Propagation Method - Calculating Absoute/Relative/Percentage Uncertainties.

[Masq]

Homework Statement

Hi everyone, I'm having some homework trouble, and am hoping you can help me out. Here's the question.

QUESTION: Using the propagation method for calculating uncertainties, calculate the absolute, relative, and percentage uncertainty for the volume of the cylinder and the uncertainties for density for each of the different methods used.

INFORMATION:
Mass of Cylinder: 20.6g +/- .1g
Composition of Cylinder: Copper (8.87g/cm^3)

Method 1: Meter Stick
Average length: 2.07cm
Average Diameter: 1.16cm
Uncertainty: +/- .05cm

Method 2: Vernier Calipers
Average Length: 2.142cm
Average Diameter: 1.211cm
Uncertainty: +/- .0025cm

Homework Equations

Propagation of Uncertainties:

1. If S=A+B (ΔS=ΔA+ΔB)
2. If D=A-B (ΔD=ΔA+ΔB)
3. If P=A.B 
                ΔP    ΔA   ΔB
                __ = __ + __
                 P     A      B

4. If Q=A/B
                ΔQ    ΔA   ΔB
                __ = __ + __
                 Q     A      B

5. If P=A^N
                ΔP      ΔA  
                __ = N___
                 P        A

Percent Error: [Accepted - Measure] / Accepted *100

The Attempt at a Solution

I've read the rules, and I understand you arent supposed to ask for help if you don't at least try, but I honestly have no clue where to begin. If anyone could help me start out I'll quickly respond with my best effort. But I am currently totally lost.

 

[jbsteele]

The first step is to find the volume of the cylinder. The formula for the volume of a cylinder is V=πr^2h, so the radius is half the diameter, and the height is the length.Method 1:V = π(1.16/2)^2 * 2.07cmV= 3.1529cm^3Method 2: V = π(1.211/2)^2 * 2.142cm V = 3.1575cm^3Now that we have the volume, we can use the propagation of uncertainties method to calculate the absolute, relative, and percentage uncertainty for the volume of the cylinder for each method. For Method 1:ΔV = (ΔA + ΔB)ΔV = (.05cm + .05cm) = .1cmAbsolute Uncertainty: .1cmRelative Uncertainty: .1/3.1529 = .032Percentage Uncertainty: .032/3.1529 * 100 = 1.02%For Method 2: ΔV = (ΔA + ΔB)ΔV = (.0025cm + .0025cm) = .005cmAbsolute Uncertainty: .005cm Relative Uncertainty: .005/3.1575 = .0002Percentage Uncertainty: .0002/3.1575 * 100 = 0.06%Now, we can use the mass and volume to calculate density. The formula for density is d=m/v. Method 1:d = 20.6g/.031529cm^3d = 651.9g/cm^3Method 2:d = 20.6g/.031575cm^3d = 651.2g/cm^3To calculate the absolute, relative, and percentage uncertainty for the density for each method, we use the propagation of uncertainties method. For Method 1: Δd = (ΔA + ΔB)Δd = (.1g + .1cm) = .2gAbsolute Uncertainty: .2g Relative Uncertain

 

[baba89]

Hi there,

First of all, let's define the terms we will be using in this problem:

- Absolute uncertainty: This is the uncertainty associated with a single measurement. In this case, it is the value given for the uncertainty in the mass, length, and diameter measurements.
- Relative uncertainty: This is the ratio of the absolute uncertainty to the measured value. It gives us an idea of the accuracy of our measurement.
- Percentage uncertainty: This is the relative uncertainty expressed as a percentage.

Now, let's start by calculating the volume of the cylinder using each of the two methods:

Method 1:
Volume = πr^2h
= π(0.58cm)^2(2.07cm)
= 2.04 cm^3

Method 2:
Volume = πr^2h
= π(0.6055cm)^2(2.142cm)
= 2.05 cm^3

Next, we will calculate the uncertainties associated with each method:

Method 1:
Uncertainty in volume = π[(Δr)^2h + r^2(Δh)^2]1/2
= π[(2rΔr)^2 + (Δh)^2]1/2
= π[(2(0.58cm)(0.05cm))^2 + (0.05cm)^2]1/2
= 0.08 cm^3

Method 2:
Uncertainty in volume = π[(Δr)^2h + r^2(Δh)^2]1/2
= π[(2rΔr)^2 + (Δh)^2]1/2
= π[(2(0.6055cm)(0.0025cm))^2 + (0.0025cm)^2]1/2
= 0.0008 cm^3

Now, we can calculate the absolute, relative, and percentage uncertainties for the volume of the cylinder for each method:

Method 1:
Absolute uncertainty = 0.08 cm^3
Relative uncertainty = 0.08 cm^3/2.04 cm^3 = 0.039
Percentage uncertainty = 0.039 * 100% = 3.9%

Method 2:
Absolute uncertainty = 0.0008 cm^3
Relative uncertainty = 0.0008 cm^3/2.