Physics Notes Plus1

Chapter 1 (Units and Measurements)

Important Formulas and Notes

Fundamental Quantities are Length(m), Mass(kg), Time (s), Current(A), Temperature(K), Luminous Intensity(cd for candela), Amount of  substance(mol)

Supplementary Quantities are Plane angle(radian), solid ange(Steradian)

Scientific Notation

a × 10^b  where 1< a < 10 and ‘b’ can be +ve or –ve

eg 6.022 × 10²³

Order of Magnitude

Order of magnitude of a number or measured value is the power of ten when expressed in scientific notation

Following examples can illustrate it more precisely.

eg 8 must be expressed as 0.8 × 10¹ ( since 8>5) there fore the Order of Magnitude is 1 (power of 10)

eg 203 must be expressed as 2.03 × 10² (since in 203  the digit 2 is less than 5) there fore the Order of Magnitude is 2(the power of 10)

eg 0.004 must be scientifically expressed as 4 × 10^-3 ( Order of Magnitude is -3)

Significant Figures

The number of significant figures gives the accuracy of measurement

The certain digits or reliable digits and first uncertain digit are known as significant digits ie the number of certain digits plus one is

the number of significant digits

Rules for Significant digits

1. Digits 1 to 9 is significant (eg number of significant digits in 35 is 2)
2. Zero/zeroes between non zero digits is significant (eg number of significant digits in 2105 is 4)
3. Trailing zero/zeoroes with decimal are significant but leading zeroes with decimal are not(eg number of significant digits in 0.049 is 2 but in 0.450 is 3)
4. Zero at the trailing position is significant for measured values or with units(eg number of significant digits in 640 is 2 but in 640 cm is 3)

When different measurements involve in a calculation, the signifcant figures of the final result should posses minimum number of significant

figures uesd in it

eg if C=A+B

let A=8.192 (number of significant figure is 4), B=7.98 ( number of significant figure is 3)

C=8.192+7.98=16.172

But the value of C must be corrected to significant figure is 16.2 (rounded to 3 significant figures ie least of A and B is that B so it is

the number of significant figure of C

Rules of Rounding off

12.43 rounded to 3 significant figure as 12.4 (3 is dropped since it is less than 5)

18.69 is rounded to 3 significant figure as 18.7 ( 1 is added to 6 since 9 is greater than 5)

14.85 is rounded to 3 significant figure as 14.8 ( digit is retained since 8 is even and last digit= 5)

56.75 is rounded to 3 significant figure as 56.8 ( 1 is added to the uncertain digit since 7 is odd and last digit=5)

Practical units of length

Parsec=3.08 × 10¹⁶ m=3.26 light year (It is defined as the distance at which an arc of length one astronomical unit subtends an angle of one second )

Light year=9.46 × 10¹⁵m (It is the distance travelled by light in one year)

Astronomical unit=1.496 × 10¹¹ m (It is the average distance between sun and the earth)

Angstrom=10^-10m

Practical units of mass

Metric ton=1000kg

Solar mass=2 × 10³⁰ kg

Chandrasekhar limit=1.4 times solar mass=2.8 × 10³⁰ kg

Practical units of time

Solar day=24 hours

Solar year=365 and quarter days

Lunar month=27.3 days

Shake=10^-8 s

Measuremnets of large distances

Parallax Method

Combination errors

If x=a+b or a-b

∆x=absolute error in x

∆a=absolute error in a

∆b=absolute error in b

Maximum absolute error in x , ∆x=∆a+∆b

Maximum relative error in x, ∆x/x=(∆a+∆b)/(a-b) (If x=a-b )
Maximum relative error in x, ∆x/x=(∆a+∆b)/(a+b) (If x=a+b )

If x=a x b or a/b

Maximum relative error in x, ∆x/x=∆a/a + ∆b/b

Percentage error in x is the sum of the percentage errors of a and b

If x=aⁿ× b^m

Maximum relative error in x, ∆x/x=n(∆a/a) +m( ∆b/b)

Dimensions of physical quantites

The fundamental quantities have dimensions as M,L,T,A,θ for mass,length,time,intensity of current, absolute temperature respectively

Dimension Formulae (Table)

Quantity	Formula	DF
Speed	Distance /time	M0L1T-1
Accelaration	Change in velocity/time	M0L1T-2
Force	mass x accelaration	M1L1T-2
Work	Force x distance	M1L2T-2
Energy	Work energy principle (W=E)	M1L2T-2
Gas Constant	PV/T	M1L2T-2θ-1
Resistance	Volt/Current	M1L2T-3A-2

The important applications of dimensional analysis are

• Convert one physical quantity from one system to another (eg SI to cgs etc)
• Check the correctness of a given relation (Checking Homogeneity of equation)
• Derive the relation between various physical quantities

eg: Check the correctness of the equation using Dimensional analysis

v=u+at

Where v and u have dimensions of velocity ie M⁰L¹T^-1

M⁰L¹T^-1= M⁰L¹T^-1+ M⁰L¹T^-2 x T¹

M⁰L¹T^-1= M⁰L¹T^-1+ M⁰L¹T^-1

M⁰L¹T^-1= M⁰L¹T^-1

Dimension of LHS= Dimension of RHS

Hence proved

eg Establish a relation between workdone, force and displacement

w α f^a d^b

ML²T^-2=M^1aL^1aT^-2a  L^1b

By comparing the powers of M,L,T

 a=1

a+b=2

so b=1

workdone=f¹× d¹

work done= force × displacement

eg Derive an expression for planck’s constant

E=hυ

Plancks constant, h= energy/frequency

Dimension of h= DO energy/DO frequency

                          =DO work/DO frequency

                          =DO force× displacement/DO frequency

                          =MLT^-2×L/T^-1

                                       =ML²T^-1    

Limitations Of Dimension analysis

1. The dimensionless constants can not be related to the equation
2. Not possible to derive relations including more than one term (ie we can not derive the relations of the form C=A+B)
3. Not possible to establish relation connectig more than 3 unknowns
4. Triganometric functions, logarthamic function, exponential functions etc can not be derived
5. Many physical quantities have same dimension so it is not possible to distinguish them by simple dimension analysis alone

 

^{****End of chapter 1*****}