3 model report

MAE 3005 LAB 3

Uncertainty in mass, volume and density

Dr John Adams

Abstract Finding the composition of U.S. pennies.

MAE 3005 Measurements Lab Page 1 Lab 3

Contents Objective .................................................................................................................................................................1

Procedure ...............................................................................................................................................................1

Apparatus ...............................................................................................................................................................1

Theory .....................................................................................................................................................................2

Data/Results ...........................................................................................................................................................3

Part 1 - Coins .......................................................................................................................................................3

Part 2 - Mass .......................................................................................................................................................3

Part 3 – Volume ..................................................................................................................................................3

Questions ................................................................................................................................................................4

More data/results ...................................................................................................................................................5

Part 4 – Density ..................................................................................................................................................5

Calculations ............................................................................................................................................................5

Composition .......................................................................................................................................................5

Accuracy .............................................................................................................................................................5

More questions ......................................................................................................................................................5

Conclusion ..............................................................................................................................................................6

Discussion ...............................................................................................................................................................6

Bibliography ................................................................................................................................................................6

Objective

The objective of this lab is to determine the composition of pennies minted pre-1982 and from 1982-onwards.

Procedure

See manual. No alterations necessary.

Apparatus

10 pre-1982 pennies electronic balance electronic caliper

10 1982-on pennies rule graduated cylinder


Theory

Volume 𝑉 = 𝜋 (𝐷

2)

2𝐻

Uncertainty (Δ𝑉)2 = (𝜋 (𝐷

2)

2Δ𝐻)

2

+ (𝜋𝐷

2𝐻 Δ𝐷)

2

True values for the compositions are [1]

Composition (fraction)

Date Copper (Cu) Zinc (Zn)

Pre 1982 0.95 0.05

1982 - present 0.025 0.975

Density calculation 𝜌 =𝑚

𝑉 where ρ = density in g/cm3

m = mass in g

V = volume in cm3 (note that 1 cm3 = 1 mL)

Uncertainty (Δ𝜌)2 = (1

𝑉Δ𝑚)

2+ (

𝑚

𝑉2 Δ𝑉)2

Density of metals [2] Cu 8.960 g/cm3 Zn 7.133 g/cm3

Composition 𝜌 =% Cu

100 𝜌Cu +

% Zn

100 𝜌Zn

Let 𝑥 =% Cu

100= fraction of Cu, then 1 − 𝑥 =

% Zn

100= fraction of Zn

𝜌 = 𝑥 𝜌Cu + (1 − 𝑥) 𝜌Zn

𝑥 =𝜌−𝜌Zn

𝜌Cu−𝜌Zn

Uncertainty (Δ𝑥)2 = (Δ𝜌

𝜌Cu−𝜌Zn)

2


Data/Results

Part 1 - Coins

1. Pre-1982 coins = 10

1982-on coins = 10

Part 2 - Mass

2. Mass using a single coin.

Pre-1982 1982-On

Single coin mass (± 0.05 g) 3.1 2.5

3. Mass using average of 10 coins.

Pre-1982 1982-On

Mass of 10 coins (± 0.05 g) 31.0 24.9

Average mass of 1 coin (± 0.005 g) 3.10 2.49

Part 3 – Volume

4-7. Volume using a rule and averaging 3 coins.

Pre-1982 1982-On

Thickness (± 0.5 mm) 1.2 1.2

1.3 1.1

1.4 1.3

Average thickness (± 0.5 mm) 1.3 1.2

Diameter (± 0.5 mm) 18.5 18.9

18.5 18.5

18.5 18.7

Average diameter (± 0.5 mm) 18.5 18.7

Volume (mm3) 349 ± 136 330 ± 138

Volume (cm3) 0.35 ± 0.14 0.33 ± 0.14


8-10. Volume using a rule and 10 coins together.

