Chapter Unit and measurement

UNIT AND MEASUREMENT

Units of measurement are vital parts of any physical quantity. Just as the person is known by his or her name, the physical quantities are known by the units of measurement.

System of unit

There are four categories of system of unit.

1.C.G.S :

In the CGS system of measurement length is measured in centimeters, the mass of the body or substance is measured in grams and time is measured in seconds. Thus centimeter, gram and seconds are the fundamental units of measurement in the CGS system.

2.F.P.S :

In this system the unit of measurement for length is foot, for mass it is pound and for time it is second. This system is used commonly in Britain and the countries that were under its rule.

3.M.K.S :

In MKS system or metric system, the unit of measurement for length is metre, for mass it is kilogram and for time it is seconds. Thus in this system meter, kilogram and seconds are fundamental units of measurement. This system was used in France and number of other European countries.

International system of unit (SI unit)

It is the system of units, which is accepted internationally for measurement.
It consisting of seven base unit and two supplementary units.

Seven Base units
S.No. Fundamental Quantities Fundamental Units  Symbol
1. Length metre m
2. Mass kilogram kg
3. Time second s
4. Temperature kelvin K
5 Electric current ampere A
6 Luminous intensity candela cd
7 Amount of substance mole mol

Supplementary units

Until 1995, the SI classified the radian and the steradian as supplementary units, but this designation was abandoned and the units were grouped as derived units.

S.No. Supplementry Quantities Supplementry Units  Symbol
1 Plane angle Radian rad
2 Solid angle Steradian sr

 

Radian(rad)

It is the unit of angle in plane.One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius of the circle.
If an arc of length ​\( ds \)​​ subtends an angle \( d\theta \)​ at the center O of a circle of radius r.

\[ d\theta=\dfrac{ds}{r}radian \]

Steradian(sr)

It is the unit of solid angle.One steradian is the solid angle subtended at the centre of a sphere,by that surface of the sphere, which is equal in area, to the square of radius of the sphere.

If a surface area ​\( dA \)​ of a spherical surface subtends a solid angle ​\( d\Omega \)​ at the center of the sphere of radius r .

\[ d\Omega=\dfrac{dA}{r^2}steradian \]

 

Prefixes

Prefixes are added to unit names to produce multiples and sub-multiples of the original unit. All of these are integer powers of ten, and above a hundred or below a hundredth all are integer powers of a thousand.

SI prefixes
Prefix Base 1000 Base 10 Decimal English word Adoption
Name Symbol Short scale Long scale
yotta Y  10008  1024 1000000000000000000000000  septillion  quadrillion 1991
zetta Z  10007  1021 1000000000000000000000  sextillion  trilliard 1991
exa E  10006  1018 1000000000000000000  quintillion  trillion 1975
peta P  10005  1015 1000000000000000  quadrillion  billiard 1975
tera T  10004  1012 1000000000000  trillion  billion 1960
giga G  10003  109 1000000000  billion  milliard 1960
mega M  10002  106 1000000  million 1873
kilo k  10001  103 1000  thousand 1795
hecto h  10002/3  102 100  hundred 1795
deca da  10001/3  101 10  ten 1795
 10000  100 1  one
deci d  1000−1/3  10−1 0.1  tenth 1795
centi c  1000−2/3  10−2 0.01  hundredth 1795
milli m  1000−1  10−3 0.001  thousandth 1795
micro µ  1000−2  10−6 0.000001  millionth 1873
nano n  1000−3  10−9 0.000000001  billionth  milliardth 1960
pico p  1000−4  10−12 0.000000000001  trillionth  billionth 1960
femto f  1000−5  10−15 0.000000000000001  quadrillionth  billiardth 1964
atto a  1000−6  10−18 0.000000000000000001  quintillionth  trillionth 1964
zepto z  1000−7  10−21 0.000000000000000000001  sextillionth  trilliardth 1991
yocto y  1000−8  10−24 0.000000000000000000000001  septillionth  quadrillionth 1991

 

SOME PRACTICAL UNITS

Light year(ly)

One light year is the distance travelled by ligth in vacuum in one year.

As we know that velocity of light in vacuum is ​\( 3\times10^8~ms^{-1} \).
And 1 year is equal to ​\( 365\times24\times60\times60 \)​ seconds.
Therefore,

\[ 1ly=3\times10^8\times(365\times24\times60\times60) \]

\[ 1ly=9.46\times10^{15}m \]

Astronomical unit(AU)

One astronomical unit is the average distance between the center of the sun to center of the earth.

\[ 1AU=1.496\times10^{11}m \]

Parallactic Second(Par sec)

One par sec is the radius of a circle at the center of which an angle is of 1′ ‘ and also an arc of the circle is 1 AU long.

