exam Tip
Remember that ρ in the buoyancy force equation is the density of the fluid and not the object itself!