Pre-1982 1982-On

Height of 10 coins (± 0.5 mm) 14.8 14.5

Average height (± 0.05 mm) 1.48 1.45

Length of 10 coins (± 0.5 mm) 190.2 190.1

Average diameter (± 0.05 mm) 19.0 19.0

Volume (mm3) 420 ± 14 411 ± 14

Volume (cm3) 0.42 ± 0.02 0.41 ± 0.02

11-12. Volume using electronic calipers and a single coin.

Pre-1982 1982-On

Height of 1 coin (± 0.005 mm) 1.49 1.47

Diameter of 1 coin (± 0.005 mm) 19.06 19.06

Volume (mm3) 425 ± 1.4 419 ± 1.4

Volume (cm3) 0.425 ± 0.002 0.419 ± 0.002

13-15. Volume by displacement of water

Pre-1982 1982-On

Volume without coins (± 1 mL) 66.2 40.0

Volume with coins (± 1 mL) 70 43.9

Difference (± 2 mL) 3.8 3.9

Average volume of 1 coin (± 0.2 mL) 0.38 0.39

Volume (cm3) 0.4 ± 0.2 0.4 ± 0.2

Questions

1. Mass measurement using 10 coins has the least uncertainty

as the resolution of the balance is averaged over the 10 coins used.

2. Mass measurement using 1 coin has the most uncertainty.

3. Volume measurement using the calipers has the least uncertainty

as the resolutions of the caliper is much better (finer measurement) than using the rule.

4. Volume measurement using displacement of water has the most uncertainty

(due to poor resolution of graduated cylinder).


More data/results

Part 4 – Density

Density calculated using most imprecise measurements

Pre-1982 1982-On

Density using most imprecise measurements (g/cm3)

7.8 ± 3.9 6.3 ± 3.1

Density using most precise measurements (g/cm3)

7.29 ± 0.04 5.94 ± 0.03

Calculations

Composition

Pre-1982 1982-On

Composition using most imprecise measurements 0.4 ± 2.1 Negative value

Composition using most precise measurements 0.09 ± 0.02 Negative value

Accuracy

Percentage error for pre-1982 = 70 % (for the most precise estimate)

This is very poor accuracy.

More questions

5. The true value of the composition for pre-1982 lies outside the experimental range. 0.95 ∉ [0.07, 0.11].

The experimental value is too small, indicating that the measured density is too small.

This could indicate that the measured mass was too small or the measured volume was too large.

The true value of the composition for 1982-on also lies outside the experimental range, since the

experimental result was negative.

The experimental value is too small, indicating that the measured density is too small.

This could indicate that the measured mass was too small or the measured volume was too large.


Conclusion

The measured density for pre-1982 coins was 7.29 ± 0.04 g/cm3 for the most precise measurements.

The measured density for 1982-on coins was 5.94 ± 0.03 g/cm3 for the most precise measurements.

The calculated composition for pre-1982 coins was (9 ± 2) % for pre-1982.

The calculated composition for 1982-on coins gave a negative result.

Discussion

The precision in this experiment was very high due to the good resolution of the instruments used, and also due

to the ability to average over a large number of coins.

The accuracy of this experiment was very poor, with the true values lying outside of the experimental range for

both set of coins.

A large source of error in the volume of the coins is that the thickness is measured at the edge, which is raised.

This causes the volume measurements to be too large and the density to be too small.

To improve the accuracy, a very large number of coins (100) should be used with the volume measured by

displacement using a graduated cylinder with much greater resolution.

One must be careful to avoid creating bubbles when placing the coins in the water, as these will also add to the

measured volume of the coins.

The error in the mass is likely to be insignificant in comparison to the error in the volume, so no improvement in

the method is necessary. However, averaging over 100 coins would increase the precision.

Bibliography

[1] "United States Mint," [Online]. Available:

http://www.usmint.gov/about_the_mint/fun_facts/?action=fun_facts2.

[2] "National Instite of Standards and Technology," [Online]. Available: http://physics.nist.gov/cgi-

bin/Star/compos.pl.

3 model report

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