Kevin Tran
Lab Partners: Kevin Nguyen, Jose Rodriguez
March 11, 2017

      Propagated uncertainty in measurements

       Purpose: We are trying to the find the density of an object by calculating the propagated of uncertainty.

       Introduction: We will be finding the density of two objects: cylinder I and cylinder II. In order to find the approximate density of the two cylinders, we need to calculate the propagated uncertainty since there is a propagated error. In order to calculate uncertainty, we use the following formula (displayed on the right side)

This formula allows us to find the uncertainty of an object.

The curly d symbols is a partial derivative. Also, dp/dm in the figure means a 

derivative in respect to m(mass) in this case. This case would
apply to the others. dp/dh would be the derivative in
respect to h(height) and dp/dD would be the derivative in respect
to D(width). Also, each of the fractional uncertainty is multiplied by their corresponding propagated error (dm, dh, and dD)

During our lab, we used a caliper to find the measurements of cylinder I. (Figure below) With the caliper, we were able to find the measurements and uncertainty of the object. 

Measurements of cylinder I:

Diameter: 1.88 cm +/- 0.01 cm

Height: 3.45 cm +/- 0.01 cm

Weight: 70.18 g +/- 0.01 g

In addition, we measured cylinder II as well.

Measurements: 

Diameter: 1.889 cm +/- 0.01 cm

Height: 8.59 cm +/- 0.01 cm

Mass: 69.92 g +/- 0.01 g

With the collected measurements: We found the density of both cylinder I and cylinder II. (Picture below).

After we have collected our data, we took the derivative of the density formula in respect to each variable: (picture below).

After taking the derivative, we started to plug in each fractional uncertainty into the formula.(Formula displayed at the far top). We start with cylinder I.

Next, we use our collected measurements from cylinder I and we plugged in the data to each value. (pictures below) note: This is one equation displayed in two pictures.

After simplifying everything, we have found the uncertainty of +/- 0.044 g/cm^3.


With the density that we have found for cylinder II, we can come to a conclusion that the density of cylinder I is 7.33 g/cm^3 +/- 0.044 g/cm^3. (picture below).

Next, we approached cylinder II the same way.

We use our collected measurements from cylinder II and we plugged the data in to each value. (pictures below).

After simplifying everything, we have the uncertainty of cylinder II of +/- 0.0379 g/cm^3. With the density that we have found for cylinder II, we can come to a conclusion that the density of cylinder II is 2.9 g/cm^3 +/- 0.0379 g/cm^3. (picture below).

In conclusion, the lab taught us how to find the uncertainty of both cylinder I and II. In addition, by applying the formula for uncertainty, we are able to get the uncertainty of both objects. With the uncertainty, we can approximately guess what the density of the objects are.