LAB 2: NICKELS
DESIGN (D)
PURPOSE
To determine the mass, dimensions, and density of an average Canadian nickel.
To practice using absolute and relative uncertainty in measurements and calculations.
To predict the mass and dimensions of a large number (n > 50) nickels.
To compare experimental values with “true” values by calculating absolute and relative errors.
THEORY
With every measurement, x, there is some uncertainty, Δx, due to the reading limits of the measuring device or the
fluctuations in the value itself (noise). This is also called the absolute uncertainty or reading error.
The relative uncertainty, x , is the ratio of the absolute uncertainty of the measurement, Δx, to the measured
x
value itself, x.
When the same set of conditions are repeated for several trials, the results should be constant. The precision of
these results canbe calculated as half of the maximum range of values. The maximum range is the difference
between the largest value, xmax, and the smallest value, xmin, for a set of repeated trials under the same conditions.
Δ𝑥 =
Uncertainty in calculations (example)
𝑥𝑚𝑎𝑥 − 𝑥𝑚𝑖𝑛
2
Consider the following rectangle whose length, l, is 40.0  0.5 mm and whose width, w, is 10.0  0.5 mm.
When adding or subtracting values
The absolute uncertainty ∆x of the result is the sum of the absolute uncertainties of the values.
The perimeter, P, of a rectangle is the sum of the sides:
P = 2 (l + w) = 2 (40.0 mm + 10.0 mm) = 100.0 mm
The absolute uncertainty of the perimeter, ∆P, could be calculated as follows
∆P = 2 (∆l + ∆w) = 2 (0.5mm + 0.5mm) = 2.0 mm = 2 mm (uncertainty only has one digit)
The perimeter would be expressed as P = (100.  2) mm or (10.0  0.2) cm or (0.100  0.002) m
When multiplying or dividing values
The relative uncertainty, x , of the result is the sum of the relative uncertainties of the values used.
x
The area, A, of a rectangle is the product of the length, l, and width, w, of the sides. A = l  w = (40.0 mm) (10.0 mm) = 400.0 mm2
The relative uncertainty of the area, A , could be calculated as follows

A
l
w
A
0.5 0.5
 


 0.01 0.05  0.06 = 1% + 5% = 6% relative uncertainty
A
l
w 50.0 10.0

A 
The absolute uncertainty of the area, ∆A, could be determine as follows: A    A  0.06(500)  30 mm2
 A 

Therefore, area, A = 500  30 mm2 = (5.0  0.3) cm2
NB
The mean value and its absolute uncertainty must have the same smallest place value (precision).
HYPOTHESIS


As the number of coins increases, their measured values should also increase. The measurements should be directly proportional
to the number of coins measured. Directly proportional relationships are linear that pass through the origin (0,0).

By using a large sample size, the relative uncertainty is reduced. The use of trendlines also reduces the effect of outliers and
further reduces the effects of random errors. The absolute error of each variable (d, h, m) is expected to be less than the
absolute uncertainty of the measuring devices.
VARIABLES
TYPE
VARIABLE NAME, SYMBOL / UNITS
Diameter, d / cm
Thickness, h / cm
Mass, m / g
Independent Number of nickels , n
Constants
Nickels
Dependent
Measuring devices
Environment
METHOD AND COMMENTS
metric ruler, 0.5 mm
metric ruler, 0.5 mm
200 g digital scale,  0.01g
Trials using 1, 5, 10, 15, and 20 nickels
All nickels were Canadian of the same period (1982-present).
Nickels that were noticeably distorted were not used
Nickels were randomly selected for each trial
The same ruler and scale was used throughout.
The scale was zeroed before measuring and the calibration was verified
All measurements were conducted in a closed classroom during a short (2 hour)
interval in which the temperature, air pressure and other environmental factors
remained within a small range.
These factors are expected to have insignificant effects on results.
MATERIALS
Measuring Devices
Other
SAFETY
There are minimal risks in this investigation.
As with all experiments, students are to behave courteously and carefully. Students should wash their hands well
immediately after the completing the experiment. No food or drink should be brought into the lab area.
30 cm ruler (± 0.5 mm) digital scale (200 ± 0.01g)
Canadian nickels (more than 20)
PROCEDURE
Part 1: Examination and separation of nickels
 To ensure similar conditions, all nickels were inspected by group members for age and for severe variations.
 Nickels that were older than 1981 or were visible distorted were excluded from trials.
Part 2: Mass and dimensions of one nickel
 One nickel was selected at random by each group member
 The diameter and height of each nickel was measured using a 30-cm ruler. The data was recorded using ink in Table 1a.
 The mass of each nickel was measured using a 200-g digital scale. The data was recorded using ink in Table 1b.
 This was repeated so that at least 4 different nickels were measured. The data was recorded using ink in Table 1c.
Part 3: Groups of nickels
 Sets of 5, 10, 15, and 20 nickels were selected randomly.
 The diameter and height of each set of nickels was measured using a 30-cm ruler. The data was recorded using ink in Table 1a
and Table 1b, respectively. The diameter was the maximum straight line distance of the set of nickels.
 The mass of each set of nickels was measured using a 200-g digital scale. The data was recorded using ink in Table 1c.
 This was repeated so that at least 4 different sets of each amount of nickels were measured.
Part 4: Data Analysis – using Excel Spreadsheets
 The mean (average), standard deviation (absolute uncertainty), and relative uncertainty was calculated for each data set and
presented in Table 2.
 Graphs of each measurement versus number of nickels were constructed. The graphs were given the following labels: Figure 1:
Width versus Number; Figure 2: Thickness versus Number; and Figure 3: Mass versus Number.
o Data points were plotted with uncertainty bars
o For each, a linear trendline was constructed and its gradient (slope) determined. These values were presented in Table3.
DATA COLLECTION AND PROCESSING
RAW DATA
Table 1a.
Diameter of various numbers of nickels for several (at least 4 repeated trials).
The uncertainty for all diameters was 0.05 cm.
Number of nickels
1
Diameter, d / cm
5
10
15
20
Table 1b.
Thickness of various numbers of nickels for several (at least 4 repeated trials).
The uncertainty for all diameters was 0.05 cm.
Number of nickels
1
Thickness, h / cm
5
10
15
20
Table 1c.
Mass of various numbers of nickels for several (at least 4 repeated trials).
The uncertainty for all diameters was 0.01 g.
Number of nickels
1
5
10
15
20
Mass, m / g
PROCESSED DATA
Table 2.
Mean values and uncertainties of diameter, thickness, and mass for each number of nickels.
Diameter, d
Number of
nickels
Mean Diameter,
d / cm
Absolute
Uncertainty,
∆d / cm
Thickness, h
Relative
Uncertainty,
d / %
d
Mean
Thickness,
h / cm
Mass, m
Relative
Uncertainty,
h / %
Absolute
Uncertainty,
∆h / cm
Mean Mass,
m/g
h
Relative
Uncertainty,
m , %
Absolute
Uncertainty,
∆m / g
m
1
5