\( l=1AU=1.496\times10^{11} \)​ and ​

\[ \theta=1second=\dfrac{1}{60}minutes=\dfrac{1}{60\times60}degree=\dfrac{1}{60\times60}\times\dfrac{\pi}{180}rad \]

\[ r=\dfrac{l}{\theta} \]

\[ 1parsec=\dfrac{1AU}{1second}=\dfrac{1.496\times10^{11}\times(60\times60\times180)}{\pi} \]

\[ 1parsec=3.1\times10^{16}m \]

Measurement of large distances

PARALLAX METHOD

Parallax method is used to measure the large distances i.e. distance of star or planets from earth.[Only used for distances less than 100 light years.]

paralax

 

 

ERRORS OF MEASUREMENT

Nothing is perfect in this word because of errors.Measuring is just an another form of comparison.
Error is the difference in the true value and measured value.

Types of errors 

1.CONSTANT ERRORS

When a physical quantity is measured a number of times and each time same error is repeated , then the error is said to be constant.

2.SYSTEMATIC ERRORS

These errors tends to be in one directional either positive or negative.Systematic error can be reduced as we know causes.These errors are caused due to following reasons:

(i)Shortcoming of the measuring method used.

(ii)Imperfection in the theory of the physical phenomenon which the quantity being measured.

(iii)Lack of accuracy of the formula used for calculations.

The five main types of systematic errors classified are

(a)Personal errors: This type of error occurs due to our carelessness. i.e. lack of proper setting of apparatus, taking observations without observing proper precautions etc.

(b)Instrumental errors: It arises due to imperfect design or manufacturing of calibration of the measuring instrument. i.e. an ordinary scale may be worn off at the ends.

(c)Errors due to external sources: External conditions like pressure, temperature, wind speed, humidity may affects experiments.

(d)Errors due to internal sources: These errors are caused due to limitations of experimental arrangements. i.e. to obtain human body temperature thermometer is placed under arm pits will always give a temperature less than the body temperature, buoyancy error in weighting is usually ignored, radiation error in experiments on heating usually ignored.

(e)Least count errors: The smallest value that can be measured by a measuring instrument is called the least count of that instrument. i.e a meter scale having graduations at 1mm division scale spacing has a least count of 1mm.etc

Least count error is the error associated with the resolution of the instrument.By using better instrument and experimental techniques least count error can be reduced.

Least count of screw gauge/vernier callipers

\[ =\dfrac{One~main~scale~division(1~M.S.D)}{No.~ofdivision~on~circular~scale/~vernier~scale} \]

Least count of vernier callipers\( =\dfrac{1~mm}{10}=0.1~mm \)

If screw gauge has pitch of 1mm and 50 division then,

Least count of screw gauge\( =\dfrac{1~mm}{50}=0.02~mm \)

3.RANDOM ERRORS

The random errors may arise due to random and unpredictable variations in experimental condition i.e. voltage supply, pressure, mechanical vibrations , etc.Random error also known as chance error .

The random error can be reduced by repeating the experiment at large number of times and taking the arithmetic mean of all experimental observations.

Let a physical quantity is measured n times, and the obtained values are ​\( a_1,a_2,a_3….n \)​.Then the best possible value can be given by arithmetic mean(​\( \overline{a}\quad{or}\quad{a_m} \)​) of all obtained values:

\[ a_m=\dfrac{a_1+a_2+a_3……a_n}{n} \]

\[ =\dfrac{1}{n}\sum_{i=1}^{i=n}a_i \]

4.GROSS ERRORS

These error asises on the account of shear carelessness of the observer.It arises due to following reasons:

(i)Reading an instrument without setting instruments properly.

(ii)Taking the observation wrongly without caring for the sources of errors and the precautions.

(iii)Recording the observations wrongly.

(iv)Using wrong values of the observations in calcualtions.

METHOD OF EXPRESSING ERROR

ABSOLUTE ERROR

The magnitude of the difference between the mean value and measured value of the physical quantity in the observation is called absolute error ​\( \lvert\Delta a_n\rvert \)
Absolute errors in the individual measured values of the quantity are:

\[ \lvert\Delta{a_1}\rvert=\lvert a_m-a_1\rvert\\\lvert\Delta{a_2}\rvert=\lvert a_m-a_2\rvert\\………….\\………….\\\lvert\Delta{a_n}\rvert=\lvert a_m-a_n\rvert \]

Absolute error can be positive or zero.

