Review of Lecture #1
Every measurement has some uncertainty due to the limits of the measuring tool.
Sig Fig Rules:
1.
All non-zero digits (1,2,3,4,5,6,7,8,9) are significant.
2.
All zeros between non-zero digits are significant.
3.
Zeros that are used purely to set the decimal point (place holders) are generally NOT significant.
4.
Trailing zeros that are not needed to set the decimal point (place holders) are significant.
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!!!
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: d q
=
B d x .
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 d d q
=
( d x ) q
£ d x
+
...
2
+
+
...
+
( d z
+ d d z ) u
+
2
...
+
(
+ d u ) d w
2 +
...
+
( d w )
2 of the original uncertainties. In any case  q is never larger than their ordinary sum,
.

A few more rules in determining the propagated error from the error on independent measurements:
Suppose that x, . . . , w are measured with uncertainties  x, . . . ,  w, and the measured values are used to compute: q
= x
´
...
´ z u
´
...
´ w
.
If the uncertainties in x, . . . , w are and independent and random, then the fractional uncertainty in q is the sum in quadrature of the original fractional uncertainties, d q q
=
( d x x
)
2 +
...
+
( d z z
)
2 +
( d u u
)
2 +
...
+
( d w w
)
2
.
In any case, it is never larger than their ordinary sum, d q
£ d x d z d u d w q x
+
...
+ z
+ u
+
...
+ w
.
Example:
Suppose you want to measure the area of a field on which you will plant a crop of wheat. You measure the length of one side of the field to be L = 63.2 + 0.5 ft, and the width to be W = 45.9 + 0.7 ft. What is the calculated area and uncertainty in the area of the field?
A
=
L
´
W
=
63.2
ft
´
45.9
ft
=
2900.88
ft
2 d
A
A
=
( d
L
L
=
0.0172
ft
2
) 2 +
( d
W
W
) 2 =
(
0.5
63.2
) 2 +
(
0.7
45.9
) 2
This gives the fractional uncertainty in A. To determine the absolute uncertainty in  A we multiply the fractional uncertainty by A to get  A. d
A
=
2900.88
ft
2 ´
0.0172
ft
2
=
49.84
ft
2
The answer in Standard Format is:
A = 2900 + 50 ft 2 = (2.90 + 0.5) x 10 3 ft 2 .

If x is measured with uncertainty  x and is used to calculate the power q = x n (where n is a fixed known number), then the fractional uncertainty in q is: d q q
= n d x x
.
The next two rules are general case scenarios and work for every function of either one or more variables.
If x is measured with uncertainty  x and is used to calculate the function q(x), then the uncertainty  q is: d q
= dq d x .
dx
Suppose that x, . . . , z are measured with uncertainties  x, . . . ,  z and the measured values are used to compute the function q(x, . . . , z). If the uncertainties in x, . . . , z are independent and random, then the uncertainty in q is d q
=
(
¶ q
¶ x d x )
2 +
...
+
(
¶ q
¶ z d z )
2
.
In any case, it is never larger than the ordinary sum d q
£
¶ q
¶ x d x
+
...
+
¶
¶ q z d z .
Equation 3.47 in Taylor can always be used to propagate the uncertainty in a function due to random and independent uncertainties.
Examples:
1. (Equation 3.23) Suppose we measure the time for a rock to fall down a canyon so that we can measure the depth of the canyon. We use a stopwatch and measure the time to be 7.38 + 0.02 seconds for the time it takes the rock to drop from our hand and reach the bottom of the canyon. What is the depth of the canyon? Include the uncertainty and state the answer in Standard Format.

d ( t )
=
1 gt
2
2
=
266.876
m
=
1
2 m
(9.81
sec
2
)(7.38 sec)
2 d d ( t )
= d ( d ( t )) d t
= gt d t dt
=
(9.81
m sec
2
)(7.38 sec) (0.02 sec)
=
1.447
m
From this information we would write the depth of the canyon in Standard Format as follows: d (7.38sec)
=
266
±
1.4
m (exception rule if uncertainty begins with 1)
2. (Equation 3.47) Now suppose we want to calculate the acceleration due to gravity. There is an experiment in the lab where we drop a steel marble from a given height and the time to fall is measured. We use the same equation as above to then determine the acceleration due to gravity. If we measure the distance fallen to be
82.61 + 0.02cm and the time to fall that distance to be 0.411 + 0.001sec what is the computed value and uncertainty of the acceleration due to gravity? d ( t )
=
1
2 gt
2 Þ g ( d , t )
=
2 d t
2 g ( d , t )
=
2(0.8261
m )
(0.411sec)
2
=
9.781
m sec
2 d g ( d , t )
=
(
¶ g
¶ d d d )
2 +
(
¶ g
¶ t d t )
2
Before we go any farther we must discuss the idea of partial derivatives. These are taken when a function depends on more than one variable. In this example the acceleration due to gravity g, is dependent on two variables; distance d, and time t. Taking the partial derivative takes place when we take the derivative of the function with respect to one of the variables, holding everything else constant. In this case: g ( d , t )
= t
2
2
2 d t
2
¶ g
¶ d
¶
¶ t g
=
= -
4 t
3 d
We take the derivative of g with respect to d (holding time constant) and then we take the derivative with respect to t (holding the distance constant). When we hold the other variables constant, they act as any numerical constant would act when you take the partial derivative.
Applying this knowledge we discover that the uncertainty in g is:

