Measurement Uncertainties
Winter 2016
Contents
Equipment
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Goals of the experiment
Background
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Warm-up Questions
The measurements
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Equipment
Metric ruler, calipers, 2 rectangles with different lengths, 2 circles wih
different diameters, ball bearing, string, digital balance, sextant, tape
measure, demonstration vernier1 .
Not all the equipment listed will be
used in this lab.
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Goals of the experiment
To make simple measurements with various instruments, some of which
may be unfamiliar, and report the results with an estimate of the uncertainty, following established procedures.
Background
The instruments
Calipers
Calipers are used to measure the width of an object or a gap. The interesting feature of this instrument is the Vernier scale attached, which
allows measurements to be made with a precision of 0.1 mm. See the
Wikipedia article http://en.wikipedia.org/wiki/Vernier_scale for
a description of its usage.
Sextant
A sextant is an instrument used to measure angles with respect to the
horizontal. The sextant we will be using in this lab is shown in Figure
1.
It requires some assembly, because it actually consists of an inclinometer, to measure the angle with respect to the horizontal, and a
Figure 1: A sextant
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sighting tube which fits into a v-shaped groove in the base of the inclinometer.
Digital balance
Figure 2 shows the digital balance to measure masses with.
The instrument should be zeroed (big button labelled “ZERO”) before making a measurement.
Electrical measuring instruments
One of the challenges proposed is to measure a resistance by measuring the voltage across, and the current through, a resistor, and then
using Ohm’s law to calculate the resistance.
A voltage, or potential difference between the ends of a resistor
drives an electric current through it, in the same way that a height
difference between the ends of a slope drives a toboggan current down
the slope.
A voltmeter measures the voltage between the ends of an element
in a circuit, like a resistor. A voltmeter has two terminals, and each
one is connected to one end of the resistor. Figure 3 shows the setup.
An ammeter measures the current through an element in a circuit.
The ammeter also has two terminals, but instead of connecting them to
each end of the resistor, they are connected in such a way that the ammeter effectively replaces a wire in the circuit. Current passes through
it as if it was just a wire, and the ammeter measures it. Figure 4 shows
the setup.
A multimeter contains an ammeter and a voltmeter packed in a
single instrument. So in this lab we’ll be using two instances of an
Anatek multimeter, shown in Figure 5.
Notice the three holes immediately to the right of the display, below the buttons. To measure voltage, the two relevant terminals are
the middle one, labelled “Common”, and the one to the right of that.
To measure current, the two relevant terminals are again the “Common” one, and the one to the left of that. The buttons are used to
select what to measure, and the appropriate range. In the figure, the
button labelled V is pressed, and the button labelled “20”, to measure
a voltage of up to 20 V.
Although the multimeter does have a digital display, you don’t have
to use half the smallest significant digit to estimate the uncertainty. On
the back of the instrument, the manufacturer specifies the measurement uncertainties.
Figure 2: A digital balance
Figure 3: Measuring a voltage with a
voltmeter
Figure 4: Measuring a current with an
ammeter
Figure 5: A multimeter
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The voltage source
To establish a voltage across the resistor, a voltage source is required.
This could be, for example, a battery like the one shown in Figures 3
and 4. A disadvantage of a battery, however, is that it only offers one
voltage value. In an experiment, we would like to make the voltage
variable, so we use a variable voltage source like the one shown in
Figure 6. The black and red terminals are equivalent to the ends of a
battery and should be connected to each end of the resistor to provide
a voltage across it. The left knob controls the voltage, and the dial
above it shows the value of the voltage currently being supplied. It is
prudent to turn the knob all the way counterclockwise before powering up the voltage source, to start off with a value of zero volts, and
gradually turn it up to the desired value. The dial on the right shows
the current being supplied. In principle, we could measure the voltage
across the resistor and the current through it using these dials, but as
you will see, the precision of the dials on the voltage source is much
less than that of the multimeters, which is why we use them instead.
Estimating and reporting measurement uncertainties
Please see the handout "Measurement Uncertainties" by A. Louro.
Warm-up Questions
1. If you measure a length of 7.5cm with an uncertainty of 1mm, what
is the relative uncertainty?
2. If you measure a volume of 1.2m3 with a relative uncertainty of
1.5%, what is the absolute uncertainty of the volume measurement?
3. You have been asked to measure the period of a pendulum. Explain
why it is better to measure the time of 100 periods, and divide by
100.
4. A paint can contains a volume V = (1.00 ± 0.03)L of paint. The
label on the can says that 1 coat of paint covers an area A = 21m2 , to
which we can assign a 10% margin of error. Estimate the thickness
t of 1 coat of paint, with its uncertainty.
The measurements
Here is a list of measurements to make. In this section, we just list the
objects and the instruments to use, and in some cases some comments
on the procedure.
Figure 6: A variable voltage source
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measurement uncertainties
Rectangular plates and metal sphere
You will be provided with a container filled with a few objects of various shapes and sizes. The contents are shown in Figure 7.
• Pick one of the rectangles and measure its area using the calipers.
• Measure the density of the sphere using the calipers and the digital
balance.
Figure 7: Objects to be measured
The height of a tall object
You can measure the height of a tall object by the simple method illustrated in Figure 8.
Measure θ with the sextant, and L with a measuring tape, then
calculate h by trigonometry.
Figure 8: Measuring a tall object
• Pick a tall object (could be a building) and measure its height using
the sextant and the measuring tape.
A resistance
• Pick one of the three resistors supplied and measure its resistance
by applying a voltage across it and measuring the voltage and the
current with a multimeter. Consult with your TA about the magnitude of the voltage to apply, and make sure your TA approves your
wiring before powering on the voltage source.
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