In Physics 1C03, we explored different types of uncertainties (if you did not take Physics 1C03, or you need a refresher, have a look at the uncertainty labs in the Physics 1C03 Lab Manual). In this lab, we will be exploring the propagation of uncertainties, i.e. if we do calculations using measured values with uncertainties.

Suppose that you wanted to measure the density of a metal washer.   A quick check with a Vernier caliper reveals that the washer is not uniformly round.  In order to get a good estimate of its dimensions, each person in your group is going to take an independent measurement of the mass of the ring, and the outside diameter, inside diameter, and thickness, in different places around the ring.  You then pool your data as described below… (equipment reading (rdg) uncertainty is 0.02 mm for calipers and 0.05 g for the scale)

Outside diameter                                

Measure the outer diameter of the washer:

 

[latex]D_1 =[/latex] ___________,   ___________,   ___________ mm

 

First, we have a source of error from reading the equipment:     ±    __________ mm

 

Instead of giving the full uncertainty interval, let’s give the equipment standard uncertainty (remember the different probability distribution functions?):            ±    __________ mm

 

Another source of error comes from the spread in the data when taking repeated measurements (think about rolling dice):         ±    __________ mm

 

Now let’s average the three measurements and calculate the error on this mean (remember the standard uncertainty on the mean calculation!):

 

Average D₁    =    __________    ±    __________ mm

 

 

Now measure the inner diameter of the washer:

 

Inside diameter                                          

[latex]D_2 =[/latex] _______,    _______,    _______

equipment reading uncertainty = ± _______ mm

equipment standard uncertainty = ± _______ mm

repeated measurement uncertainty = ± ________ mm

 

Average D₂    =    __________    ±    __________ mm

 

Next measure the thickness of the washer:

 

Thickness                                                                            

[latex]t =[/latex]_______,    _______,    _______

equipment reading uncertainty = ± _______ mm

equipment standard uncertainty = ± _______ mm

repeated measurement uncertainty = ± ________ mm

 

Average t       =    __________    ±    __________ mm

 

 

And finally the mass of the washer:         

 

Mass                                                                                               

[latex]m =[/latex]________,    ________,    ________

equipment reading uncertainty = ± _______ g

equipment standard uncertainty = ± _______ g

repeated measurement uncertainty = ± ________ g

 

Average m     =    __________    ±    __________ g

 

 

Change the length units to cm and summarize your average values here:

Average D₁    =    __________    ±    __________ cm

 

Average D₂    =    __________    ±    __________ cm

 

Average t       =    __________    ±    __________ cm

 

Average m       =    __________    ±    __________ g

 

The volume of the annulus will be calculated as      [latex]V=\frac{\pi}{4}(D_1^2-D_2^2)t[/latex]

 

[latex]D_1^2 =[/latex] ___________    ±    ________   cm²

 

[latex]D_2^2 =[/latex] __________    ±    ________ cm²

 

[latex]D_1^2 - D_2^2 =[/latex] __________     ±    ________ cm²

 

[latex]V =[/latex] __________     ±    ________ cm³ 

 

[latex]\rho =[/latex] __________      ±    ________ g/cm³

 

The densities of various metals are provided in the table below.  Comparing to these values, what would you surmise that your washer is made of? (Have your answer ready to discuss with your lab instructor.)

 

material               density (g/cm³)

aluminum                    2.702

arsenic                        5.727

antimony                    6.685

zinc                              7.134

tin                                7.287

iron                             7.875

brass                           7.913

nickel                           8.912

copper                         8.933

bismuth                       9.800

silver                           10.501

lead                             11.342

gold                             19.282

platinum                     21.460