Name:
Lab Partner(s):
Date Lab Performed:
Dr. Julie J. Nazareth
Physics 121L/131L
Section:
Measurements and Uncertainty
Part A: Copper Cylinder
Measure the mass, length and diameter of a copper cylinder and record the data in Table
1. Record the data in grams (g) and centimeters (cm), NOT millimeters. Be sure to record the
measurement to the proper number of decimal places to reflect the precision of the instrument
used (See Do’s and Don’ts of Physics Lab Reports #22). Leave 1-2 extra non-significant digits
for the average because this is an intermediate step. You will round the best estimate in Table 3.
Lab partners should trade off repeating the measurements, starting of from “scratch” each
time like it is the first measurement (making sure the instrument is zeroed). Measurements are
repeated to estimate the sample uncertainty. Although the cylinder may look like it was cut
straight, unless you measure it on both ends and in the middle, you wouldn’t know for certain.
Also, different people read instruments slightly differently, leading to small variations in the
various measurements. Be careful when reading the micrometer – it is easy to misread the
instrument by 0.5 mm (=0.05 cm) causing a large sample uncertainty given the precision of the
instrument.
A mass balance is used to measure the mass (smallest division on instrument = 0.1 g).
The length is measured using a dial caliper (smallest division on instrument = 0.1 mm), and the
diameter is measured using a micrometer (smallest division on instrument = 0.01 mm).
Remember that you can measure to one half of the smallest division if the tick marks are
somewhat widely spaced and you can tell if the measurement is closer to one tick mark, halfway
in-between, or closer to the following tick mark. Many students measure the mass and the length
to one-half of the smallest division. If the tick marks on an instrument are close together, then
you can only measure to the whole of the smallest division. Most students measure the diameter
to the whole of the smallest division. You and your lab partner should decide whether to use one
half or the whole smallest division for each instrument and both use the same precision.
Table 1: Multiple Measurements of the Physical Dimension of a Copper Cylinder
Measurements***
1
2
3
4
5
6
Average+++
Mass,
m (g)
diameter,
d (cm)
length,
l (cm)
***
These measurements should be recorded with the proper number decimal places to reflect the precision of the
instrument used for the particular measurement.
+++
This value is an intermediate step, so keep 1-2 extra non-significant digits. You will round properly when using
the average as the best estimate in Table 3.
Now, take the difference between the average value in Table 1 and each individual measurement.
And record the results in Table 2. You need the differences to estimate the sample uncertainty.
Lab: Measurements and Uncertainty
09/23/2010
Data & Reporting score:
Table 2: Difference between Each Measurement and Average Value to Find Sample Uncertainty
Differences
1
2
3
4
5
6
m – mave (g)
d – dave (cm)
l – lave (cm)
Now look at the six differences listed in Table 2. Choose a difference (absolute value) that
encompasses all or most six values. This is your sample uncertainty. The instrumental
uncertainty is how precisely you can measure (one-half or a whole of the smallest division on the
instrument). Look at how precisely the data are recorded in Table 1 to help you decide the
instrumental uncertainty (instrumental precision). To get the overall uncertainty, choose the
largest of the instrumental or sample uncertainty. Finally record the best estimate (average) with
the overall uncertainty, properly rounded, in the experimental value column. To properly round
a value with uncertainty, round the overall uncertainty to one significant digit. Then round the
best estimate to the same decimal place as the rounded uncertainty. (See Do’s and Don’ts #26)
Table 3: Measurement Uncertainties and Reporting of Experimental Values for Copper Cylinder
Preliminary Uncertainties
Overall
Uncertainty
Instrumental
Sample
Experimental Value

mass, m (g)

diameter, d (cm)

length, l (cm)
Calculation: On an attached sheet of paper, calculate the density of a copper cylinder using
equation 3. Use the rounded experimental values in Table 3 as input into equation 3. Include
uncertainties and units in your calculations. Note: propagating the uncertainty through the
calculation is a major component of your grade for this lab experiment. Follow the rules for
reporting experimental values to round your result and its uncertainty (See Do’s and Don’ts
#26). Put a box around your final answer and record the answer again, properly rounded in
Table 4.
Table 4: Density of Copper Results
Experimental Value
3
±
Density, ρ (g/cm )
Accepted Value
Check to see if your experimental value for the density of copper agrees with the accepted value
within uncertainty. If the result is way off, ask the instructor right away!!! If the result is pretty
close (i.e., differs by ~0.1 g/cm3 or so), but doesn’t agree within uncertainty, think about what
might have affected your experiment, but was beyond your control. Measuring mistakes are
within your control (fix them!). Purity or shape of your copper sample is beyond your control.
Don’t forget to write your summary! (Always start your paragraph with an introductory sentence
stating the goal, reason or purpose in doing the lab. Give your final result for the density of
copper and discuss whether your experimental value agrees with accepted value within
uncertainty. If it is very different, ask the instructor right away! If it is close, but does not quite
agree, then discuss what things beyond your control might have affected your experiment.)
Lab: Measurements and Uncertainty
09/23/2010