9-Mar-2015: Unknown Mass and Propagated Uncertainty Lab

Lab 4: Unknown Mass and Propagated Uncertainty

Purpose: To calculate the propagated error of measurements and to determine the mass of an unknown mass.

This lab had two parts. In Part 1, we calculated the propagated error in density measurements. In Part 2, we determined the mass of an unknown mass.

Part 1: Propagated Uncertainty of Density

Pre-Lab: For this portion of the lab, we learned how to use a caliper.  If you are new to them, please head on over to: http://sites.laverne.edu/physics/courses/general-physics-labs/measurement-lab/
It is important to learn this skill, in order to accurately record data for this lab.

 

Materials: -Scale      - Vernier Caliper       -Three Metal Cylinders (Steel, Aluminium, Copper)

In this portion of the lab, we calculated the density of each cylinder and the propagated error in each of our measurements and ultimately found the uncertainty in our calculated value of density.

Procedure: First, we used the caliper to measure each cylinder's diameter and height. Then, we weighed each cylinder to find its mass. In order to account for uncertainty, we also included a small +/- value to each of our measurements. Our results are the following:
D=Diameter, H=Height, M=Mass

• Steel: H= 5.00 +/- .01 cm, D=1.41+/- .01 cm, M=61.1 +/- 0.1 g
• Copper: H=5.12 +/- .01 cm, D=1.28 +/- .01 cm, M=57.5 +/- 0.1 g
• Aluminum: H=4.87 +/- .01 cm, D=1.45 +/- .01 cm, M=21 +/- 0.1 g

     The next step was to calculate the density of each cylinder.

For this step, we used Density= Mass/Volume, V=pi*H*(D/2)^2 (Volume of a cylinder)

We simplify the equation and get Density= 4M/(pi*H*D^2)
Now, we use this equation and calculate the density for each cylinder.

Density:

• Steel= 7.83 g/cm^3
• Copper= 8.73 g/cm^3
• Aluminum= 2.61 g/cm^3

As stated above, Density= 4M/(pi*H*D^2)
Here, we notice that density is a function of three variables (M,D,H). In order to find the uncertainty in the calculated value for density, we must add the products of the partial derivative of each variable and that variable's uncertainty.  When taking a partial derivative, you derive only one variable and treat the rest of the function as a constant.

Note: Take the absolute value of the partial derivatives.

Uncertainty in Calculate Value of Density:

• Steel: dp= .14
• Copper: dp= .17
• Aluminum: dp= .05

Final Calculated Values of Density:

• Steel: 7.83 +/- .14 g/cm^3
• Copper: 8.73 +/- .17 g/cm^3
• Aluminum: 2.61 +/- .05 g/cm^3

Calculated Values vs. Accepted Values

• Steel: 7.83 +/- .14 g/cm^3  vs.  7.75 to 8.05 g/cm^3
• Copper: 8.73 +/- .17 g/cm^3  vs.  8.94 g/cm^3
• Aluminum: 2.61 +/- .05 g/cm^3  vs.  2.7 g/cm^3

We find that the calculated values of the three cylinders are within range of the accepted values.

Part 2: Finding the mass of an unknown mass

In this portion of the lab, we will find the mass of two different unknown masses.

Materials: - Rods  -  C- clamps -    String  -     Spring Scales  -       Unknown Masses

The setups can be seen below:

Mass 8

Mass 7

 

As you can see, the masses are suspended in equilibrium. The unknown mass is held in equilibrium by two strings, each with its own spring scale. The strings are then tied to rods at a distance, resulting in a hanging mass.

 

Procedure:

In order to determine the unknown mass, we must record the angles of each string from the horizontal axis and the forces found on the spring scales. From here, we draw a FBD for each mass and only solve for forces in the y direction because in this case forces in the x direction are irrelevant, since the mass is in equilibrium. 

 

When we solve for forces in the y direction: m= [(F1*sin(theta1))+(F2*sin(theta2))]/ g

 

Mass 8:  Unknown Mass = 1.05 kg

(F1*sin(theta1)) is the y-component of F1.

(F2*sin(theta2)) is the y-component of F2.

 

Same process is done for Mass 7

Mass 7: Unknown Mass= .84 kg

(F1*sin(theta1)) is the y-component of F1.

(F2*sin(theta2)) is the y-component of F2.

Now that we have the calculated value for the unknown masses, we must find its uncertainty.

 m= [(F1*sin(theta1))+(F2*sin(theta2))]/ g

Here, we can see that m is dependent of four variables (F1,F2,theta1,theta2).
m=m(F1,F2,theta1,theta2).

Therefore, in order to find the uncertainty in the calculated value for mass, we must add the products of the partial derivative of each variable and that variable's uncertainty. 

We are essentially doing the same steps as in Part 1 of the lab.

Propagated Uncertainty

Mass 8: dm= .104
Note: d(theta1) & d(theta2) are in radians, ( 2 degrees = .035 radians)

Mass 7: dm= .099
Note: d(theta1) & d(theta2) are in radians, ( 2 degrees = .035 radians)

 Final Values:

• Mass 8: 1.05 +/- .104 kg, 10% uncertainty
• Mass 7: 0.84 +/- .099 kg, 12% uncertainty

In this lab, we learned how to use a caliper and calculated the propagated error of measurements and determined the mass of an unknown mass. In Part 1 of the lab, we measured three metal cylinders and calculated their density. Then , we found the uncertainty for the calculated values of density and compared those values to the accepted densities of steel, aluminum, and copper. In Part 2 of the lab, we determined the mass for two unknown masses by drawing a FBD and solving for m. Finally, we calculated the propagated uncertainty of the masses.