A small sphere of radius 3.5 mm and mass 2.7 g was dropped through oil of viscosity 0.24 Pas and density 880 kgm^-3. Determine its maximum speed.

Answer 1

Hii Madeleine L.,

To determine the maximum speed (terminal velocity) of the sphere, we use Stokes' Law and the concept of terminal velocity. Terminal velocity occurs when the net force on the sphere becomes zero, meaning the gravitational force is balanced by the buoyant force and the viscous drag force.

Given Data:

1. Radius of the sphere, r = 3.5 mm = 3.5 × 10³ m
2. Mass of the sphere, m = 2.7 g = 2.7 × 10⁻³ kg
3. Density of oil, ρ_oil = 880 kg/m³
4. Viscosity of oil, η = 0.24 Pa·s
5. Acceleration due to gravity, g = 9.81 m/s²
6. Density of the sphere, ρ_sphere = mass / volume
7. Volume of the sphere = (4/3) π r³
8. ρ_sphere = m / [(4/3) π r³]

Step 1: Calculate the Density of the Sphere

Volume of the sphere:

V = (4/3) π (3.5 × 10⁻³)

V = (4/3) × 3.1416 × (4.287 × 10⁻⁹)

V = 1.795 × 10⁻⁸ m³

Density of the sphere:

ρ_sphere = (2.7 × 10⁻³) / (1.795 × 10⁻⁸)

ρ_sphere = 150.5 kg/m³

Step 2: Use the Terminal Velocity Formula

The formula for terminal velocity is:

v_t = (2 r² g (ρ_sphere - ρ_oil)) / (9 η)

Substituting the values:

v_t = (2 × (3.5 × 10⁻³)² × 9.81 × (150.5 - 880)) / (9 × 0.24)

v_t = (2 × 1.225 × 10⁻⁵ × 9.81 × (-729.5)) / (2.16)

Since the density of the sphere is much less than that of oil, the result will be negative, which means the sphere will not sink but float instead.

Final Answer:

The sphere will not reach a maximum falling speed as it is less dense than oil and will float instead of sinking.

Answer 2

Falling objects will eventually reach terminal velocity and since we have not been told how far the object falls, we must assume it reaches terminal velocity (which, is the maximum speed it can fall).

At terminal velocity, acceleration equals zero or, in other words the sum of the forces equals zero.

Free Body Diagram:

where F_G is the force of gravity (aka, its weight), F_D is the drag force, and F_B is the buoyancy force.

So:

∑F_y = 0

F_D + F_B - F_G = 0

Calculate F_G:

F_G = mg = (2.7/1000)(9.8) = 0.02646 N

Calculate F_D:

For oil and other thick liquids, use Stokes Law (for a sphere) to find F_D:

F_D = 6πηrv where η is the viscosity, r = radius of sphere, v = velocity of sphere

F_D = 6π(0.24)(3.5/1000)v

F_D = 0.01583v

Calculate FB:

The buoyancy force equals the weight of the displaced liquid:

Liquid weight (LW) = Volume (V) multiplied by density (ρ) multiplied by "g"

LW = Vρg

Volume of a sphere = 4/3πr³

LW = (4/3)πr³ρg

LW = (4/3)π(3.5/1000)³(880)(9.8)

LW = 0.001549

F_B = 0.001549 N

So, using F_D + F_B - F_G = 0 and plugging in what we calculated for each:

0.01583v + 0.001549 - 0.02646 = 0

0.01583v = 0.02487

v = 1.57 m/s