MEAN ABSOLUTE ERROR

Mean absolute error ​\( \Delta\overline{a}~or~(\Delta a)_{mean} \)​ is the total arithmetic mean of all absolute error.

\[ \Delta\overline{a}~or~(\Delta a)_{mean}=\dfrac{\lvert\Delta a_1\rvert+\lvert\Delta a_2\rvert+\dots\lvert\Delta a_n\rvert}{n} \]

OR

\[ =\dfrac{1}{n}\sum_{i=1}^{n}\lvert\Delta a_i\rvert \]

RELATIVE ERROR OR FRACTIONAL ERROR

Relative error or fractional error(​\( \delta a \)​) is the ratio of absolute error to mean value.

\[ \delta a=\dfrac{mean~absolute~error}{mean~value}=\dfrac{(\Delta a)_{mean}}{a_{m}} \]

OR

\[ \delta a=\dfrac{(\Delta\overline{a})}{a_{mean}} \]

PERCENTAGE ERROR

When relative error\fractional error is expressed in percentage, we call it percentage error.Thus,

\[ \delta a=\dfrac{\Delta{a}_{mean}}{a_{mean}}\times100\% \]

COMBINATION OR PROPAGATION OF ERRORS

The final error depends upon error present in individual measurement and measurements contains many operations(+,-,​\( \times \)​,/ etc).
We discuss below some propagation of error in different mathematical operations.

(i)Error in addition:

Suppose two quantities are a and b are there whose sum is x.

Suppose ​\( x=a+b \)

We know that True value = Measured value + Error

So,

\( x=a+b \) is a True value respectively.Which can be written as sum of measured value and error respectively.\( x\pm\Delta{x}=(a\pm\Delta{a})+(b\pm\Delta{b}) \)

for illustration x can be written as x(measured value)+Δx(error)

where,

\( \Delta{a}= \)​mean absolute error in measurement of a.

\( \Delta{b}= \)​mean absolute error in measurement of b.

\( \Delta{x}= \)​ mean absolute error in calculation of sum (a+b) which is x.

SO,

\[ x=a+b\\x\pm\Delta{x}=(a\pm\Delta{a})+(b\pm\Delta{b})\\x\pm\Delta{x}=(a+b)\pm\Delta{a}\pm\Delta{b}\\x\pm\Delta{x}=x\pm\Delta{a}\pm\Delta{b}\qquad[a+b=x]\\\pm\Delta{x}=\pm\Delta{a}\pm\Delta{b} \]

The four possible values of ​\( \Delta{x} \)​are ​\( (+\Delta{a}+\Delta{b}) \)​; ​\( (+\Delta{a}-\Delta{b}) \);\( (-\Delta{a}+\Delta{b}) \);\( (-\Delta{a}-\Delta{b}) \);

The maximum value of ​

\[ \Delta{x}=\pm(\Delta{a}+\Delta{b}) \]

Hence,maximum mean absolute error in sum of the two quantities is equal to sum of the mean absolute errors in the individual quantities.

(ii)Error in subtraction:

Suppose two quantities are a and b are there whose difference is x.

Suppose ​\( x=a-b \)

\( x=a-b \) is a True value respectively.Which can be written as sum of measured value and error respectively.\( x\pm\Delta{x}=(a\pm\Delta{a})-(b\pm\Delta{b}) \)

where,

\( \Delta{a}= \)​mean absolute error in measurement of a.

\( \Delta{b}= \)​mean absolute error in measurement of b.

\( \Delta{x}= \)​mean absolute error in calculation of difference (a-b) which is x.

SO,

\[ x=a-b\\x\pm\Delta{x}=(a\pm\Delta{a})-(b\pm\Delta{b})\\x\pm\Delta{x}=(a-b)\pm\Delta{a}\mp\Delta{b}\\x\pm\Delta{x}=x\pm\Delta{a}\mp\Delta{b}\qquad[a-b=x]\\\pm\Delta{x}=\pm\Delta{a}\mp\Delta{b} \]

The four possible values of ​\( \Delta{x} \)​are ​\( (+\Delta{a}+\Delta{b}) \)​; ​\( (+\Delta{a}-\Delta{b}) \);\( (-\Delta{a}+\Delta{b}) \);\( (-\Delta{a}-\Delta{b}) \);

The maximum value of ​

\[ \Delta{x}=\pm(\Delta{a}+\Delta{b}) \]

Hence,maximum mean absolute error in difference of the two quantities is equal to sum of the mean absolute errors in the individual quantities.

(iii)Error in multiplication

Suppose two quantities are a and b are there whose product is x.