 d g ( d , t )
=
(
2 d d )
2 +
(
-
4 d d t )
2 =
(
2
.411
2
´
0.0002)
2 +
(
-
4
´
0.8261
0.411
3
´
0.001)
2 t
2 t
3
=
0.04759
m sec
2
It is important to note that the general equation for finding the uncertainty in a function will ALWAYS work for EVERY function. Sometimes it is easier though to simply use the special cases where the function is only addition/subtraction, or multiplication/division, rather than take the partial derivative every time.
Discrepancy = difference between two measured values of the same quantity.
Discrepancy

A

B , where A and B are measured values of the same quantity.
The discrepancy between two measured quantities can be either Significant or Insignificant depending on how it relates to the uncertainties in the two measured quantities. For example suppose one student measures the velocity of a baseball to be A = 3.2 + 0.2 m/sec, and a second student measures the velocity of the same baseball to be B = 5.3 + 0.2 m/sec. The discrepancy here is significant. Why?
Discrepancy
=
A
-
B
=
3.2
m sec
-
5.3
m sec
=
2.1
m sec
The question we have to ask is… Is the discrepancy between these two numbers within the estimated error of each measurement?
Notice that the discrepancy is greater than the uncertainties in both of the measured values combined by quite a lot. There is no value between A and B that simultaneously falls within the probable measurements of A and B. This discrepancy is significant.
Now two measurements of another baseball are taken and the measurements are; C = 4.1 + 0.5 m/sec, and
D = 5.1 + 0.7 m/sec. The discrepancy here is 1.0 m/sec but this discrepancy is not significant in this case because there is a value between C and D that simultaneously falls within the probable measurements for C and
D. See below:

The discrepancy here is more than the uncertainty of either measurement but the measurements overlap with their uncertainties. There are values that fall between C and D that simultaneously falls within the probable measurements of C and D. Thus this discrepancy is not significant and the measurements are both reasonable.
Often times we will compare our measured values with a “Known” or “Accepted” value. In this case the discrepancy is significant if it is larger than the uncertainty in the measurement. This is not to say that the discrepancy is just fine if the uncertainty is huge. For example we may measure the acceleration due to gravity to be 15.8 + 7.9 m/sec 2 . The discrepancy from the known value of the acceleration due to gravity, 9.8 m/sec 2 , is
6 m/sec 2 . This is within the uncertainty but is pretty useless because the uncertainty in our measurement is
50% of the measurement. So our goal is to reduce the uncertainty in any measurement as much as possible while also getting as close as possible to the “known” or “accepted” value.
1.
What is the value of the function f = 2A – B if A = 69.3 + 0.4m and B = 23.6 + 0.7m in standard format?
A.
115.0 + 1.06 m
B.
115 + 0.81 m
C.
115.0 + 0.8 m
D.
115 + 1 m
2.
Calculate the fractional uncertainty if two measured values, A = 69.3 + 0.4m and B = 23.6 + 0.7m are multiplied together.
A.
3%
B.
0.0009
C.
0.000913
D.
0.09%

3.
What is the algebraic expression for the absolute error (  E) of the function for kinetic energy
KE = (1/2)mv 2 , given uncertainties in the mass (  m) and velocity (  v)? (Use the General Rule when determining this answer.)
A. d
E
=
B. d
E
=
C. 
E

è
1
2 v
2 d v
ø
+ ( mv d m
) 2
è
1
2

1
2 v
2 d m
ö 2
ø v
+ ( mv d v
) 2
2
 v


 mv
 m
 d
E
=
¶
¶
E m d v
ö 2 ¶
¶
E v d m
ö 2
D.
è ø
+
è ø

4.
Two research groups make a measurement of a new elementary particle. The two measurements are m
1
= (7.8 + 0.1) x 10 -27 kg and m
2
= (7.0 + 0.2) x 10 -27 kg. Based on the reported masses, did the two research groups measure the mass of the same particle? Explain.
A.
Yes, the discrepancy is only 0.8 x 10 -27 kg which is very small so they must have measured the same particle
B.
No, the discrepancy is large, such that the uncertainties of either measurement do not overlap so the discrepancy is significant.
C.
No, the uncertainties are much too large which means the measurements are too imprecise to be considered at all.
D.
Yes, measuring small quantities is very hard to do and any measurements that are even remotely close to each other indicates each group measured the same particle.