To best estimate the uncertainty in the length of such a pendulum, measured with a metre stick with cm markings, we should takeQuestion 2Answera.the reading uncertainty of the metre stick.b.a multiple of the reading uncertainty to account for the non ideal use of the metre stickc.the standard deviation of many measurements of the length divided by the square root of the number of measurements we maked.no uncertainty

The length of an object is measured three times using a meter stick with smallest graduation of 0.1 cm on the stick. The measurements are 6.15 cm, 6.15 cm and 6.15 cm. The measurement error is:

Let us first define the standard deviation s. Suppose weperform N measurements 1, 2, · · · , N with the average¯. Then the deviation of each measurement is given byδ =  − ¯ with  = 1, 2, · · · , N. The standard deviation s iss =√√√ ∑N=1(δ)2N − 1When we report the average value of N measurements,the uncertainty we should associate with this averagevalue is the standard error.σ = spN =√√√√∑N=1(δ)2N(N − 1)The standard error is smaller than the standard deviationby a factor of 1/pN, since the statistical uncertainty canbe reduced by large number of measurements. Also it isuseful to write σ2 = δ2 ≡ 1N(N − 1)N∑=1δ2 .Suppose we want to determine a quantity  = ƒ (, ),which depends on  and . We want to know the error in = ƒ (, ) if we measure  and  with errors σ and σ .Using the Taylor expansion, we can obtain the law of theerror propagation as follows(δ)2 = ∂ƒ∂2(δ)2 + ∂ƒ∂2(δ)2 + 2 ∂ƒ∂∂ƒ∂(δδ)If the measurements of  and  are uncorrelated, then,on the average, we should expect to find equal distribu-tions of positive and negative values for this term, and weshould expect (δδ) = 0. At the end of the day, using thedefinition of the standard error σ, we can obtainσ =√√√ ∂ƒ∂2σ2 + ∂ƒ∂2σ2Exercise problems: Now find the standard error σ in = ƒ (, ) as a function of the errors in σ and σ for thefollowing functions:(a)  =  +  (0.5 pts)You can find the absolute uncertainty of the sum(or difference) is the root square sum of theindividual absolute uncertainties when adding (orsubtracting).(b)  =  ×  (0.5 pts)(c)  = /  (1 pt)You can find that the relative uncertainty of the prod-uct (quotient) is the root square sum of the individualrelative uncertainties.(d)  = 2 (1 pt)(e)  =  exp(c) with c constant. (0.5 pts)(f)  = 1/ 

What is the relationship between the accuracy and uncertainty of a measurement?

Suppose you determine the acceleration of gravity g =4π2LT2 by measuring the oscillation period T of a pendu-lum with length L. Determine the value and uncertaintyof g from the following measurementT = 2.01 ± 0.02 s, L = 1.000 ± 0.002m.If you wish to improve the above measurementsignificantly, which part of the measurement (T or L) doyou want to improve? Why?

Which one of the following statements is true?Group of answer choicesOnly the quantities we measure directly have uncertainties; once we perform calculations with these quantities our answer is exact.If a height is recorded as h = (12 ± 3) mm, then it is absolutely certain that the height is between 9 mm and 15 mm.When we repeat a measurement many times and take the average value of these measurements, the uncertainty on this average (i.e. the SEM) is generally expected to be smaller than the uncertainty on the individual measurements.We only have uncertainties in first year lab because our equipment is not precise enough; in research labs there is no uncertainty.