Physics 1140 Summer 2011
Lecture #1
Any measured quantity has an uncertainty, which propagates through any computation from measured
quantities. A meter stick, for example, can only measure lengths to within a millimeter to be certain and some
fraction of a millimeter is uncertain.
The measurement of the arrow is 25.45cm. Because the
meter stick is divided into centimeters and millimeters we
know for certain that the arrow is between 25.4 and 25.5cm
but the last digit is an estimate and thus uncertain due to the
limitation of the measuring device. So the measurement
should be reported as 25.45 + 0.05cm where 0.05cm is the
estimated uncertainty in the measurement.
In the measurement 25.45cm we have 4 Significant Figures (Sig Fig’s). We use the number of Sig Fig’s to let
our reader know the rough estimate of the relative accuracy of the measurement. We only write down as many
digits that actually might mean something. So we would not write the above measurement as 25.451873cm
because we cannot say anything beyond the 4th Sig Fig.
Sig Fig Rules to Follow:
1.
2.
3.
4.
All non-zero digits (1,2,3,4,5,6,7,8,9) are significant.
All zeros between non-zero digits are significant.
Zeros that are used purely to set the decimal point (place holders) are generally NOT significant.
Trailing zeros that are NOT NEEDED to set the decimal point are significant.
X = 3.28
Y = 0.0071
F = 89.00
G = 2800
Q = 2800 + 10
(Three Sig Fig’s)
(Two Sig Fig’s, the zeros in front don’t count)
(Four Sig Fig’s, the zeros after the decimal count)
(Ambiguous, the zeros may be significant or not)
(Three Sig Fig’s, The uncertainty indicates the Sig Fig’s)
“Standard Format” for reporting results
z = 3 8 . 2 + 0 . 2 cm



Precision (decimal place) of answer and uncertainty must match.
1 Sig Fig only in uncertainty (possible exception if first digit is 1: x = 0.15).
Include units always.
D = (6.118697 + 0.008394) cm
WRONG!!!
(Rule 1)
(Rule 1,3)
(Rule 1,4)
(Rule 1,3,4)
(Rule 1,4)
Example: Measuring the length of a pendulum. (Lab M1)
1.
We will begin by first measuring the length from the middle of
the support rod to the top of the mass () while it is hanging on
the pendulum.
2. Next we will measure the height of the mass (h).
3. To find the length of the pendulum (L) from the middle of the
support rod to the middle of the mass we will add:  + h/2.
The measurements for h and  have uncertainties due to the limits of our
measuring device, in this case a 2-meter stick. If we were to use a more
precise device, such as a laser guided system, we may be able to decrease
the uncertainty but we would never be able to eliminate it completely.
Thus due to the uncertainty in our two measurements we will inherently
have an uncertainty in the length (L) of our pendulum.
Suppose I make the following measurements:
 = 50.65 + 0.05 cm
The + 0.05cm is the uncertainty in our measurement of  based on the limit of
our measuring device. In this case a 2-meter stick. I estimate the 5 at the end
of the measurement but I know that the measurement could be anywhere
between 50.6 and 50.7cm so I estimate the uncertainty to be 0.05cm.
h = 3.60 + 0.05 cm
In the calculation of L, I must consider the estimated uncertainty in both h and
. We will show this now.
The calculation of L is as follows:
L =l+
h
3.60
= 50.65 +
= 52.45cm
2
2
This is an easy calculation to do. The uncertainty is a bit more complex. Before we get to the calculation of the
uncertainty in L, a few rules must be discussed:
Uncertainty in a Measured Quantity Times an Exact Number.
Equation 3.9 in Taylor page 54.
If the quantity x is measured with uncertainty x and is used to compute the product
q = Bx
where B has no uncertainty, then the uncertainty in q is:
dq = B dx.
Uncertainty in Sums and Differences.
Equation 3.16 in Taylor page 60.
Suppose that x, . . . , w are measured values with uncertainties x, . . . , w and the measured values are used to
compute
q = x + ...+ z - (u + ...w).
If the uncertainties in x, . . . , w are known to be independent and random, then the uncertainty in q is the
quadratic sum
dq = (dx)2 + ...+ (dz)2 + (du)2 + ...+ (dw)2
of the original uncertainties. In any case q is never larger than their ordinary sum,
dq £ dx + ...+ dz + du + ...+ dw.
Now to calculate the uncertainty in the length L:
1
2
1
2
dL = (dl) 2 + ( dh) 2 = (0.05) 2 + ( 0.05) 2 = 0.0559cm
So the final measurement of the length of the pendulum is
L = 52.45 + 0.06 cm
(Standard Format)
Notice I rounded the uncertainty to 0.06 so the decimal place in the uncertainty matches the significant
figures of the measurement. The calculator will almost always show more decimal places than are appropriate so
rounding the calculator answer is almost always done when reporting your measurement in Standard Format.
More Examples:
A = 4.39 + 0.05 m
1.
B = 9.840 x 10 ^3 + 1.4 m
C = 7.81 x 10^1 + 2.8 m
Calculate f = A + C:
A + C = 4.39 + 7.81x101 = 8.249x101
df = dA2 + dC 2 = (0.05)2 + (2.8)2 = 2.8004
f + f = ( 8.2 + .3) x 101m = 82 + 3m
(Standard Format)
2. Calculate A – B
f = A - B = 4.39 - 9.840x103 = -9.83561x103
df = dA2 + dB2 = (0.05)2 + (1.4)2 =1.40089
f + f = (-9.836 + 0.001) x 103 m
=
(-9,836 + 1) m
=
(-9,836 + 1.4) m (Standard Format)
Note the calculation for the uncertainty in f (f) is the same for both addition and subtraction. Also notice
that the final answer in Standard Format can be written in multiple ways depending on the scientific notation.
Sometimes it is preferable to write the result in a certain way as opposed to another but either is correct.
Clicker Questions for Lecture 1:
1.
2.
3.
4.
5.
How many Sig Fig’s does 50,391 have?
A. 4
B. 6
C. 1
D. 5
How many Sig Fig’s does 0.0028 have?
A. 2
B. 4
C. 5
D. Ambiguous
How many Sig Fig’s does 40,300 have?
A. 2
B. 3
C. 5
D. Ambiguous
Which of the following are written in “Standard Format”
A. 2.323 + 0.009 m
B. 8.41 + 0.2 sec
C. (7.3891 + 0.01) x 102 m/sec
D. 9.8 + 0.002 Watts
If f = 2.39 + 0.02 m and x = 8.21 + 0.04 m, calculate g = f + x along with the propagated error and put
your answer in Standard Format.
A. 10.6 + 0.04 m
B. 10.60 + 0.044 m
C. 10.60 + 0.04 m
D. 10.60 + 0.04