Reduced Submission Requirements: Submit only the items listed in the Results and Questions sections with a title page and your lab data.

Purpose

To experimentally determine the coefficient of drag on a smooth sphere.

Theory

The drag force experienced by an object in a moving fluid is given by:

FD = ½ ρ A CD v²

where:

F_D is the drag force on the object (N)

CD is the coefficient of drag unique to the object (1.0 for a smooth sphere) (unitless)

ρ is the density of the fluid (kg/m³)

A is the reference area of the object, typically the projected cross-section (m²)

v is the velocity of the fluid. (m/s)

We can approximate airflow in a smooth tunnel as experiencing a linear decrease in velocity due to turbulence and frictional forces. The tunnel used in the lab session also has staggered openings to allow the air flow rate to drop more quickly than it would in a closed system.

If the air velocity at both ends of the tube, and the length of the tube are known, then the velocity at any height can be approximated using the fact that a fractional increase in rest height is matched by a fractional decrease in air velocity. For example, a ball resting at a height of 50% of the length of the tube would experience an air velocity 50% of the way between V₁ and V₂.

Height / Length = (V₁ - V) / (V₁ - V₂)

or

y = - (ΔV / L) x + V₁ in y = mx + b form.

where:

Height is the measured height where the ball rests above the blower. (m) : Independent variable x.

Length is the total length of the tube. (m)

V₁ and V₂ are the air velocities at the entrance and exit respectively. (m/s)

V is the velocity at the height in question. (m/s) : Dependent variable y.

Potentially Useful Properties:

• Tube Diameter: 5.10 cm
• Ball Diameter: 4.15 cm
• Ball Mass: 0.55 g
• Reference Area: 13.50 cm²
• Air Density: 1.2 kg/m³
• Air Kinematic Viscosity: 1.60x10^-5 m²/s

Procedure

1. Using the Hot Wire Anemometer, test the air velocity at the tube entrance and exit for each of the given fan speeds.
2. Record the ball diameter and mass as well as the tube length.
3. Place the Styrofoam ball in the tube and find the resting height for each of the given fan speeds (measure against the center of the ball)
   • Record the value when the ball is at the lowest resting height if it bounces up and down.

Results

1. Calculate the gravitational force experienced by the ball.
2. Calculate the air velocity around the ball for each height.
3. Solve for the coefficient of drag assuming that the drag force balances the gravitational force when the ball is at rest for each height.
4. Compare this value to the expected coefficient for a perfectly smooth and round sphere in a laminar flow.

Questions

1. Increasing which of the following properties by 10% would cause the greatest increase in drag force on the same size/mass sphere, and why:
   • Tube Diameter
   • Coefficient of Drag
   • Air Temperature
   • Air Velocity
   • Air Density
2. BONUS (1): A penny is dropped from a great height (eg: the top of the CN Tower) at what speed will it hit the ground?
   • Is there any mathematical truth to the legends of people being killed by falling coins dropped by those on the observation deck?