A steel sphere of radius 6.5 mm is dropped into a lake. What is the terminal speed of the sphere?
To calculate the terminal speed of the steel sphere, we need to consider the forces acting on it. The key force here is the drag force, which opposes the motion of the sphere through the liquid due to its velocity.
The drag force can be expressed as:
F_drag = 0.5 * ρ * A * C_d * v^2
Where:
- F_drag is the drag force
- ρ is the density of the fluid (in this case, the density of water)
- A is the cross-sectional area of the sphere
- C_d is the drag coefficient
- v is the velocity of the sphere
In order to calculate the terminal speed, we need to find the velocity at which the drag force is equal to the gravitational force acting on the sphere:
mg = F_drag
Where:
- m is the mass of the sphere
- g is the acceleration due to gravity
The mass of the sphere can be calculated using the formula:
m = ρ * V
Where:
- V is the volume of the sphere
The volume of the sphere can be calculated using the formula:
V = (4/3) * π * r^3
Where:
- r is the radius of the sphere
Let's plug in the values and calculate the terminal speed step by step:
1. Calculate the volume of the sphere:
V = (4/3) * π * (6.5 mm)^3
2. Convert the radius to meters:
r = 6.5 mm / 1000 (1 meter = 1000 mm)
3. Calculate the volume:
V = (4/3) * π * (6.5 mm)^3
4. Calculate the mass of the sphere:
m = ρ * V
5. Calculate the gravitational force on the sphere:
mg = m * g
6. Set the gravitational force equal to the drag force and solve for the terminal speed:
mg = 0.5 * ρ * A * C_d * v^2
By solving this equation for v, you can find the terminal speed of the steel sphere.