Lab 6: Part 1
(Measuring Density of Metal Cylinders)
Purpose: In this lab, the main purpose is to determine just how uncertainty in measurements can lead to uncertainty in final results of calculations which utilize said measurements. This is done using methods of differentiation learned in calculus. The first half of this lab focuses on determining density of three different cylindrical shaped objects (as well as determining our uncertainty in our density calculations).
Materials: For the first portion of this lab, we were given three cylindrical pieces of metal. We then used a Vernier Caliper to measure their height and diameter, and we used a scale which we recorded their masses from. These items are depicted below.
(Vernier Caliper and the three unknown cylindrical shaped metals)
(Measuring the mass of cylinder #1)
Procedure: Firstly, we measured and recorded each metal cylinders height (cm), diameter (cm), and mass (g), these results are shown below.
(Measurements of unknown cylindrical metals)
While we can calculate the density of each cylinder given these measurements, it is unclear just how certain our results are. To get a more precise result, we need to take into account the uncertainty in each individual measurement and then determine our uncertainty in our density calculations.
We begin determining the uncertainty in our density calculations with the simple density definition.
D = (m / V)
We know the volume of a cylinder is given by:
V = pi* r^2 * h Where r = (diameter / 2)
Combining these yields:
D = [ m / (pi * (d/2)^2 * h) ] --- > D = (4m / pi * d^2 * h)
We now have an equation representing density in terms of each metal's mass, diameter, and height.
Using methods of differentiation from calculus, we can determine the uncertainty in each measurement in order to determine the uncertainty in our density calculation result, one such instance is depicted below:
(Calculation of uncertainty in density measurement for cylinder #1)
So, using the method depicted above, we have:
D1 = 8.80 g/cm^3 (+ -) 0.201 g/cm^3
D2 = 2.79 g/cm^3 (+ -) 0.0561 g/cm^3
D3 = 6.64 g/cm^3 (+ -) 0.145 g/cm^3
Comparing these outcomes with a table of accepted values of densities of various metals,
we see that metal 1 corresponds to Copper (Cu), metal 2 with Aluminum (Al), and metal 3 with Cerium (Ce).
Conclusion: This lab is meant to develop the skills and procedures necessary to calculate just how uncertain experimental results can be. We use methods of differentiation in order to find a range of values for our final result, which should (and did in this case) land within the scientifically accepted values for such calculations.
Lab 6: Part 2
(Determination of an unknown mass)
Purpose: The purpose of this section of the lab is essentially identical to the part prior, only difference being tools of measurement, and instead of solving for density we will be solving for the mass of a hanging object placed in a system of static equilibrium. Our final result will have the mass of the object, as well as a range of uncertainty associated with such result.
Materials: The systems in which the hanging masses belonged were set up prior to us beginning this lab. Depicted below are photos of 2 out of the 3 set ups.
(Unknown mass #7)
(Unknown mass #1)
Procedure: The measurements required to determine the mass of the object are 1) The forces (N) and 2) The angles between a horizontal line and said forces (degrees). It's important to remember, that when using these recorded angles in calculations, they must be converted into radians.
Depicted below is the work done in order to determine the unknown mass along with the uncertainty in it's determination:
(Solving for propagated uncertainty for unknown mass #7)
Using the same method above for unknown mass #1, we have:
m #1 = 0.743 kg (+ -) 0.414 kg
Conclusion: This second portion of the lab demonstrates how the propagated error becomes larger when the measurement tools become less and less precise, hence the larger range of uncertainty.
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