August 31, 2000
PHY 113 { Additional notes for Chapter 1 (Problem Set # 1)
In class, we did not quite have enough time to discuss error analysis. This will be discussed
also in your laboratory work. Some (hopefully) helpful comments follow:
Some degree of error is associated with any measurement. For example, Suppose your ruler
has centimeter and millimeter markings. If you measured one side of yourtextyou could say
that its length is
l
1
Æl
1
(for example 22
:
2
0
:
2) cm. Suppose the second length is measured
as
l
2
Æl
2
, while the thickness is
t
Æt
.If you now wanted to compute the volume of your
text, that would be
V
=
l
1
l
2
t:
To get an idea of the error in your calculation you need to think about the error in each
length measurement. Precisely,
ÆV
(
l
1
Æl
1
)(
l
2
Æl
2
)(
t
Æt
)
l
1
l
2
t:
If we want an estimate of the error, then we can make the following approximations to the
above formula for
ÆV
.
Æl
1
is small so that terms like
Æl
1
Æl
2
and
Æl
1
Æl
2
Æt
can be neglected.
Since we want to estimate the maximum possible error, we should replace
with +.
Therefore,
ÆV
Æl
1
l
2
t
+
l
1
Æl
2
t
+
l
1
l
2
Æt:
If we divide this result by
V
,weget the very compact result:
ÆV
V
=
Æl
1
l
1
+
Æl
2
l
2
+
Æt
t
:
This shows that in this case the fractional error is equal to the sum of the fractional errors
in each of the length measurements. Not all derived quantities will have this simple result,
but often one can estimate the error as a function of fractional errors. Homework problem
#4makes use of some of these ideas.
