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?In physics, uncertainty designates the margin of imprecision on the value of the measurement of a quantity.The relative uncertainty of a product or a quotient is equal to the sum of the relative measurement uncertainties: Z=XY -> ?Z/Z= ?X/X+ ?Y/Y Z=X/Y -> ?Z/Z ?= /X+ ?Y/Y.?In the case where we have a quantity Z which is a function of measurement X: Z= F(X), the absolute uncertainty of Z is equal to the derivative of the function multiplied by the absolute uncertainty of the measurement:Z= F (X) -> ?Z=F'(x) ?X where F'(X) is the derivative with respect to X. ?Absolute uncertainties are added for addition and subtraction.?This writing means that the true value of g is included in the interval: [g- ?g; g+ ?g] ?In the case where the uncertainty on a quantity is not explicitly given, we accept the level of the last significant figure as the order of magnitude of the uncertainty.1-3Principle of the 1/10th the vernier caliper For a precise measurement, a Vernier caliper is combined with the graduated ruler.L = 1.37 cm this means that L= (1.37 ?0.01) cm M = 350 Kg this means that M= (350+-1) Kg. Significant Digits In a physical measurement, the number of significant figures determines the accuracy of the measurement.?The devices used give a precision of around 0.1% to 0.2% : it is therefore illusory, in the experimental exercises, to hope for more than 03 significant figures.Rounding a number It is an approximate value of this number, obtained from its decimal expansion, by reducing the number of significant figures.This method limits the accumulation of errors during successive calculations.The relative uncertainty is therefore 0.02 or 2% .???
In physics, uncertainty designates the margin of imprecision on the value of the
measurement of a quantity. The concept is related to that of error, which is the
difference between the measured value and the true value.
During a measurement, it requires taking into account the precision of the
instruments used and the influence of this precision on the calculated results.
Consider a length measurement X= (25.1 0.5) cm
The central value of the measurement X=25.1 cm
The absolute uncertainty of this measurement is designated by X=0.5cm .
The relative uncertainty of the measurement is the ratio of the absolute uncertainty
to the central value and is expressed by X/X .
In our example X/X=0.5 25.1=0.02 . The relative uncertainty is therefore 0.02 or 2%
. This means that the measurement is accurate to within 2% .
We can write as well (25.1 0.5) cm or (25.1 2%) .
The absolute uncertainty of a sum or a difference is equal to the sum of the
absolute uncertainties:
Z=X+Y → Z= X+ Y
Z=X-Y → Z= X+ Y.
The relative uncertainty of a product or a quotient is equal to the sum of the
relative measurement uncertainties:
Z=XY → Z/Z= X/X+ Y/Y
Z=X/Y → Z/Z = /X+ Y/Y.
In the case where we have a quantity Z which is a function of measurement
X: Z= F(X), the absolute uncertainty of Z is equal to the derivative of the function
multiplied by the absolute uncertainty of the measurement:Z= F (X) → Z=F'(x) X where F'(X) is the derivative with respect to X.
Absolute uncertainties are added for addition and subtraction. The relative
uncertainties are added for multiplication and division.
The value of a quantity g must always be accompanied by an uncertainty
g : g = [g g ].
This writing means that the true value of g is included in the interval:
[g- g; g+ g]
In the case where the uncertainty on a quantity is not explicitly given, we accept the
level of the last significant figure as the order of magnitude of the uncertainty.
Ex: If we write 8.32 N this means that the uncertainty is of the order of 0.01N .
x = 23.0 cm this means that x is known to within 0.1 cm (we will say that the distance
is known to within 0.2m).
L = 1.37 cm this means that L= (1.37 0.01) cm
M = 350 Kg this means that M= (350±1) Kg.
Significant Digits
In a physical measurement, the number of significant figures determines the accuracy
of the measurement. These are the numbers known with certainty plus the first
uncertain number.
Ex : 1234 to four (4 ) significant digits. The first uncertain number is 5 .
The number of significant figures in a result depends first of all on the precision and
quality of the measuring instruments available: Precision is expensive.
The devices used give a precision of around 0.1% to 0.2% : it is therefore illusory, in
the experimental exercises, to hope for more than 03 significant figures.
The absolute uncertainty of a result should be rounded to one significant figure.
In practice, uncertainty generally influences the last digit of a result.
It is essential that the measurement and the uncertainty have the same number of
digits after the decimal point.
The 0 placed to the left of the number do not count: 0.06 is expressed with a single
significant digit.
The 0 placed to the right of the number count: 2,800 is expressed with 4 significant
digits.
The position of the decimal point does not play a role: 1.25 and 12.8 are expressedwith 3 significant figures.
Rounding a number
It is an approximate value of this number, obtained from its decimal expansion, by
reducing the number of significant figures.
In practice, the method consists of separating the ten decimal digits (0, 1,…,9) into two
parts:
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