Title: The formula for error analysis'
1The formula for error analysis.
Error propagation step by step is a long a
tedious procedure Are there any shortcuts?!
Actually, there are. For any function
of independent variables
I am not sure, you see THIS as a shortcut, but it
is a kind of.
2Step by step or partial derivative? Which way of
suffering is nobler?
Going step by step would be pretty easy, BUT we
CANNOT do it!
Example
Because at least one variable, x, appears in the
equation more than once! And the two xs are NOT
independent variable, because they are the SAME
variable.
In a case like that, you absolutely have to use
the partial derivative way!
3Experiment 2
- Devise a simple, fast, and non-destructive method
to measure the variation in thickness of the
shell of a large number of racquet balls in
shipments arriving at a number of stores, to
determine if the variation in thickness is much
less than 10. - Devise a method to measure the thickness and
density of the inner and outer cylinders without
damaging them so that rods outside 5 tolerance
will not be used in a machine.
Relative
Enough accuracy with better speed
Absolute
best accuracy
4Whats the Point
- This is an experiment about using repeated
measurements to determine the accuracy of a
measurement technique. - Experimental methods can be modified and improved
in light of the result of repeated measurements. - We should learn to use averages to improve the
accuracy of our results.
5Racquet Balls
R
d
The counterfeiters make balls with the same mass
and the same average moment of inertia, I, but
have worse quality control on the thickness, d,
and hence on I. We are looking for a larger
spread in d implying a spread in I.
6The Rods
The rods have materials of two densities. The
radii can be measured with a caliper. The total
mass can be measured. What we are checking is
the density. We want our measurement to be
accurate, even if it is time consuming.
7Moments of Inertia
- Both problems can be solved by measuring the mass
and moment of inertia of the objects. - For the balls, we only need to measure the
variation in thickness but do it quickly. - For the rods, we need absolute measurements but
have more time.
We want the error of measurement of the wall
thickness, R-r, to be much less than 10.
You should try to exercise propagating errors of
these quantities while preparing for the lab.
We want to measure both densities to 5.
8Rolling ball
9Rolling ball
You will test a number of balls, measure radius
of each of them and calculate
For each ball, you will carry out multiple
measurements of the time interval, t1, between
the two photogates.
How do we correct for the non-zero distance,
x1? The motion along the rail is with an even
acceleration, a. Therefore, for displacement as
a function of time we have
t1 relates to x - x1 t relates to x therefore
10How to use the measured parameters (time and
geometry) to calculate I?
Energy conservation.
Rolling radius.
For uniform acceleration.
We finally arrive at
Plug.
Solve for I.
11Repeating Measurements
- The errors on rolling time and pendulum period
will likely be bigger than the smallest division
on your timer. - You will need to measure repeatedly to find out
what the error is. - You may also have to measure repeatedly to reduce
the random error.
Mean time for one ball after n measurements, t1,
t2, , tn, - our best estimate of the actual time.
How many times should you measure to get the SEM
to 30 (0.3) of the SD?
Standard deviation, SD, a measure of scatter of
the individual data points, t1, t2, , tn, and
our estimate of error of an individual measurement
Standard deviation (error) of the mean (SEM)
our estimate of error of the calculated mean
value, based on the random uncertainties of a
large number of individual measurements.It can
be reduced to minimum by repeating the
measurements.
12We have
Typical value for the balls are d 4.5 mm and R
28.25 mm, r 23.75 mm
Lets take d 4.4 mm we obtain for it r 23.85
mm and
13We have
We needed to propagate errors for a complex
function
The method we used was to perturb the argument
near a known value, to calculate the change in
the function and the relative values of the two.
It can be applied in general and is exactly the
same as calculating
We needed to know and plug in the actual
approximate values of d and R. The answer would
be substantially different for d 10 mm.
14Propagate Error from Time to I
We have got and
Go straightforward
Finally, we obtain
The error analysis indicates that to obtain d
with a precision of 5, we need a precision of
0.2 in t.
15It is easy to calculate from
It is convenient to introduce ,
plot and solve the equation graphically
corresponds to
and
16In practical terms as long as the setup and the
intended ball starting point of do not change,
only random errors of photogate time are
important. To make a conclusion, whether the
thicknesses of different balls are uniform within
10, it is desirable to measure individual shell
thicknesses to 3 error. The error of 3 in ball
thickness, d, implies an error of mean time as
small as 0.1.
We are only trying to find differences between
balls, therefore, many errors can be
ignored. Only the measured rolling time or
pendulum period are important. We must measure
one ball many times to determine the measurement
error. We must measure many balls of each type
experimentally determine the spread in
thickness. Propagate error on I into error on
thickness. There are physical limits on I.
Should be ltlt 10