Even #1
Physics 23 Laboratory Report
Experiment E1:
Measurement and Errors
Prepared By: ____________________
Partner: _______________________
Lab Instructor:
Lab Section:
Recitation Instructor:
Date:
Recitation Section
Sp 07
Remarks by Grader:
Grade:
Lab E1 Measurement and Errors
This laboratory exercise differs from the ones to follow. This exercise reviews tools we
will need for the rest of the semester.
The Goals for this laboratory are:
Learn to use Vernier Calipers
Learn the precision of measuring instrument
Review Standard Deviation
Learn Least squares Fitting using EXCEL
Learn how to calculate the error in calculated quantities
Review use of Excel
1. Using your Vernier calipers, measure the three dimensions of your aluminum block
(record with units)
L=____________________
W=___________________
H=____________________
2.
a) What is the smallest difference in length you could measure with a meter
stick?
∆ℓ=_________________________
b) What is the smallest difference in length you could measure with your Vernier
calipers? ∆ℓ=_________________
3. Often we make only a few or even just one measurement of a quantity. In this case, we
shall use the smallest difference we can measure as an estimate of our measurement error,
∆x in the quantity x. We will write x = 1.03 + .02 cm, for example, when we measure x
with a device which has 0.02 cm as its smallest measurable difference.
If we measure a reasonable number of samples, x1, x2,…xn, with say n > 10, then we
can use the standard deviation, s, as the measurement error. The standard deviation is
defined by
 x
n
sx =
i 1
i
x

2
nn  1
Where x is the average of the n measurements. Excel uses the function STDEV to
calculate sx. We shall use sx as estimate of our measurement error: x +sx .
Usually, we will not make enough measurements to calculate sx and will use the
method in the paragraph above.
4. How do measurement errors affect quantities calculated using the measurements?
For example, we measure two sides of a metal plate and calculate the area. What
is the estimated error in the area?
a
a
b
b
We have measured the length of the two sides of a rectangular plate and wish to calculate
its area. One side was found to be length a and the smallest division on our ruler was a,
so we claim that the length of this side is a ±a. Similarly, for the horizontal side we find
a length b±b. We suspect that our best estimate for the area of the plate is A = ab, but
what error bars should we attach to A? At look at the diagram indicates that the smallest
area would be AMIN = (a-a)*(b-b)=ab - ab - ba + ab  A- ab - ba  Similarly,
the maximum expected value for A is AMAX A + (ab + ba). We have assumed that a
<< a and b << b and hence we have neglected the product ab. If this is not the case,
we do not really need to worry about errors in A. They are large. We can summarize
with
A   A  ab  (a b  b a ) eq (1)
so that
 A  a b  b a eq (2)
It is convenient to use the fractional or relative error in quantities, such as the fractional
error in a: a = (a/a). Dividing eq (2) by A = ab
A    a  b    
A
b
A a
a b
Hence, we do a flying generalization and conclude that the relative errors add for
multiplication. It can be shown the same is true for division. Convince yourself that for
addition, the errors, NOT the relative errors, add.
For your aluminum block measurements, what is ∆ℓ? Δℓ = _______________
Hence, the measurements with your error estimates can be written in the form x±∆ℓ and
are:
L=____________________
W=___________________
H=____________________
Now calculate the relative errors:
ρL = ∆ℓ/L = _________________________
ρW = ______________________________
ρH = _____________________
Calculate the volume of your aluminum block: V = _____________________
Calculate the relative error in your calculate volumes: ρV = _________________
Calculate the error in volume: ∆V = ______________________________
Hence you would report the volume you determined to be V±∆V = _________________
PART TWO: Free Fall
Use the EXCEL file E1-Part 2-FreeFall.xls for part two and then staple the two separate
print outs together.