Measurement

Fundamental Units
Dimensional Analysis Significant Figures Error Analysis  

     

Fundamental Units - S. I. System


To measure a physical quantity we require a standard of that physical quantity. This is standard is called Unit of that physical quantity.

There are a large number of physical quantities and every quantity needs a unit.

Fundamental Units : The units of fundamental physical quantities are called Fundamental units.
In mechanics, unit of length, mass and time are m(meter), kg(kilogram) and s(second), respectively.,which are fundamental units.

Derived Units: The units of all other physical quantities, which can be obtained from fundamental units, are called Derived Units.

For example unit of velocity, density and force are m/s, kg/m3, kg/ms2 respectively.

 

Who decides the Units

In 1960, an international committee established a set of standards for these fundamental quantities. This is called the International System (SI) of Units.

The International System of Units (abbreviated SI from systeme internationale , the French version of the name) is a scientific method of expressing the magnitudes or quantities of important natural phenomena.

There are seven base units in the system, from which other units are derived. This system was formerly called the meter-kilogram-second (MKS) system.




































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Dimensional Analysis

The dimensions of a physical quantity refer to the type of units that must be used in order to obtain the measure of that quantity. Dimensions are expressed in terms of the base quantities and are denoted as shown :


The dimensions are not dependent on the actual units used or on the details of the shape of the object whose are is being measured.

The formulae for the areas (A) of the triangle and the circle are different but the dimensions of the areas in both cases are the same, [L2].

Examining the dimensions of physical quantities is useful because it enables one to check whether equations are incorrect. Quantities on both sides of an equality must have the same dimensions. Only quantities with the same dimensions can be added or subtracted.

For example, imagine that you (rather hastily!) derived an equation describing the motion of an acceleration body in terms of its final velocity, v, its initial velocity, u, its acceleration, a, and time, t.:

v = ut + at


Can this be correct? Examine the dimensions of each term:

[ LT-1 ] = [ LT-1 ][ T ] + [ LT-2 ][ T-1 ]
[ LT-1 ] = [ L ] + [ L ][ T-1 ]


The term on the left hand side of the equation does not have the same dimensions as the terms on the right hand side, and therefore it must be wrong.


Dimensions of Fundamental Units:


for Length, L
for Mass, M
for Time, T
for Electric Current, A
for Temperature, K
for Luminious Intensity, cd
for Amount of substance, mol


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Significant Figures

No measurement of a physical quantity can be entirely accurate. It is important to know, therefore, just how much the measured value is likely to deviate from the unknown, true, value of the quantity. The art of estimating these deviations should probably be called uncertainty analysis, but for historical reasons is referred to as Error Analysis.

The brief discussions about how errors are reported, the kinds of errors that can occur, how to estimate random errors, and how to carry error estimates into calculated results.


Absolute and Relative Errors



The Absolute error in a measured quantity is the uncertainty in the quantity and has the same units as the quantity itself.

For example if you know a length is 0.428 m ± 0.002 m, the 0.002 m is an absolute error.

The Relative error (also called the fractional error) is obtained by dividing the absolute error in the quantity by the quantity itself. The relative error is usually more significant than the absolute error.

For example a 1 mm error in the diameter of a skate wheel is probably more serious than a 1 mm in a truck tire.

Note: that relative errors are dimensionless. When reporting relative errors it is usual to multiply the fractional error by 100 and report it as a percentage.

Systematic Errors


Systematic errors arise from a flaw in the measurement scheme which is repeated each time a measurement is made. If you do the same thing wrong each time you make the measurement, your measurement will differ systematically (that is, in the same direction each time) from the correct result.

Some sources of systematic error are:

Errors in the calibration of the measuring instruments.

Incorrect measuring technique: For example, one might make an incorrect scale reading because of parallax error.

Bias of the experimenter: The experimenter might consistently read an instrument incorrectly, or might let knowledge of the expected value of a result influence the measurements. It is clear that systematic errors do not average to zero if you average many measurements. If a systematic error is discovered, a correction can be made to the data for this error.

If you measure a voltage with a meter that later turns out to have a 0.2 V offset, you can correct the originally determined voltages by this amount and eliminate the error.

One must simply sit down and think about all of the possible sources of error in a given measurement, and then do small experiments to see if these sources are active. The goal of a good experiment is to reduce the systematic errors to a value smaller than the random errors.



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