Dimensional Analysis Significant Figures Error Analysis

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

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

In mechanics, unit of

Derived Units

For example unit of

The

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.

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

The dimensions are not

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,

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.:

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

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

The term on the

Dimensions of Fundamental Units:

for Length,

for Mass,

for Time,

for Electric Current,

for Temperature,

for Luminious Intensity,

for Amount of substance,

Whenever you make a **measurement**,
the number of meaningful digits that you write down implies
the error in the measurement. For example if you say that the
length of an object is **0.428 m**, you imply an uncertainty of
about **0.001 m**. To record this measurement as either **0.4 or
0.42819667** would imply that you only know it to **0.1 m** in the
first case or to **0.00000001 m** in the second. You should only
report as many **significant figures** as are consistent with the
estimated error. The quantity **0.428 m** is said to have **three
significant figures**, that is, three digits that make sense
in terms of the measurement. Notice that this has nothing to
do with the **"number of decimal places"**.

The same measurement in centimeters would be**42.8 cm** and still be a
**three significant figure number**. The accepted convention is
that only one uncertain digit is to be reported for a measurement.
In the example if the estimated error is **0.02 m** you would report
a result **of 0.43 ± 0.02 m, not 0.428 ± 0.02 m**.

Students frequently are confused about**when to count a zero as a significant figure**.

The rule is:

**If the zero has a non-zero digit anywhere to its left,
then the zero is significant, otherwise it is not**.

For example :**5.00** has **3 significant figures**; the number **0.0005** has only
**one significant figure**, and **1.0005** has **5 significant figures**.

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Error Analysis

The same measurement in centimeters would be

Students frequently are confused about

The rule is:

For example :

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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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.

The

For example if you know a length is

The

For example a

Systematic errors arise from a flaw in the measurement scheme which is

Some

Errors in the

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