Finding Propagation of Uncertainty?

[Monkey618]

You measure the mass of the cylinder to be m = 584.9 +- 0.5 grams, and you measure the length of the cylinder to be L = 18.195 +- 0.003 cm. Just like in the lab you performed, you now measure the diameter in eight different places and obtain the following results.
Diameter (cm)
2.125
2.090
2.065
2.240
2.110
2.100
2.080
2.240

This gives an average of 2.131 +- 0.0695

This makes the density = 9.01 g/cm^3 +- propagation of uncertainty

Trying to calculate this, I have: sqrt( ((1*0.5) / 584.9)^2 + ((-2 * 0,0695) / 2.131)^2 + ((-1 * 0.003) / 18.195)^2 ) = 0.0652

However the online grading system say's that I'm wrong. So where have I gone wrong with the uncertainty of the density. All of the other values have been graded and marked correct, so where did I mess up with the uncertainty.

I'm not really clear where the "1", "-2", or "-1" came from in the formula above, I was basing it on my notes from class.

 

[gneill]

You measure the mass of the cylinder to be m = 584.9 +- 0.5 grams, and you measure the length of the cylinder to be L = 18.195 +- 0.003 cm. Just like in the lab you performed, you now measure the diameter in eight different places and obtain the following results.
Diameter (cm)
2.125
2.090
2.065
2.240
2.110
2.100
2.080
2.240

This gives an average of 2.131 +- 0.0695

This makes the density = 9.01 g/cm^3 +- propagation of uncertainty

Trying to calculate this, I have: sqrt( ((1*0.5) / 584.9)^2 + ((-2 * 0,0695) / 2.131)^2 + ((-1 * 0.003) / 18.195)^2 ) = 0.0652

However the online grading system say's that I'm wrong. So where have I gone wrong with the uncertainty of the density. All of the other values have been graded and marked correct, so where did I mess up with the uncertainty.

It looks like you forgot to multiply the "sqrt" by the calculated value of the function being evaluated (the density value).

I'm not really clear where the "1", "-2", or "-1" came from in the formula above, I was basing it on my notes from class.

They are values that depend upon the exponent of the variable in the function. Write out the function being evaluated in one line (promote the variables in the denominator to the numerator and adjust exponents accordingly):

$$f(m,d,L) = \frac{4}{\pi}M^1 d^{-2} L^{-1}$$

You can pick out the values as the exponents of the variables. Thus, for example, the value "-2" is associated with the diameter variable d.

 

[mfb]

You calculated the relative uncertainty, and both the formula and the result look good. Similar to gneill, I think the online grading system wants the absolute uncertainty.

 

[Monkey618]

I've been meaning to get back here to say Thank You! Multiplying by the density was exactly what I needed to do for WebAssign (the online grading system) to accept the answer.

 

[Nickie]

I would say that your approach to calculating the uncertainty of the density is correct. However, it is important to understand the meaning of the numbers in the uncertainty formula.

The "1" in the formula represents the uncertainty in the mass measurement (0.5 grams), the "-2" represents the uncertainty in the average diameter measurement (0.0695 cm), and the "-1" represents the uncertainty in the length measurement (0.003 cm). These numbers are used to account for the variations in the different measurements that contribute to the final value of density.

It is possible that the online grading system is using a different approach or formula to calculate the uncertainty. I would suggest double-checking the formula or reaching out to your instructor for clarification. It is also possible that there was a mistake in entering the values into the system. Overall, it is important to understand the concept of propagation of uncertainty and how to apply it in your calculations, rather than just relying on a specific formula.