Suppose ​\( x=a\times{b} \)

\( x=a\times{b} \) is a True value respectively.Which can be written as sum of measured value and error respectively.\( x\pm\Delta{x}=(a\pm\Delta{a})\times(b\pm\Delta{b}) \)

where,

\( \Delta{a}= \)​mean absolute error in measurement of a.

\( \Delta{b}= \)​mean absolute error in measurement of b.

\( \Delta{x}= \)​ mean absolute error in calculation of product (​\( a\times{b} \)​) which is x.

SO,

\[ x=a\times{b}\\x\pm\Delta{x}=(a\pm\Delta{a})\times(b\pm\Delta{b})\\on~taking~common\\x\Big(1\pm\dfrac{\Delta{x}}{x}\Big)=a\Big(1\pm\dfrac{\Delta{a}}{a}\Big)\times{b}\Big(1\pm\dfrac{\Delta{b}}{b}\Big)\\x\Big(1\pm\dfrac{\Delta{x}}{x}\Big)=(a\times{b})\Big(1\pm\dfrac{\Delta{a}}{a}\Big)\times\Big(1\pm\dfrac{\Delta{b}}{b}\Big)\\we~know~that~a\times{b}=x\\x\Big(1\pm\dfrac{\Delta{x}}{x}\Big)=(x)\Big(1\pm\dfrac{\Delta{a}}{a}\Big)\times\Big(1\pm\dfrac{\Delta{b}}{b}\Big)\\\Big(1\pm\dfrac{\Delta{x}}{x}\Big)=\Big(1\pm\dfrac{\Delta{a}}{a}\Big)\times\Big(1\pm\dfrac{\Delta{b}}{b}\Big)\\\Big(1\pm\dfrac{\Delta{x}}{x}\Big)=\Big(1\pm\dfrac{\Delta{b}}{b}\pm\dfrac{\Delta{a}}{a}\pm\dfrac{\Delta{a}}{a}\times\pm\dfrac{\Delta{b}}{b}\Big)\\\pm\dfrac{\Delta{a}}{a}\times\pm\dfrac{\Delta{b}}{b}~is~neglected~because~their~value~is~too~small\\\Big(1\pm\dfrac{\Delta{x}}{x}\Big)=\Big(1\pm\dfrac{\Delta{b}}{b}\pm\dfrac{\Delta{a}}{a}\Big)\\\pm\dfrac{\Delta{x}}{x}=\pm\dfrac{\Delta{b}}{b}\pm\dfrac{\Delta{a}}{a} \]

The four possible values of ​\( \dfrac{\Delta{x}}{x} \)​is ​\( \Big(+\dfrac{\Delta{b}}{b}+\dfrac{\Delta{a}}{a}\Big) \)​; ​\( \Big(+\dfrac{\Delta{b}}{b}-\dfrac{\Delta{a}}{a}\Big) \)​; ​\( \Big(-\dfrac{\Delta{b}}{b}+\dfrac{\Delta{a}}{a}\Big) \)​; ​\( \Big(-\dfrac{\Delta{b}}{b}-\dfrac{\Delta{a}}{a}\Big) \)​;

The maximum value of ​

\[ \Big(\pm\dfrac{\Delta{x}}{x}\Big)=\pm\Big(\dfrac{\Delta{b}}{b}+\dfrac{\Delta{a}}{a}\Big) \]

Hence,maximum fractional error or relative error in product of quantities is equal to sum of the fractional or relative errors in the individual quantities.

(iv)Error in division

Suppose two quantities are a and b are there whose quotient is x.

Suppose ​\( x=\dfrac{a}{b} \)

\( x=\dfrac{a}{b} \) is a True value respectively.Which can be written as sum of measured value and error respectively.\( x\pm\Delta{x}=\dfrac{(a\pm\Delta{a})}{(b\pm\Delta{b})} \)

where,

\( \Delta{a}= \)​mean absolute error in measurement of a.

\( \Delta{b}= \)​mean absolute error in measurement of b.

\( \Delta{x}= \)​ mean absolute error in calculation of quotient ​\( \dfrac{a}{b} \)​ which is x.

