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I’m working on a worksheet that has to be answered based on information given. I provided all the documents that might help and also the Lab steps. Please notice that Q6 will be solved based on the excel picture given. Please read carefully before you accept the assignment
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Measurements & Uncertainties
Physics Lab for Engineers
function. The probe is used in conjunction with an interface box and software to
allow for rapid computer based data collection. The system reads a voltage from the
probe, which is converted into a resistance, and finally using an established equation
it computes a temperature from this resistance (originally a measured voltage).
The physical process enabling us to record temperature is the changing resistance of
our sensor to changes in temperature. This is of course a continuous, analogue,
process. However, in order for the computer to store these readings it must convert
that analogue process into discrete, digital data. This is called analogue-to-digital
conversion.
Numbers are stored by the computer in combinations of bits, where a bit is either 0
or 1. 2-bit data could store four, 22, different number states as either 00, 01, 10, 11.
Most data storage systems will have something like 8-bit (28), 212, 216, 232 capability.
The total amount of storage states, or counts/numbers, we can record is given the
term analogue-digital units or ADUs, these then can be arbitrarily assigned to
whatever real numbers we like.
For example, if you have a sensor that has 8-bit capability you can “sense and store”
28=256 discrete values. This could give you all the whole numbers between and
including 0-255. If you wanted to increase precision to one decimal point you would
end up with a much smaller range, only 0.0-25.5, but still have a total of 256 discrete
numbers.
1. Make sure the Pasco interface box is turned on. Then use the Windows Search
bar to locate and open the Capstone software. Seek help from your instructor to
initialize the software setup. You will choose either the “stainless steel
temperature probe” or the “temperature probe” which appears black.
2. Adjust the sensor sampling frequency to 10-30Hz so we can get a lot of data in
a short amount of time. In the Capstone window choose to display a “Table”
and “Graph” view of the temperature readings (this will populate once you
start collecting).
3. Collect some room temperature data for about 3-6 seconds and stop collecting.
4. Increase the displayed precision of the temperature data table to at least 6
significant figures. If the room and probe are both at a steady state temperature
and in equilibrium you should notice a lot of very similar data- nearly constant.
5. Scroll through the data to find the minimum measurable digital step change
(resolution), in temperature. Record this information on the in-lab worksheet,
and complete the remaining parts of the worksheet for this section.
Remember that the sensor and interface system is converting an analogue physical
sampling process into digital numbers to be displayed and stored. As you should
have realized above, this inevitably always results in some decrease or loss in
resolution and precision since not every possible value can be represented and
stored. Each stored value, ADU, must cover a finite range of analogue, real world
states. In practice, if you must keep the same computing hardware a tradeoff must be
made between resolution (range covered by a bit) and sample space (absolute range
you can measure across).
-2-
Measurements & Uncertainties
Physics Lab for Engineers
Density of a Solid
1. Inspect the solid object provided for this experiment.
2. At your table you have a ruler, Vernier caliper, and triple-beam balance.
Determine the resolution of each device. Also find a good working value for
the precision, or uncertainty, of each device. Record these on the worksheet.
3. Measure the object height and diameter with the caliper and record them.
4. Record your estimate of the uncertainty in the dimension measurements. Units
must be included in all data records.
5. Based on your diameter measurement and uncertainty, determine the object’s
radius and the uncertainty in the radius. Keep in mind this technically involves
propagation of uncertainties (very easy this time).
6. Adjust your triple-beam balance so that it reads zero when it is empty. What
type of error does this eliminate?
7. Measure the mass of the solid object on the triple-beam balance. Estimate and
record the uncertainty along with the units of this measurement.
8. Complete the remaining parts of the in-lab worksheet for this section.
Period of a Pendulum
1. Record 10 measurements of the time it takes for a pendulum to swing away
from its highest point and then to return to that point (one period) using a
stopwatch. Try to keep the same swing arc and of course the same pendulum
string length for each measurement.
2. Using a meter stick, measure and record the pendulum length from the point
where it is fastened at the top down to the best estimate of the pendulum bob’s
center of mass. Don’t forget to include units and your estimate on the
measurement uncertainty.
3. Complete the remaining parts of the in-lab worksheet for this section.
Make sure you complete the in-lab worksheet and turn it in to the instructor before
leaving lab. Also, do not forget to look on Canvas for the post-lab worksheet and
complete it before next week’s lab.
-3-
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Basic Terms and Definitions for Experimental Physics
Updated: 12 January 2017
Uncertainty: Unavoidable imprecision or variation in a value that can be minimized to
some extent depending on available materials, methods, and circumstances. Natural
uncertainty can never be completely eliminated. Uncertainty and error are interchangeable
terms, but using the term error often leads to confusion in everyday speech. Among
scientists the word error is always meant to mean an uncertainty.
Mistake: Examples can include misinterpreting instructions, forgetting a critical step in
procedure, incorrect mathematical calculations, problems with units, etc. Mistakes are
avoidable and fully correctable once discovered; this is in contrast to uncertainties. When
most people hear the word “error” in everyday speech they generally take it as mistake. It
is important to mention mistakes if they occurred and could not be corrected as it helps
explain unexpected results.
