The following three problems represent typical error propagation exercises you w

Question

The following three problems represent typical error propagation exercises you will be performing practically during every lab period. it is important that you understand how to do them. I will schedule the same 3 question into your first 3 pre-labs. First two times there will be no credit taken for the wrong answers. Try it, ask your TAs for help, and then try it again next week. For the third time it'll be a full fledged pre-lab question.

Problem #1: Assume you measured the mass of the cart used in an experiment to be (1.00 ± 0.01) lb (that is American pounds). Roughly there's 500 g per pound. Calculate the mass and the standard deviation for the mass of the cart in grams. (HINTS: Mathematically (functionally) what does conversion from one mass unit to another do? If so, which propagation of error formula from the  Appendix C should you be using? Can you intuitively understand what the correct formula (esp. the one for the relative error) means, to the point that you won't need a calculator to get the right answer? )

mcart = 500 ± g

Problem #2: Assume you measured the mass of the cart to be (500 ±1 )g and the mass of the additional mass you put on the cart to be (500 ± 1) g as well. Since the scale you are using in the lab can not measure objects heavier than 600g you will have to sum up individual pieces and propagate the error. So what would be the mass and the standard deviation of the cart and the mass? (HINTS: Which propagation of error formula from the Appendix C should you be using? Can you intuitively understand what the correct formula means, to the point that you won't need a calculator to get the right answer? )

mcart+weight = 1000 ± g

or

mcart+weight = 1000.0 ± g

Problem #3: Assume you have measured the mass of the cart and the fan which moves the cart to be (1.00 ± 0.02) kg and the acceleration of the cart to be (0.200 ± 0.004)

m/s2. What is the force and the standard deviation of the force which acts on the cart + fan? (HINTS: Which propagation of error formula from the Appendix C should you be using? Can you intuitively understand what the correct formula means (pay attention to the simplicity of the propagation of the relative error and its similarity to Problem#2), to the point that you won't need a calculator to get the right answer? )

F = 0.200 ± N

or

F = 0.20 ± N

What is the minimum number of significant figures that can be used for the value of the force in this problem?

Explanation / Answer

#1. mass, m = 1 +- 0.01 lb

assuming 500 pounds in one gram

let M be mass in grams

then dM is error in mass

dM = k*dm [ where k is conversion factor]

so, M = 500 +- 0.01*500 = 500 +- 5 grams

#2. m = 500 +- 1 g

additional mass, m' = 500 +- 1g

so while adding these two

d(m + M) = sqroot(dm^2 + dm'^2) = sqroot(1 + 1) = sqroot(2)

hence M + m = 1000 +- 1.414 g

#3. mass, m = 1+- 0.002 kg

acceleration a= 0.2 +- 0.004 m/s/s

Force = m*a = 0.2*1 = 0.2 N

dF/F = sqroot((dm/m)^2 + (da/a)^2) = sqroot(0.002^2 + (0.004/0.2)^2)

dF = 0.004 N

so F = 0.2 + 0.004 N

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