Buoyancy (HL IB Physics)

• Buoyancy is experienced by a body which is partially or totally immersed in a fluid
  • The buoyancy force is exerted on a body due to the displacement of the fluid it is immersed in
• Buoyancy 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

• Buoyancy is calculated using:

• Where:
  • F_b = buoyancy force (N)
  • ρ = density of the fluid (kg m^–3)
  • V = volume of the fluid displaced (m³)
  • g = acceleration of free fall (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, F_b of the water will push upward on the ball
• When you let go of the ball, the buoyancy force of the water on the ball will cause the ball to accelerate to the surface
• The ball will remain stationary floating on the surface of the water
• A this point, the weight of the ball acting downward, F_g, is equal to the buoyancy force acting upwards, F_b

The ball floats when the buoyancy force and its weight are balanced

• Notice that

• Where:
  • m = mass of the ball (kg)
  • ρ = density of the ball (kg m^–3)
  • V = volume of the ball (m³)
• The buoyancy force and the weight force are equal

Drag Force at Terminal Speed

• Terminal velocity, or terminal speed, is useful when working with Stoke’s Law
• This is because, at terminal velocity, the forces in each direction are balanced

(Equation 1)

• Where:
  • W_s = weight of the sphere (N)
  • F_d = the drag force (N)
  • F_b = 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
  • V_s = volume of the sphere (m³)
  • ρ_s = density of the sphere (kg m^–3)
  • r = radius of the sphere (m)
  • g =  acceleration of free fall (m s⁻²)

• Recall Stoke’s Law

(Equation 3)

• Where
  • F_d = viscous drag force (N)
  • η = fluid viscosity (N s m⁻² or Pa s)
  • r = radius of the sphere (m)
  • v = velocity of the sphere through the fluid (ms⁻¹)
    • In this case, 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 using volume, density and acceleration of free fall

(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