Hint:
The
polyfit
command finds a polynomial of a specified order that best fits the given data. In this
example, we want to see if the experimental data falls along a straight line. We therefore choose 1 as
the degree of the polynomial.
Hint:
The
polyfit
command returns a vector containing the coefficients of the polynomial:
[
]
1
0
n
α
α
α
L
corresponding to
(
)
1
1
0
...
n
n
P X
X
X
α
α
α
=
+
+
+
Hint:
To suppress the line connecting the experimental data points, select the big arrow in the figure’s
menu and double-click on the line. A property editor window will appear. Change the line style to
‘no line’ and choose circles for the marker style.
!
SOLUTION
close all
clear all
% Experimental data
Re = [0.05875 0.1003 0.1585 0.4786 3.020 7.015];
Cd = [492 256.2 169.8 58.88 10.86 5.5623];
% Take the logarithm of these data
lnRe = log(Re);
lnCd = log(Cd);
% Construct the line that fits best data. As we want a
% line, the degree of the polynomial is set to 1.
P = polyfit(lnRe,lnCd,1)
% Define the fitting line, so it can be plotted
lnRefit = linspace(log(0.05875),log(7.015),20);
lnCdfit = P(1)*lnRefit + P(2);
Re
0.05875
0.1003
0.1585
0.4786
3.020
D
C
492
256.2
169.8
58.88
10.86

% Define the theoretical line : ln(Cd) = - ln(Re) + ln(24)
lnRetheo = linspace(log(0.05875),log(7.015),20);
lnCdtheo = log(24) - lnRetheo;
plot(lnRe,lnCd)
hold on
plot(lnRefit,lnCdfit,'--')
plot(lnRetheo,lnCdtheo,'-.')
xlabel('ln(Re)')
ylabel('ln(Cd)')
title('drag coefficient on a sphere as a function of the
Reynolds number')
grid on
P =
-0.92980905266427
3.45384753973349
If the experimental data were exactly following Stokes’ law,
P would be :
Ptheo =
-1.0000
3.17805383
The experiment are thus in good agreement with the theory
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
1
2
3
4
5
6
7
ln(Re)
ln(Cd)
drag coefficient on a sphere as a function of the Reynolds number
Experimental data
Best fit
Theoretical line
:
YOUR TURN
The same students have repeated their experiments at Reynolds numbers between 10
4
and 10
5
and have found a constant drag coefficient of 0.4775. Assume the density of air
ρ
is 1.25 kg/m
3
.
The following table give the drag force on the sphere as a function of the air speed.
V
(m/s)
4
6
7
8
9
D
F
(N)
0.036
0.09298
0.10682
0.1568
0.193
Find the radius of the sphere used in the experiments. Follow these steps:
a)
In the definition of the drag force:
2
2
1
2
D
D
F
C
V
r
ρ
π
=
the group
2
1
2
D
C
r
ρπ
is now a constant, since the drag coefficient is assumed to be constant. That means
that the plot of
D
F
versus
2
V
should be close to a straight line. Using
polyfit
, find an equation of a line
that best fits the data given in the table
b)
Using the results in part a) extract the value of
2
1
2
D
C
r
ρπ
c)
Using the numerical values provided, estimate the radius of the sphere
20. Vectors of random numbers
Command :
randn
generate a vector whose components are chosen at random
hist
plot a histogram

At the assembly line of a big aircraft company, bolts are stored in boxes containing 1000 parts
each. The bolts are supposed to have a nominal diameter of 5 millimetres. But due to the
uncertainty in the manufacturing process, some turn out to be larger and some turn out to be
smaller than nominal.

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- Fall '08
- DARVE,KHAYMS
- Derivative, Acceleration, Velocity, Vdiode