PHY132
Experiment
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Measurements, uncertainties and graphs
This lab is a collection of concepts and techniques you will use in all physics labs. It is presented as
a series of exercises.
Uncertainties or errors
Infinitely precise measurements do not exist. All measurements have an intrinsic uncertainty or
error. Uncertainties or errors are not mistakes, but have to do with instrumental and experimental
conditions. The uncertainty of a measurement x is called x. In laboratory, we always strive to
present measurements as x  x.
Estimating the uncertainty of a measurement
Uncertainty is usually estimated from:
A) the limitation of the instrument used to make the measurement: if this is the most important
factor affecting the measurement, then the uncertainty will be given by the value of the
smallest measurement that can be made with the instrument. For example, the smallest
measurement that can be made with a meter stick is x = 0.001 m. Therefore, any
measurement you make with a meter stick will have a  0.001 m uncertainty associated with
it.
or
B) the deviation of repeated measurements: this is best applied when measurements are repeated
many times (>10). In our labs, we will repeat some measurements about three times, which is
not quite enough. But nonetheless, we will perform the standard deviation calculation to
practice the technique. With repeated measurements, the uncertainty is the standard
deviation of the mean, represented by  (so, x =  in this context):
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∑𝑁
𝑖=1(𝑥𝑖 − 𝑥̅ )
𝜎=√
𝑁(𝑁 − 1)
where 𝑥̅ is the average of all measurements xi, and N is the total number of measurements. Then we
̅ ± 𝝈.
can write, for repeated measurements, that our final measurement is 𝒙
Significant digits
Your calculator will often return many more digits than are justifiable. The number of significant
figures in a measurement ends where the uncertainty begins. For example: a ruler has an uncertainty
of  0.1 cm, so your reading can be 20.3 cm, but never 20.35 cm, nor 20.3567 cm, etc! When you
perform mathematical operations with your measurements, it is a general rule of thumb to keep the
same number of significant figures of the least precise quantity, in the final result.
When you change units, you should always preserve the number of significant digits of the original
measurement. Use scientific notation (powers of 10) when necessary to preserve the number of
significant digits.
Rounding numbers: If the digit to be abandoned is 5 or greater, add one unit; if less than 5,
preserve the previous digit.
Sum and subtraction: Keep the number of decimal places of the least precise quantity.
Product and division: Keep the number of significant digits of the least precise quantity.
1. Determine the number of significant digits in each measurement:
a) 5.08 kg
b) 20,570 lb
d) 2.00 × 10-4 s
c) 0.060 cm
2. Determine the precision (uncertainty) of each measurement:
a) 30.6 ft
b) 0.0500 s
c) 18,000 mi
d) 4 × 105 N
3. Round the following numbers until the digit in bold:
a) 12.7558 =
b) 25.34523 =
c) 0.02378 =
e) 0.0547×10-3 =
d) 2789 =
f) 5602.573 =
4. Change the units as indicated (remember to keep the same number of significant digits):
a) 2.5 km →
_____________ m
b) 25 g →
c) 0.1 m →
_____________ km
d) 0.01 kg → _____________ g
_____________ kg
5. Consider the following numbers:
A = (2.34 ± 0.02) m
B = (12.5 ± 0.5) m
C = (2.4 ± 0.2) m
a. Which has less significant digits (less accurate)?
b. Which is more precise?
c. Calculate the following (remember to follow the rules of operations with significant
digits):
A + B + C=
A/C =
A×B=
2
Caliper
To use a caliper, close the jaws lightly on the object to be measured. The bold numbers in the fixed
top scale are in cm, the tick marks are in mm. The bottom sliding scale has tick marks in tenths of
mm. A) Read the top scale in mm at the zero mark of the bottom scale. B) Read the bottom scale:
find where the top and bottom tick marks align, and make a reading in tenths of mm on the bottom
scale at the alignment mark. Thus, we will consider the caliper’s uncertainty as ± 0.1 mm.
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A
2
3
Example: 16.3 ± 0.1 mm
0
5
align
0
B
Read the following measurements, and remember to write the uncertainty.
Reading: ______________________
0
1
0
5
2
0
5
6
7
Reading: ______________________
0
2
0
5
3
4
Reading: ______________________
0
9
0
5
10
11
Reading: ______________________
0
3
5
0
Micrometer
To use a micrometer, gently close the spindle lightly on the object to be measured. Micrometers may
have different graduations (precision scale). In this example, the horizontal numbers in the sleeve are
in mm, the bottom tick marks are in 0.5 mm. The vertical scale in the thimble has tick marks in 0.01
mm. Thus, we will consider the micrometer’s uncertainty as ± 0.01 mm.
Reading: 11.57 ± 0.01 mm
Reading: 3.48 ± 0.01 mm
Reading: 1.32 ± 0.01 mm
Reading: 16.02 ± 0.01 mm
Read the following measurements, and remember to write the uncertainty.
Reading: _____________________
Reading: _____________________
Reading: _____________________
Reading: _____________________
Reading: ______________________
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Exercise: area, volume, uncertainties
For this exercise you will use the percent uncertainty (or error) in measurements. Percent uncertainty
is given by:
𝑒𝑟𝑟𝑜𝑟
% 𝑒𝑟𝑟𝑜𝑟 =
× 100%
𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡
You will be given a certain small rectangular object. Using either the micrometer or the caliper,
perform the following measurements and calculations (remember to write uncertainties and units):
height
Base
Dimensions of the base:
Area of the base:
________ ± ________
% error dimension 1 =
________ ± ________
% error dimension 2=
________ ± ________
% error for area =
How to get the uncertainty for the area? You will use the following approximation: the % error for
the area will be the sum of the % errors in the dimensions. So, error of area = (% error of dimensions
added / 100%)  (value of area).