10
15
20
Table 3.
Average diameter, thickness, and mass of one nickel taken from gradients of graphs of each measurement versus the number of nickels.
Diameter, d / cm
Mean Diameter,
d / cm
Thickness, h / cm
Relative
Uncertainty,
d / %
d
Absolute
Uncertainty,
∆d / cm

Mean Thickness,
h / cm3
Mass, m / g
Relative
Uncertainty,
h / %
h
Absolute
Uncertainty,
∆h / cm

Mean Mass,
m/g
Relative
Uncertainty,
m , %
Absolute
Uncertainty,
∆m / g
m

SAMPLE CALCULATIONS


For each TYPE of calculation, ONE sample calculation must be shown
o Generally, the first set of data that uses the calculation is used to demonstrate the calculation
o Communicate the variables and formula clearly for the reader
o Substitute using a designated sample set and solve
o Show proper number of significant digits and appropriate units in the final answer
Sample calculations should be shown near the table where the data is presented
Calculations to be shown include, but are not limited to:
Mean diameter, d, for 1 nickel (from data in Table 1a)
x = (sum of all time intervals, xi ) ÷ (the number of samples taken, n)
x
xi

n
Absolute uncertainty for the diameter, d, of one nickel (from data in Table 1a)
Δ𝑑 =
𝑑𝑚𝑎𝑥 −𝑑𝑚𝑖𝑛
2
=
Relative Uncertainty for 1 nickel (from data in Table 1b)
relative uncertainty
= (absolute uncertainty of the measurement) ÷ (measurement value) = x
x
Gradient for the change in diameter per change in number of nickels (from Figure 1: Diameter versus Number)

Use distant points on trendline. Annotate graph to clearly show the two points used.
d d2  d1
change in diameter



change in number of nickels n n2  n1
Absolute Error using published value for diameter*

Error = | Experimental – Accepted1| =
(*cite your source of the published values)
Relative Error, % Error, using published values for diameter
| AbsoluteError |
%Error 
 100%
AcceptedValue

1
Royal Canadian Mint (2014). Industrious, enduring–the 5-cent coin – Technical specifications. Accessed online:
http://www.mint.ca/store/mint/learn/5-cents-5300006#.UvRL2ij8EV8
CONCLUSIONS AND EVALUATION
ATQ (Answer the Question) 
Respond to the purpose of the investigation.

State the experimental value and its uncertainty for the mass, height, diameter and density of
an average Canadian nickel.

Compare experimental values with “true” values by stating absolute and relative errors for each
measurement,

Evaluate whether the absolute errors were within the uncertainty range for the experimental
measurement.
Trends
Identify any trends or anomalies that were observed

Does the age of the coin significantly affect any of the measurements? Explain.

Are the data points closer to the trendline when the number of coins measured is small or large?
Problems/Limitations
Identify significant sources of error & limitations of the investigation

Explain the effect of each significant source of error or limitation on your results

Be specific and provide details. “Human error” is insufficient.
Improvements
Identify reasonable changes to improve the lab. Explain their effect

Explain how the problems identified above can be solved or reduced

Improvements in procedure, materials, organization, and time management can be addressed.
Extension
Make practical predictions from this investigation

Predict the mass of a large number (n = 50) nickels.

Predict the value of 1.000 m long line of nickels