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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:

- The first five: 0, 1, 2, 3 and 4 for which we go to the lower value.
- The last five: 5, 6, 7, 8 and 9 for which we move to the higher value.

This method limits the accumulation of errors during successive calculations.

Always perform the calculations while keeping all the digits present, and round the

answer at the end of the calculations only, according to the number that has the

fewest significant digits.

The devices are not perfect and given their imperfections, a result must be more

correctly given by a “framework” which takes into account the uncertainty of the

measurement.

The absolute uncertainty must be rounded off, almost always no higher value, and

only one significant figure is kept.

It is essential that the measurement and the absolute uncertainty have the same

number of significant figures after the decimal point.

When finalizing a numerical result it is important to include the appropriate number

of significant figures. It's a simple matter of clarity of expression :

INSTEAD OF [246.462 2.754] WE WILL HAVE [246 3]

INSTEAD OF [134.753 2] WE WILL HAVE [135 2]

1.2 Graphical representation

In most cases, we resort to graphical representation of the results obtained. To draw

the graph, we first choose a coordinate system and scale in abscissa and ordinate.

Then, we trace the curves using the representative points.

Notes : All partial results must be grouped in a table. Since the measurements are

uncertain, we would obtain a broken line if we simply joined the points corresponding

to a series of measurements. But if the phenomenon is continuous, the curve must not

show any break in slope, it must have a regular appearance. If we have made a largenumber of measurements, we will obtain a cloud of points and the curve should pass

through the cloud where it is densest.

Usually, we graph the uncertainties of particular measurements g + ∆g plotted parallel

to OY and a + ∆a along OX as shown in Figure 1.

Let g= f(a) be the quantity whose variation was studied as a function of the variable a .

Its graph is given in the figure opposite. The points M, M', M" give the average values

of the particular measurements, while ∆ g and ∆ a are the uncertainties on g and a .

The curve passes as close as possible to the representative points inside the error

rectangles ∆ g, ∆ a.

Caution : The choice of scale is very important; it is necessary to ensure that there is

optimal use of the sheet of graph paper (the graph should not occupy a very small

portion of the sheet).

1-3Principle of the 1/10th the vernier caliper

For a precise measurement, a Vernier caliper is combined with the graduated ruler. It

is a movable ruler sliding (sliding) along the fixed rule, and carrying a certain number

of graduations shorter than those of the (fixed) rule. The assembly (ruler + vernier)represents the essential mechanism of the “caliper”. The simplest of the classic

verniers is the 1/10th vernier which has 10 divisions corresponding in

length to 9 graduations of the fixed ruler, i.e. 9mm (see figure 1). When an object is

positioned for measurement, the vernier graduation coinciding with any ruler

graduation indicates the number of tenths of a mm to add to the main reading

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