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Phys 212
Spring 2000
Recitation Activity #1: Error Analysis1
NAME:
Solution
DATE:
January 6, 2000
The following is a mythical introductory Physics
experiment. A mass was dropped from different
heights above a velocity detector. The students
recorded the heights and resulting velocities. After
stating that they were assuming negligible air
friction, they decided to apply energy conservation
principles to determine a relationship for their data.
Since the kinetic energy of the mass should be equal
to the change in potential energy, they arrived at the
following equation.
5 or 6
100
4.5
90
4
75
0, 1, 2 or 3
60
shoddy work 40
no paper
0
M
h
Velocity
Detector
1
mv 2  mgh
2
(1)
Where h is the initial height of the mass above the detector and v is the velocity of the
mass when it passes the detector. With some algebraic manipulation,
v  2g h
(2)
Exercise 1. Find the standard error.
For the above example, let us find the standard error for the velocity if the standard error
associated with the measurement of the initial height above the detector is known. First,
we need a formula to relate the two errors. In general,
2
 Q 
 Q 
 Q 
2
2
 SE 2 ( y )  
SE (Q)  
 SE ( x)  
 SE ( z )  ....

x

y

z






2
2
2
(3)
Where Q is a quantity for which we wish to find its standard error and is calculated from
measured variables x, y, z, etc., which have known standard errors. Do not worry if you
have not used partial derivatives before, for the problems in the this course we will
restrict our use to one term, i.e.,
2
 dQ 
2
SE (Q)  
 SE ( x)
 dx 
2
(4)
The students estimate their accuracy at measuring the initial height to be 0.1 meters.
Rewriting (4),
2
 dv 
SE (v)    SE 2 (h)
 dh 
2
and taking the derivative of equation (2) with g assumed to be a constant,
arrive at our final expression, SE 2 (v) 
dv
g
, we

dh
2h
g
SE 2 (h) .
2h
Q1. Why does the standard error of the velocity depend on the initial height of the mass?
(Compare the percentage effect of the height standard error on the velocity for h = 0.1 m
and h= 100 m.)
0.1m/0.1m is a 100 % effect and would be very noticeable effect and minor changes
in the placement of the mass would have a large percentage effect on the velocity at
the detector.
For h = 100 m, 0.1 m/ 100 m is only 0.001 % effect and for this experiment would
not be noticeable. The velocity of the mass dropped from 100.1 m would not be
noticeable different than a mass dropped from 99.9 m.
The students took measurements from 1 meter to 10 meters. Since the standard error of
the velocity is largest for the 1 meter data point, let us use that height to calculate our
standard error.
SE 2 (v) 
g
9.8 m s 2
m2
SE 2 (h) 
(0.1m) 2  0.049 2
2h
2(1m)
s

SE (v )  0.22
m
s
Q2. So, our expected measured velocity for h = 1 m is (4.4 +/- 0.2) m/s using equation
(2) and our standard error result. What is your expected measured velocity for h = 2 m?


9.8m / s 2
2

v  2(9.8m / s ) 2m 
(0.1m) 2   6.26  0.157 m / s  6.3  0.2m / s


2(2m)


Since the velocity and the height can both be measured, equation (2) allows for an
experimental determination of the acceleration due to gravity, g. If v versus sqrt(h) is
plotted, the slope of the line should equal 2 g and the standard error of the slope of
the linear fit should give us an estimate of the errors in our final value of g.
Plot of measured veloctiy versus drop height of mass
20
Velocity [m/s]
velocity = slope * sqrt(height)
15
slope
Value
Error
4.466
0.07
10
5
Velocity
0
0.5
1
1.5
2
2.5
3
3.5
1/2
Sqrt(Height) [m ]
Q3. Using the above plot, determine g including the standard error.
2
s = slope =
2g

m1 / 2 
 4.466

s 

2
 9.973 m 2
 g  s /2 
s
2

dg
s
ds
2

m1 / 2 
m1 / 2 2
m2
 dg 
2
2
2
m


SE ( s )    SE ( g )  s (0.07
)   4.466
(0.07
)  0.098 4
s2
s 
s
s
 ds 

2
SE(s) = 0.313 m/s
2
2
g  9.97  0.31 m
s2
Q4. Does your result agree with the accepted value of g = 9.8 m/s2? Why or why not?
g    9.97  0.31 m
s2
 10.28 m
s2
g    9.97  0.31 m
s2
 9.66 m
s2
Yes, because accept value of g is within the error limits, g    g  g   “or the range
of the calculated values for g.”
Exercise 2: Types of Errors
Errors arise from many causes and identifying the nature of errors is not always easy to do.
Usually observable errors fall into two classes: systematic or random (accidental) errors. A
systematic error is an error due to a definite identifiable cause. An example of a (personal)
systematic error in the above example would be a student who always measured the height of the
ball as the distance from the top of the detector to the bottom of the ball. (Instead of using the
center of mass of the ball and the center of the detector.) A random error in the above experiment
would be the reading of the height measurement. If the reader of the ruler rounded the height up
and down to the nearest increment, the errors introduced into the measurements would tend to
cancel out for a large number of measurements. If, however the student always rounded up, this
would be a source of (personal) systematic error.
Systematic errors may be subdivided into three kinds:
1.
Instrument errors are those caused by the physical design limitation and/or calibration of
the measuring instrument. Examples from this experiment include the calibration of the velocity
detector and location of the ball when it takes a velocity reading. Does the detector measure the
velocity of the ball when the bottom edge, center, or top edge of the ball hits the center of the
detector? Or return an average value as the ball passes in front of the detector?
2.
Natural errors arise from changes in ambient conditions. For example, temperature
changes may change the resistive value of the resistor used to calibrate the detector.
3.
Personal errors depend on the physical limitations and habits of the observer.
Random errors are usually due to a large number of unknown causes acting in different ways.
One way to check for random errors is to repeat several seemingly identical experiments and,
provided the measurements are sufficiently sensitive, note the variation in the measured values.
Truly random errors are just as likely to be positive as negative, so taking the average of the
measured value is likely to minimize the error. Taking the average of the measurement does not
eliminate a systematic error. In addition, systematic errors are usually cumulative, that is they
“add up.”
Q5. What type of error are we calculating in Exercise 1?
Random
Q6. Please list several possible sources of (a) random and (b) systematic errors for this
experiment.
Random: Rounding of height measurement, rounding of detector velocity reading, initial
placement of mass (sometimes slightly high or low), etc.
Systematic: Position at which velocity of ball is measured, detector readings drift during
experiment due to changes in temperature, error in calibration of detector, etc.
Several means three or more, so student needs to lists at least three errors total for full
credit and must list at least one of each type. (Example: 2 random and one systematic error
correctly listed would receive full credit.
1
Measures in Science and Engineering, their expression, relation and interpretation, B.S. Massey, Ellis
Horwood Limited, England, 1986.