Height: ________ ± ________
% error of height =
Volume = height × area of base = ___________± ________
% error of volume =
Likewise, the error for the volume will be based on the sum of the % error of area and height. Error
of volume = (sum of % error / 100%)  (value of volume).
Remember that 1 m = 102 cm = 103 mm
1 m2 = 104 cm2 = 106 mm2
1 m3 = 106 cm3 = 109 mm3
Convert the volume into m3. Answer = ______________ ± ________ m3
Before you move on, make sure the values are written with the appropriate number of significant
digits, units (cm, mm, cm2, mm3, you are free to choose) and uncertainty.
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Exercise: standard deviation of the mean
In this exercise you will make repeated time measurements to improve the uncertainty. Sometimes
repeated and careful measurements can lessen the  uncertainties involved, and therefore, increase
the precision of a determination. The new uncertainty is called standard deviation of the mean, .
An apparatus to measure centripetal force will be given to you.
Centripetal force is not the purpose of this exercise, but at the end, we will
provide you with an equation that allows you to check the final average
value obtained. You will need to spin the apparatus by hand, until the
weight is hanging vertically from the string, and making 90º with the
spring (just as in the figure). Once you have reached this condition, keep a
constant speed by giving eventual nudges in the central vertical bar, and
checking the alignment of the weight as you go.
string
spring
weight
Using a stopwatch, measure the amount of time it takes for the weight to complete 10 revolutions
with a constant speed, and keeping the 90º alignment between string and spring. Repeat the
measurement eight times: each person in your group must contribute with a measurement.
Notice that an initial uncertainty of  0.2 s has already been entered in the data table for each initial
measurement: this is the average human reaction time, which greatly overcomes the uncertainty of
the stopwatch used. Average the 8 measurements. Complete the rest of the table.
TIME FOR 10 REVOLUTIONS
N
t ± instrum. uncertainty
1
± 0.2 s
2
± 0.2 s
3
± 0.2 s
4
± 0.2 s
5
± 0.2 s
6
± 0.2 s
7
± 0.2 s
8
± 0.2 s
t - tave
Sum (Σ)=
Average (tave) =
Calculate the standard deviation (σ) as s =
S
N(N -1)
The most probable value of the repeated measurements is now given by
tave ± σ
(t-tave)2
= _____________________________
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Is the standard deviation of the mean , smaller than the initial uncertainty  0.2 s?
In this experiment, have repeated measurements improved the precision in the final determination of
time?
Now let’s compare your value of tave with what it should have been for this experiment, according to
theory:
4𝜋 2 𝑟𝑀
𝑡𝑡ℎ𝑒𝑜𝑟𝑦 = 10√
=
𝑚1 𝑔
where M is the mass of the hanging weight (measure it with a digital scale: make sure to convert to
kilograms), r is the radius of the circular trajectory of the hanging weight (measure it with a ruler
from the central vertical axis: make sure to convert to meters), and g = 9.8 m/s2.
To get 𝑚1 you will need to use the other string that is attached to the weight, and stretch it over the
pulley. Keep adding masses to the string that goes over the pulley until you have reproduced the
same deformation of the spring as when you were spinning the apparatus: in other words, until there
is a 90º alignment between vertical string and horizontal spring (see figure). That mass is 𝑚1 .
r
M
m1
In most experiments, we will want to compare an expected or theoretical value with an
experimental or measured value. The most used formula for the comparison is
% 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 =
𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 − 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑
× 100%
𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑
In general, we aim at measurements with % diff < 5% in respect to expected values.
What is the % diff between tave and ttheory?
If the % diff is negative, then it just means that tave > ttheory. If the % diff is positive, then tave < ttheory.
Which was the case for you, and give 2 possible explanations for it.
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Graphs
Distance vs. Time
Distance (m)
Δt
35
30
25
20
Δs
15
A properly executed graph must contain
all the following elements: 1) title; 2)
labeled axis; 3) units on both axes; 4) a
sensible choice of scale on the axes that
allows for FULL use of the graph paper.
The slope is defined as the change in the
vertical axis (Δs in the figure) divided by
the change in the horizontal axis (Δt in
the figure).
In the graph shown, slope = Δs/Δt.
10
5
0
0
1
2
3
4
5
6
7
8
Time (s)
(In the figure, the dashed lines and
symbols Δs, Δt are shown for teaching
purposes, but you will not show that in
your graphs.)
A car is observed to be moving in a straight line. The following positions are recorded at the times
shown. Assume the position can be observed with an uncertainty of  10 m, and the time
measurement uncertainty is  0.5 s.
Time (s)
5 10
15 20 25 30 35 40
Position (m) 124 152 175 202 225 245 275 305
Plot a position vs. time graph, including vertical and horizontal
error bars on each data point.
Find the best fit straight line through the data using a ruler. Never connect the dots. This line is a
visual estimate of the average behavior of all your data points. It is not required to pass on top of
data points; in fact, sometimes it can miss them all. Avoid adjusting a line by using only the first and
last data points as guides: you are bound to make a mistake this way, as not all distributions of points
are alike. Once a line of best fit is drawn in a graph, then the slope of your data must be calculated in
respect to positions along that line.
Calculate the slope of the line of best fit. Use positions that are exactly on the line of best fit.
Indicate on the graph the positions used to calculate the slope. Show detailed, step-by-step
calculations on the back of the graph.
Slope, with units = _________________________
ASSIGMENT DUE IN 1 WEEK
Your assignment must include the following, in this order: 1) cover sheet with title of
experiment/activity, date, your name, instructor’s name, course & CRN; 2) completed 8 pages of this
handout, 3) the manual plot required at the end.
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