Mar 4, 2015: Lab 6; Propagated Uncertainty in Measurements

1. Measuring the Density of Metal Cylinder

Purpose:

We want to measure the density and its uncertainty for 3 metal cylinders, given the uncertainty value of the measuring equipments.

Procedure:

 _Use caliper to measure the height and diameter of the cylinders
 _ Use scale to measure the mass

Red: copper; White: Aluminum; Rusty green: steel

 Data and Analysis:
_ with measured uncertainty of +/- 0.1g and +/- 0.1mm, we obtain the following:
                           Height                        Mass                   Width
Aluminum:        50.9mm                    17.2g                  12.8mm
Copper:             50.9mm                     56.6g                  12.8mm
Steel:                 50.9mm                     53.8g                  12.8mm  

_ ρ, density is calculated by mass/volume, where volume of cylinder is πh(D/2)^2:

Density: Aluminum= 2.626g/cm^3

              Copper= 8.641g/cm^3

              Steel= 8.214g/cm^3

      Where: dρ- uncertainty in density (the calculated value)

                 dm- uncertainty in mass measurement

                 dd- uncertainty in diameter measurement

                 dh- uncertainty in height measurement

_ By using partial differentiation, we obtain:

        dh=dd; since we use the same equipment for 2 measurements;
_ Calculations,

One example of a detailed calculation of propagated uncertainty

Aluminum: dρ= 0.06136 g/cm^3
Copper: dρ= 0.1672 g/cm^3
Steel:  dρ= 0.1593 g/cm^3

• Aluminum: 2.626+/- 0.0614 g/cm^3

• Copper: 8.641+/- 0.1672 g/cm^3

• Steel: 8.214+/- 0.1593 g/cm^3

_ Comparison to theoretical density: Aluminum: 2.70g/cm^3;  Copper: 8.96g/cm^3;
       Steel: 7.75-8.05g/cm^3.

Conclusion

Including uncertainty in measurements and final calculations is important since it guarantees, (in a worst case scenario), the true value of density is in the range of that uncertainty.

2. Determination of an unknown mass:

Purpose:
We want to use a given angle, force, and our knowledge of free-body diagram to find the unknown hanging mass and its propagated uncertainty.

Experiment:
We want to find 2 unknown masses and their uncertainty by using these lab setups:

One of the lab setups

• The red spring measures the force in unit of "N"
  
• The yellow compass measures angle in degree, where we have to convert to radians for calculation. (to covert: degree*(π/180) )

yellow compass; unknown mass of the bottle

Data and Analysis:

   Object#7: 1st Force= 6N; 2nd Force= 8N; first θ= 21°; second θ= 46°

   Object#8: 1st Force= 7.5N; second Force= 10.5N; first θ= 11°; second θ= 48°

From free-body diagram: ΣFy: F1Sinθ1 + F2Sinθ2  = mg 

Hence,  m=(F1Sinθ1 + F2Sinθ2) /g ; all angles are converted to radians

Free-body diagram, calculation setups, and data collected

From this equation we get object 7's mass= 0.807 kg; object 8's mass= 0.94 kg

To find the uncertainty, we follow the same procedure as the first half of this lab, partial differentiation:

partial differentiation

dF= +/- 0.5N from the red spring scale; dθ= +/- 2° from the yellow compass

equation for partial differentiation for object 7 is similar to ob.8

Answer: obj.8's mass= 0.94 +/- 0.0989 kg

              obj. 7's mass= 0.807 +/- 0.0947 kg 

The propagated uncertainty of mass~ 1%

Conclusions:

In this second half of lab, instead of a density as a calculated error, we have mass as a calculated error. Our dF and dθ assure the measured uncertainty that turn out to be in the range of 1% error (or an error that only affect from the second decimal place of the mass).