Classification of Sources of Uncertainty
Random Intrinsic
Random Measurement
The quantity itself being investigated
changes randomly independent of
whether or not it is actually being
measured.
The measured value of a quantity being
investigated changes randomly due to
uncontrollable variability in the
measurement method or device.
Examples: pulse rate, heartbeat, photons
per second from a star, microscopic
friction varying randomly across a surface
Examples: Volume of a liquid measuring
a meniscus on a graduated cylinder
(parallax effect), digital reading bouncing
between two values in the last digit,
recording from any marked scale in
general.
Systematic Intrinsic
The quantity itself being investigated
changes predictably between
measurements. Such as all equally greater,
or equally lower.
Examples: macroscopic friction causes
predictable energy loss over time,
temperature being physically hotter in
direct sunlight versus in the shade.
Systematic Measurement
The measured value of a quantity being
investigated changes predictably due to
the measurement method or device.
Examples: bias offset, a balance scale is
not properly calibrated to zero, one
thermometer always offset warmer than
another.
In most general introductory lab settings only random measurement errors are easily
accounted for in actual numbers, the other types are much more difficult to discover and
account for or minimize. Systematic measurement errors, if discovered, can be minimized
with improved procedure or finer adjustment of equipment. Random intrinsic errors can
be estimated if many samples under similar conditions are averaged together to see their
variation. Systematic intrinsic errors often go unnoticed until deeper levels of physics or
more variables are accounted for.
Precision: Describes the consistency or reproducibility of measurements among the
measurements themselves, regardless of any expected value. The degree to which
measurements show similar results under similar conditions. Can be used to describe
data recorded or an instrument itself. Can be quantified by absolute uncertainties and
standard deviations. Large random errors will easily destroy any level of precision.
Accuracy: Specifically, how true a measurement is to the ideal or expected value for that
quantity. Can be used to describe data recorded or an instrument itself. Can be
quantified by differences and percent differences. Large systematic errors or biases will easily
destroy any level of accuracy.
In more general everyday usage something said to be “accurate” is normally considered
to be both accurate and precise at the same time, making it the most ideal measurement.
Resolution: The smallest noticeable change in quantity that a measuring device by itself
can clearly display or record. Not necessarily equal to a value’s precision or uncertainty.
Example 1: A meter stick has ruled markings every 1 mm. The resolution of the meter
stick is 1 mm.
Example 2: A digital multimeter can display voltage to 0.1 Volts. The DMM
resolution is 0.1 Volts.
Absolute Uncertainty: The numerical value signifying the amount of uncertainty in
a measurement . Can be a best estimate, a value based off a measurement scale, or
determined from statistical analysis or many measurements in a process.
Example 1: Using an alcohol thermometer ruled every 1°C, you observe the alcohol
column at room temperature to be around 21.5±0.5°C. You feel you can confidently
guess between the ruled markings down to every half a ruling, or every 0.5°C, and
are confident the column lies somewhere above 21°C but below 22°C. Your absolute
uncertainty in temperature is then ±0.5°C. Your thermometer resolution is 1°C.
Example 2: Using a digital balance you measure the mass of a block of metal. The
digital reading displays in kg and displays xx.xx kg (4 digits, two kg decimals). The
display shows 25.15 kg, and every so often it changes to 25.16 kg. You would safely
record the mass to be 25.15±0.01kg and your absolute uncertainty in mass is ±0.01kg
(same as the resolution). Since this is a digital scale it is better to be conservative with
the uncertainty. You cannot read between digital numbers displayed, and you
probably do not know how the manufacturer programed the sensor to round off
numbers.
Relative Uncertainty:
The uncertainty in a value with respect to the value itself. Also called fractional uncertainty.
Percent ‘Relative’ Uncertainty:
∙ 100%
A percentage form of relative uncertainty.
Difference, Discrepancy: = − or =
−
Comparing two values by subtraction that ideally you expected to be the same, thus
finding the discrepancy between them. This could be comparing results from two methods
measuring the same thing, or comparing your result to an expected known reference
value.
Percent Difference:
∙ 100% or
|
,
|
,
∙ 100%
A percentage form of the difference. For comparing to a reference value use it in the
denominator; ideally the reference has much higher precision than your value so it makes
sense to normalize to this expected value. In the case of comparing two methods, it is up
to the experimenter to decide what works best in the denominator.
Mean: ̅ = ∑
If measuring the same thing ( ) many times ( ), and you expect to always get the same
or nearly the same result, you would average the many measurements together into one
overall recorded value ̅ called the mean. This mean value should be a more confident,
more certain, and lofty result since it combines many similar results into one.
Standard Deviation: =
∑
( − ̅ )
If measuring the same thing ( ) many times ( ), which are only affected by random
sources of uncertainty, then you would expect to see the many measurements varying
randomly but centered around a mean value ̅ . The assumption is that the mean value is
the most likely value to occur. The standard deviation is a definition to assign a
numerical value to how far the measurements fall from the mean. This results in the
classic “Bell Curve” distribution where most results (68%) are spread within ±1.0 .
Figure 1: An example “Bell Curve” or Normal Distribution
plotting probability distribution.
The N-1 term is used for small sample sizes, generally N
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