Lab 1: Measurement and Uncertainty
Name: __________________________
Goals:
- Use various instruments to measure a broad variety of physical quantities.
- State the uncertainty in measured and derived quantities.
- Compute uncertainties due to random error.
Types of Uncertainty:
Limitations of the instrument:
Measurements are only as precise as the instruments we use. The precision of every instrument – triple beam scales, tape
measures, stop watches, etc. – is limited by its smallest division. For example, our meter sticks have a smallest division of 1
mm, so we can only measure the length of an object to roughly the nearest millimeter. We might agree that the best way to
take a measurement with a meter stick is to state the length of an object to the nearest millimeter and attach an uncertainty
of 1 mm. For example, a pencil might measure 15.4 ± 0.1 cm (0.1 cm is the same as a millimeter). This uncertainty is a very
conservative estimate – we might choose instead to measure lengths to the nearest half millimeter, then state the
uncertainty as 0.5 mm. In this case, we might write 15.40 ± 0.05 cm (note that the uncertainty addresses the last place
value in the measurement).
The rules for quantifying uncertainty are somewhat arbitrary and may differ across disciplines. For example, in chemistry
courses I was taught to “eyeball” graduated cylinder measurements to the nearest fifth of a division, then state the
uncertainty as one fifth of a division. This technique is perfectly valid for a large graduated cylinder but reckless for a small
one! In this class, our general rule is “Use the uncertainty stamped on the instrument. If there is no uncertainty stamped on
the instrument, then use the size of the smallest division as your uncertainty unless otherwise noted.”
Random variation:
Another type of uncertainty arises when we take repeated measurements from a population. For example, every student
will measure the volume of a different penny in this lab. The pennies should all have the same volume, but all the pennies
are slightly different due to their different histories. When we see this kind of variation, the goal is to approximate the mean
volume of the population (all the pennies in the world) by using the sample (all the pennies measured in this classroom).
In practice, we compute the uncertainty in the mean by using technology. First we compute the standard deviation, s, for
s
the sample, then the uncertainty is crudely approximated by the formula
, where n is the sample size. Our
n
s
approximation for the mean volume of pennies is x ±
, where x is the mean of the sample. When you take an
n
introductory statistics course this will all be covered more generally when you talk about confidence intervals.
Absolute and Relative Uncertainty:
For a pencil with length 15.4 ± 0.1 cm , we say that the absolute uncertainty is 0.1 cm. The formal notation for the length
measurement is L ± D L , where L stands for the length, and D L stands for the absolute uncertainty in length.
It is also common to compute the relative uncertainty in a measurement in order to compare the uncertainty to the overall size
DL
0.1 cm
of the measurement. For our pencil, the relative uncertainty is computed as
or
» .006 or 0.6% . This means that
L
15.4 cm
our uncertainty is less than 1% the length of the pencil; i.e., we have committed less than 1% error by measuring the pencil with
a meter stick.
Uncertainty in Derived Quantities:
When two or more measured quantities are combined algebraically, their uncertainties combine according to the
formulas for propagation of error shown below. We will only use the following three formulas (each formula is written
supposing the quantities x and y have uncertainties D x and D y ).
 x 
- Error in a sum or difference:
If z = x ± y , then z 
- Error in a product or quotient:
If z = xy or z = x / y , then
- Error in a product or quotient with exponents:
If z = x m y n , then
2
  y 
2
z
 x   y 
    
z
 x   y 
2
z
 x   y 
 m   n

z
 x   y 
2
2
2
1. Use the Vernier calipers to measure the thickness and diameter of a penny. Each student should use a different penny! Be sure
to include the uncertainty of each measurement. Use an uncertainty with only one significant digit addressing the last significant
digit of the measurement.
Thickness: ______ ± ______ cm
Diameter: ______ ± ______ cm
2. Use the results from Part 1 to compute the volume of your penny. You will have to rework the volume formula in terms of
diameter, and you will have to use a propagation of error formula to find the uncertainty. Show all your calculations clearly!
Volume: ______ ± ______ cm3
3. Add your penny volume to the list on the blackboard (you don’t need to include uncertainty with your volume). Once the entire
class has added their penny volumes to the list, compute the mean volume of the pennies and compute the uncertainty due to
random variation.
Mean Volume ( V ): ______ cm3
Uncertainty in Mean Volume ( D V ): ______ cm3
4. Use a tape measure to measure the area of the west blackboard in square meters. You will have to use another propagation of
error formula to obtain the uncertainty! Again, show all the relevant calculations clearly.
Length: ______ ± ______ m
Area: ______ ± ______ m 2
Height: ______ ± ______ m
M
. Grab a metal sample from the box and use a triple beam scale
V
and graduated cylinder to measure its density. Be sure to include the correct uncertainty for each quantity, measured or
derived.
5. Density is the ratio of an object’s mass to its volume:  
Object Color: ___________________
Object Mass:
______ ± ______ kg
Object Volume:
______ ± ______ m 3
Object Density:
______ ± ______ kg/m 3
(use the “one fifth of a division” rule for the volume measurement).
Your metal sample should be one of the following:
Metal
Iron
Description
Dark grey, may show rust
Density
7874 kg/m 3
Lead
Dark grey, very soft
11340 kg/m 3
Aluminum
Light grey
2700 kg/m 3
Brass
Yellow
8600 kg/m 3 (approx.)
Identify your sample and compute a percent error. Percent error is usually calculated as
Percent error: _______________
actual - accepted
×100 .
accepted
1 2
gt , where g is the
2
acceleration caused by gravity (approximately 9.8 m/s2 ). This relationship can be used to measure your reaction time! Have a
lab partner hold a meter stick vertically so that it’s lower end is just between your thumb and forefinger. Your partner should
release the meter stick without warning, and you should catch it as fast as you can. Measure the distance that the meter stick
falls on each trial and record your results for ten trials below.
6. When an object is freely falling from rest, the fall distance is related to the fall time by the formula d 
Drop distance for ten trials (cm):
_____
_____
_____
_____
_____
_____
_____
_____
_____
Average drop distance: _____ _ cm
Solve the formula d 
1 2
gt for t so you can compute your reaction time. Show your work below:
2
Formula solved for t: _____________
Now you can plug in your average drop distance to obtain your reaction time.
Your average reaction time: ________ s
Add your reaction time to the chart on the chalkboard.
_____
7. The acceleration caused by gravity (g) can be measured by timing the free-fall of an object. I will escort one group at a time to
the 400 building, and each group will collaborate to drop 10 bean bags from the balcony. The height of the balcony and the
time for each fall need to be measured. Important note: make sure each bean bag drop is a surprise to the timer! It is much
easier to correct for the timer’s reaction time than it is to coordinate the drop perfectly.
Height (with measurement uncertainty): ______ ± ______ m
Time for fall (with uncertainty of random variation): ______ ± ______ s
Now solve the formula d 
1 2
gt for g. Show your algebra clearly.
2
Free fall formula solved for g: ___________________
Finally, compute the experimental value of g including the uncertainty. You will have to use an error propagation formula to get
the uncertainty. Show your work clearly.
Measured value of g: ______ ± ______ s
Hopefully this interval captures the accepted value of g!
Compute the percent error between the experimental and accepted values of g. Typically we are trying for less than 10% error
in this class, but 20% is acceptable. Show your work.
Percent error: __________
What would have happened to your result for g if you hadn’t corrected for reaction time? Would it be consistently bigger or
smaller than the corrected value? Justify your answer algebraically.
_____________________________________________________________________________________________
_____________________________________________________________________________________________