11- Sept. - 2014 Unknown Mass Lab

Unknown Mass Lab

Objective:

 To practice our ability to determine uncertainty through finding an unknown density and unknown mass respectively. (Lab Partners: Kevin and Ivan)

Set up (2 of them):

1) We used calipers and were given 3 metal cylinders to determine density and propagate uncertainty

as seen from (fig 1 - 3).

2) We were given a set up of an unknown mass in equilibrium in a tension set up with spring scales to label tension forces as seen in (fig 4). To find angle we used a piece of string, a small weight and a protractor.

Part 1)
 This part of the lab is to apply what we learned about propagating uncertainty through finding the volume of three metal cylinders and their mass to determine the object's density. Density is the amount of mass (kg) per cubic meter (m^3). The next three figures were our work for each cylinder to find their densities and propagated error:

Figure 1. One of the metal cylinders with work on calculating its density.

Figure 2. The second metal cylinder with work on finding its density.

Figure 3. The third cylinder and work on finding its density.

The calculated densities with their propagated error for each figure are:

fig. 1= 7.49 ± .26 g/(cm^3) 

3.0% uncertainty 

fig. 2= 8.005 ± .714 g/(cm^3)

8.9% uncertainty

fig. 3= 17.574 ± .404 g/(cm^3)

2.3% uncertainty

Part 2)

During this part of the lab, we had to find the mass of an unknown on two pieces of string connected to spring scales at unknown angles. To find the angles we used a protractor, a piece of string and a small weight. We tied the weight to the string  and tied the other end on the hook (we were sure to only use enough weight to keep our string straight, but not too much to bend the string too much. We then used the protractor along the string connected to the unknown mass and read the angle between the two strings.

Figure 4. An unknown mass hanging on string attatched to spring scales each at an angle.

This is our work and set up to find the equation that could help us find our mass along with uncertainties in the find measurements to use for propagating our uncertainty for the mass.

Figure 5. The process in which we solved for our mass function in terms of:
 gravity, angle 1,  angle  2, Tension 1, and Tension 2.

We then plugged in our values into our new found equation and got:

m= (2.3 N)(cos(.4363) - sin(.4363)) + (3.9 N)(sin(.6458) + cos(.6458)) = .670 kg

(9.81 m/(s^2))

From there we had to find the partial of each independent variable (except gravity) and find our uncertainty from this example as shown below in (fig 6).

Figure 6. Calculations in which to find uncertainty of our unknown mass by finding the partial in respect to each independent variable then square rooting the dot product of the derivative of  our mass function (treating it as a vector) to find dm or uncertainty of mass. 

Through that we find that our mass with our uncertainty is:

m= .670 ± .034 kg

5.1% uncertainty

Source of uncertainty lie mostly with the reading of angles because we don't know if there was enough change in angle that could effect the results.