SO,

\[ x=\dfrac{a}{b}\\x\pm\Delta{x}=\dfrac{(a\pm\Delta{a})}{(b\pm\Delta{b})}\\on~taking~common\\x\Big(1\pm\dfrac{\Delta{x}}{x}\Big)=\dfrac{a\Big(1\pm\dfrac{\Delta{a}}{a}\Big)}{b\Big(1\pm\dfrac{\Delta{b}}{b}\Big)}\\x\Big(1\pm\dfrac{\Delta{x}}{x}\Big)=\dfrac{a}{b}\Big(1\pm\dfrac{\Delta{a}}{a}\Big)\Big(1\pm\dfrac{\Delta{b}}{b}\Big)^{-1}\\we~know~that~\dfrac{a}{b}=x\\x\Big(1\pm\dfrac{\Delta{x}}{x}\Big)=x\Big(1\pm\dfrac{\Delta{a}}{a}\Big)\Big(1\mp\dfrac{\Delta{b}}{b}\Big)\\\Big(1\pm\dfrac{\Delta{x}}{x}\Big)=\Big(1\mp\dfrac{\Delta{b}}{b}\pm\dfrac{\Delta{a}}{a}\pm\dfrac{\Delta{a}}{a}\times\mp\dfrac{\Delta{b}}{b}\Big)\\\pm\dfrac{\Delta{a}}{a}\times\mp\dfrac{\Delta{b}}{b}~is~neglected~because~their~value~is~too~small\\\Big(1\pm\dfrac{\Delta{x}}{x}\Big)=\Big(1\mp\dfrac{\Delta{b}}{b}\pm\dfrac{\Delta{a}}{a}\Big)\\\pm\dfrac{\Delta{x}}{x}=\pm\dfrac{\Delta{a}}{a}\mp\dfrac{\Delta{b}}{b} \]

The four possible values of ​\( \dfrac{\Delta{x}}{x} \)​is ​\( \Big(+\dfrac{\Delta{b}}{b}+\dfrac{\Delta{a}}{a}\Big) \)​; ​\( \Big(+\dfrac{\Delta{b}}{b}-\dfrac{\Delta{a}}{a}\Big) \)​; ​\( \Big(-\dfrac{\Delta{b}}{b}+\dfrac{\Delta{a}}{a}\Big) \)​; ​\( \Big(-\dfrac{\Delta{b}}{b}-\dfrac{\Delta{a}}{a}\Big) \)​;

The maximum value of ​

\[ \Big(\pm\dfrac{\Delta{x}}{x}\Big)=\pm\Big(\dfrac{\Delta{b}}{b}+\dfrac{\Delta{a}}{a}\Big) \]

Hence,maximum fractional error or relative error in division of quantities is equal to sum of the fractional or relative errors in the individual quantities.

(v)Error in raised power of the measured quantity

Suppose two quantities are a and b are there whose quotient is x.

Suppose ​\( x=\dfrac{a^n}{b^m} \)

\( x=\dfrac{a^n}{b^m} \) is a True value respectively.Which can be written as sum of measured value and error respectively.\( x=\dfrac{a^n}{b^m} \)

where,

\( \Delta{a}= \)​mean absolute error in measurement of a.

\( \Delta{b}= \)​mean absolute error in measurement of b.

\( \Delta{x}= \)​ mean absolute error in calculation of quotient ​\( \dfrac{a}{b} \)​ which is x.

SO,

Before we further go i want to remind you some basics of log.

(i)​

\[ y=x^m\\log_a~y=m~log_a~x \]

(ii)

\[ y=a\times{b}\\log_a~y=log_a~a+log_a~b \]

(iii)

\[ y=\dfrac{a}{b}\\log_a~y=log_a~a-log_a~b \]

So,

\[ x=\dfrac{a^n}{b^m}\\takin~log~both~side\\log~x=log~a^n-log~b^m\\log~x=n~log~a-m~log~b\\on~ differentiating~both~the~sides\\\dfrac{dx}{x}=n\dfrac{da}{a}-m\dfrac{db}{b} \]

In terms of fractional error we rewrite the formula as:

\[ \pm\dfrac{\Delta{x}}{x}=\pm{n}\dfrac{\Delta{a}}{a}\mp{m}\dfrac{\Delta{b}}{b}\\ \]

The four possible value of ​\( \dfrac{\Delta{x}}{x} \)​is ​\( \Big(+n\dfrac{\Delta{b}}{b}+m\dfrac{\Delta{a}}{a}\Big) \)​; ​\( \Big(+n\dfrac{\Delta{b}}{b}-m\dfrac{\Delta{a}}{a}\Big) \)​; ​\( \Big(-n\dfrac{\Delta{b}}{b}+m\dfrac{\Delta{a}}{a}\Big) \)​; ​\( \Big(-n\dfrac{\Delta{b}}{b}-m\dfrac{\Delta{a}}{a}\Big) \)​;

The maximum value of

\[ \Big(\pm\dfrac{\Delta{x}}{x}\Big)=\pm\Big({n}\dfrac{\Delta{a}}{a}+{m}\dfrac{\Delta{b}}{b}\Big) \]

Hence,fractional error or relative error in a quantity raised to power (n) is n times the fractional or relative error in the individual quantity.

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