L03%20Overheads

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L03 Overheads

• An Introduction to Uncertainties

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Computing Uncertainties

• There are two types of uncertainties
• Uncertainties associated with measurements.
  These uncertainties are estimated.
• Uncertainties associated with calculations using
  experimental data. These uncertainties are
  computed. To do this, we will use an approximate
  method using Fractional Uncertainties.

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A Simple Example
A bug crawl across a table top. You start your
stopwatch when it starts moving, and take a 10
measurements of its distance of travel, D (in
cm), and its time of travel, t (in s). An
(approximate) average distance value is estimated
from your data to be 26.0 cm, and a corresponding
(approximate) average time value is 37.4 s. You
estimate the uncertainties in these values to be
d(D) 0.2 cm and d(t) 0.4 s. The typical (or
about average) values are thus D 26.0 0.2 cm
and t 37.3 0.4 s.
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Using your data, you then plot a graph of D vs.
t. From the slope of the best-fit straight line
you compute the average speed of the bug to be v
0.6627 cm/s. To find the uncertainty in this
computed value, d(v), we must first compute the
Fractional Uncertainty of each type of quantity
used in the calculation of v, as follows.
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We first find the Fractional Uncertainty in D D
26.0 0.2 cm .
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We then find the Fractional Uncertainty in t t
37.3 0.4 s .
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Note that any type of uncertainty is kept to only
one significant figure. That is, uncertainties
are only recorded to one non-zero digit. Note
also that fractional uncertainties have no
unitsthe units in the numerator must always be
the same as the units in the denominator!
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We have thus far computed the fractional
uncertainties of the quantities involved in our
calculation of v FU(D) 0.008 and FU(t)
0.01. We will always pick the largest fractional
uncertainty to be the fractional uncertainty of
our computed answer. Thus, for v FU(v) 0.01.
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We then use the definition of fractional
uncertainty to solve for the desired uncertainty
in v, d(v) from which we get that d(v) FU(v)
. v 0.01v.
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We then use the previously obtained value of v, v
0.6627 cm/s, to obtain d(v) FU(v) . v 0.01v
0.006627 cm/s. Since all uncertainties are
rounded to only one significant figure, we get
that d(v) 0.007 cm/s. We thus have that v
0.6627 cm/s and d(v) 0.007 cm/s.
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v 0.6627 cm/s and d(v) 0.007 cm/s. Since it
makes no sense for our answer for v to be carried
to decimal places beyond the position of the
non-zero digit in our uncertainty (if were
uncertain here we know nothing about whats
beyond it!), we must round the value for v to the
third decimal place (in this example). We thus
get our final answer v 0.663 0.007 cm/s.
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Our final answer is v 0.663 0.007 cm/s. Note
that the units at the end apply to both the
answer and its uncertainty, so you need not write
the units twice if they are put at the end. Note
also once more that the last decimal place of the
answer is the same as that for the uncertainty.
This must always be the case for experimental
answers.
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ALL uncertainty calculations this semester (and
next!) will work this way. It will take a little
while for you to get used to this procedure, but
it is very straight forward and you will catch on
quickly if you put effort into trying to
understand how it works.
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ALL of our tests will cover fractional
uncertainties in error calculations like what
youll be doing in lab, so the sooner you
understand them and see how to compute them the
better off youll be! (Theyre really not hard!)
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The back of your lab manual has a summary of an
uncertainty calculation very similar to the one
we just worked out. Refer to that summary when
working on your own uncertainty calculations.
Discuss how to compute your uncertainties with
your group partners, and call over your
instructor if youre having difficulty
understanding what to do.