A steel sphere of radius 6.5 mm is dropped into a lake. What is the terminal speed of the sphere?
To determine the terminal speed of the sphere, we need to consider the forces acting on it.
The sphere experiences two main forces: gravity (its weight) pulling it downwards, and drag (a resistive force) opposing its motion as it falls through the lake.
The weight of the sphere can be calculated using the formula:
Weight = mass * gravity
To find the mass of the sphere, we need its volume and density:
Volume = (4/3) * π * (radius)^3
Density = mass / volume
Since the density of steel is typically around 7850 kg/m³, we can use this value to calculate the mass of the sphere.
Once we have the weight, we can determine the net force acting on the sphere:
Net force = Weight - Drag
At terminal speed, the net force on the sphere is zero. This means that the weight and drag forces balance each other out and the sphere falls at a constant speed.
The drag force can be approximated using the Stokes' law for a spherical object in a fluid:
Drag = 6π * viscosity * radius * velocity
Here, we need to know the viscosity of water, which is approximately 0.001 Pa·s.
To find the terminal speed, we can set the net force to zero and solve for velocity:
Weight = Drag
mass * gravity = 6π * viscosity * radius * velocity
Now, let's calculate the terminal speed step-by-step:
1. Calculate the volume of the sphere:
Volume = (4/3) * π * (radius)^3
= (4/3) * π * (6.5 mm)^3
2. Convert the volume to m^3:
Volume = ((4/3) * π * (6.5 mm)^3) / (1000^3)
3. Calculate the mass of the sphere using density:
mass = density * volume
4. Calculate the weight of the sphere:
Weight = mass * gravity
5. Calculate the drag force:
Drag = 6π * viscosity * radius * velocity
6. Set the drag force equal to the weight:
mass * gravity = 6π * viscosity * radius * velocity
7. Solve for velocity:
velocity = ((mass * gravity) / (6π * viscosity * radius))
Now, plug in the values into the equations, and solve for velocity.