Download Part 4: Dropping A Ping-Pong Ball - MET 213 Dynamics

FEATURE
My Favorite Experiment Series
by R.M. French
Part 4: Dropping A Ping-Pong Ball
The technical literature contains a number of proposed drag
coefficients,2–4 but we can take Cd 5 0.445 as approximately
correct. The drag coefficient of a smooth sphere is actually
a function of Reynolds number at very low speeds. The Reynolds number is basically the ratio of momentum to viscosity.
NOMENCLATURE
a
C
Cd
Fa
L
m
Re
s
t
t0
v
v0
w
m
acceleration
proportionality constant relating aerodynamic drag
to square of velocity
non-dimensional drag coefficient
aerodynamic force
reference length, ball diameter
mass
Reynolds number
reference area, cross-sectional area of ball
time
initial time
velocity
initial velocity
weight
viscosity of air
Re 5
rvL
m
ð4Þ
where r is air density, v is velocity, L is a reference length, and
m is the viscosity of air. For the ping-pong ball, the reference
length is the diameter, 40 mm. The viscosity of air is 1.789 3
105 kg/ms, so Re 5 2739v.
For 0.5 < Re < 750 that corresponds to 0.00018 m/s < v <
0.276 m/s, the approximate expression is
The familiar exercise of calculating the motion of a body in
freefall usually ignores the effect of aerodynamic forces. For
dense or streamlined bodies, this is often a good assumption.
However, not all bodies can be so approximated. An easy
example is a ping-pong ball. A sphere has high drag, and
the weight of the ball is very small. The result is that aerodynamic forces cannot be ignored even at low speeds. The
opposing forces acting on the ball are shown in Fig. 1.
The aerodynamic force is written in terms of a nondimensional
drag coefficient. Basic aerodynamic theory assumes the drag
coefficient is constant, and this is often correct. While the flow
over a smooth sphere is surprisingly complex,1 the drag coefficient is approximately constant over a useful range of velocities. The density of air at sea level in the standard
atmosphere is 1.225 kg/m3, and the reference area is simply
the cross-sectional area of the ball.
Cd 5
24 1 1 0:15Re0:687
Re
ð5Þ
For Reynolds numbers between 750 and 200,000, the drag
coefficient is approximately constant at Cd 5 0.445.
An expression that accounts for both Eq. 5 and the portion
of the velocity range during which the drag coefficient is constant would allow a continuous analysis of the problem.
Accordingly, a rational polynomial of order 34 was developed
to approximate the drag coefficient for Reynolds numbers 0.5
to 200,000.
Figure 2 shows the predicted velocity of a dropped ping-pong
ball using both the constant drag coefficient and the variable
coefficient. The difference is small enough that it hardly
justifies the additional complication. The ball is predicted to
reach a final, constant velocity (terminal velocity) when
Consider the equation of motion for a dropped ball.
ma 5 mv_ 5 w 2 Cv2
ð1Þ
where C 5 ½ Cdrs. This is a simple differential equation in
velocity and can be solved by direct integration.
dv w 2 Cv2
m
dv 5 dt
5
or
dt
w 2 Cv2
m
The solution to Eq. 2 is
rffiffiffiffi
pffiffiffiffiffiffi
w
v0 c
wc
v5
tanh
ðt 2 t0 Þ 1 tanh2 1 pffiffiffiffiffiffi
c
m
wc
ð2Þ
ð3Þ
Editor’s Note: ET created a Feature series to focus on short, educational/teaching–related articles under the title ‘‘My Favorite Experiment.’’ The short articles
demonstrate experimental techniques that can be applied directly to the classroom
and laboratory to enhance both the teaching process and the conveyance of various
apparatus and measurement methods to the students. Series editor: Kristin
B. Zimmerman, General Motors Corporation.
Mark French (SEM member) is a professor at Purdue University, West Lafayette,
IN, formerly with Robert Bosch Corporation.
doi: 10.1111/j.1747-1567.2006.00017.x
Ó 2006, Society for Experimental Mechanics
Fig. 1: Forces acting on ball
March/April 2006 EXPERIMENTAL TECHNIQUES
59
DROPPING A PING-PONG BALL
Fig. 3: Experimental and analytical results
Fig. 2: Comparison of drag approximations
SUMMARY
aerodynamic force and gravitational force are in balance. This
is certainly supported by experience and intuition. Of course,
the analytical prediction should be verified experimentally.
EXPERIMENTAL DATA
In order to get velocity data at several points, a ball drop was
recorded using a high-speed video camera. The ball was dropped while being recorded at 250 frames/s. Figure 3 shows good
correlation between the analysis using a constant drag coefficient and the experimental results.
Note that after about 0.5 s, the slope of the position curve is
approximately constant. This is exactly what would be expected
of a body moving at approximately constant velocity (approximately zero acceleration). Constant slope thus suggests that
the gravitational and aerodynamic forces are nearly in balance
and the ball is approaching terminal velocity.
60
EXPERIMENTAL TECHNIQUES March/April 2006
A differential equation is presented accounting for the effect of
aerodynamic drag on a falling sphere along with its analytical
solution. The resulting expression for the velocity of the ball
predicts a terminal velocity of approximately 8 m/s. A falling
ping-pong ball was recorded using a high-speed video, and the
results correlate well with the analytical prediction.
References
1. Van Dyke, M., An Album of Fluid Motion, Parabolic Press,
Stanford, CA (1982).
2. Abraham, F.F., ‘‘Functional Dependence of Drag Coefficient of
a Sphere on Reynolds Number,’’ Physics of Fluids 13(8):2194
(August 1970).
3. Johnson, T.A., and Patel, V.C., ‘‘Flow Past a Sphere Up to
a Reynolds Number of 300,’’ Journal of Fluid Mechanics 378(17):
19–70 (1999).
4. Liao, S.J., ‘‘An Analytic Approximation of the Drag Coefficient
for the Viscous Flow Past a Sphere,’’ International Journal of NonLinear Mechanics 37(1):1–18 (2002). n