PHYSICSPHYSICS

VIKASANA - VIGNANA PATHADEDEGE NIMMA NADIGEBridge Course Program for SSLC Students who want to take up Science in I PUC in 2012

MATHEMATICAL TOOLSMATHEMATICAL TOOLS

Mathematics is the TOOL of Physics.Mathematics is the TOOL of Physics.

A good knowledge and applications ofA good knowledge and applications of fundamentals of mathematics ( which are

d i h i )used in physics ),

helps in understanding the physical phenomena and their applications.p pp

Equations

A quadratic equation is an equation equivalent to f h fone of the form

Wh b d l b d 0

2 0ax bx c+ + =Where a, b, and c are real numbers and a ≠ 0

To solve a quadratic equation -- factorise.

2 5 6 0x x− + =

( )( )3 2 0( )( )3 2 0x x− − =

3 0 or 2 0x x− = − = 3x = 2x =3 0 or 2 0x x= = 3x = 2x =

2 4b b ac− ± −2 b 42

b b acxa

− ± −=

2 0ax bx c+ + =

This formula can be used to solve any quadratic equation

1 2 6 3 0x x+ + =

2 4b b acx − ± −=

(16 6 (3) 6 36 122

− ± −=

2x

a= )(1) 2

6 36 12 6 2424 4 6 2 6= × =

6 2 66 36 122

x − ± −=

6 242

− ±=

6 2 62

− ±=

( )2 3 6±

2 in common

( )2 3 6

2

− ±= 3 6= − ±

3 6 3 6− +3 6− − 3 6+

R tRoots

2 5 0x x+ − =( 3)( 2) 0x x+ − =

1 212

s ut at= +2

1 2150 202

t gt= − +2

Binomial Theorem

X 

1

2 3( 1) ( 2)(1 ) 1n n n n n− −2 3( ) ( )(1 ) 1 .......2! 3!

n n n n nx nx x x+ = + + + +

2 3( 2)( 2 1) ( 2)( 2 1)( 2 2)1 ( 2) .....2! 3!

x x x− − − − − − − −= + − + + +(1+x)-2

2 36 241 2 .....2! 3!

x x x= − + − +

1 2 x= −

2! 3!

( x << 1)( )

⎛ ⎞⎜ ⎟ 2

12

2gR hg = g

Rh

−⎜ ⎟⎛ ⎞⎜ ⎟′ = +⎜ ⎟⎜ ⎟ ⎝ ⎠⎛ ⎞12 RhR

R⎜ ⎟ ⎝ ⎠⎛ ⎞+⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠

2h⎞⎛ h2' (1 )hg g

R= −

⎟⎠⎞

⎜⎝⎛ <<1Rh

Introduction To Logarithms

Logarithms developed to simplifyLogarithms developed to simplifyLogarithms developed to simplify

complex arithmetic calculations

Logarithms developed to simplify

complex arithmetic calculationscomplex arithmetic calculations. complex arithmetic calculations.

Transform multiplicative processesTransform multiplicative processes

into additive ones. into additive ones.

MultiplyMultiply 2,234,459,912 and 3,456,234,459.

Without a calculator !

lot easier to add these two numbers.

What is a logarithm ?What is a logarithm ?What is a logarithm ?What is a logarithm ?

Definition of LogarithmDefinition of LogarithmDefinition of LogarithmDefinition of LogarithmSuppose b>0 and b≠1, Suppose b>0 and b≠1, there is a number ‘p’ such that:there is a number ‘p’ such that:logb n p=

pb n=If and only ifb n

fThe first, and perhaps the most important step, in os po a s ep,understanding logarithms is to realize that they alwaysto realize that they always relate back to exponential equations.

Example 1:

3Write 2 8 in logarithmic form.=

Solution:

log2 8 = 3 We read this as: ”the log base 2 of 8 is

equal to 3”equal to 3 .

Standard Formulae of logarithmsStandard Formulae of logarithms

1. loge mn = loge m + loge n

2. loge (m/n)= loge m - logen e e e

3 l n l3. loge mn = n loge m

Two Systems of LogarithmsNatural Logarithm. Logarithm of a number g g fto the base e (e = 2.7182) is called natural logarithm.Common Logarithm. Logarithm of a number to the base 10 is called commonnumber to the base 10 is called common logarithm. In all practical calculations, we always use common logarithm.always use common logarithm.

Conversion of Natural logarithm to Common logarithmCommon logarithm

Natural logarithms can be converted into l i h f llcommon logarithms as follows:

loge N = 2.3026 log10 Nge g10

≅ 2.303 log10 N

ExamplepWork done during an isothermal process is

V2

1

logeVW RTV

=

This can be written as

2 210 10log 2.0303 logV VRT RT

V V≈W = 2.3026

1 1V V

Introduction Trigonometric RatiosIntroduction Trigonometric Ratios

Trigonometry Trigonometry

means “Triangle” and “Measurement”means “Triangle” and “Measurement”

Vikasana – Bridge Cource 2012

OOppositte side

Adjacent sideθθ

Adjacent side

φφAdjacent sidde

O it idOpposite side

Three Types Trigonometric RatiosThree Types Trigonometric Ratios

There are 3 kinds of trigonometric ratios we will learnratios we will learn.

sine ratio

cosine ratio

tangent ratio

Definition of Sine Ratio.

θθ

Sinθ =Opposite side

Sinθ hypotenuses

Definition of Cosine RatioDefinition of Cosine Ratio.

θθθθ

CosθAdjacent Side

Cosθ = hypotenuses

Definition of Tangent Ratio.

θθ

t θOpposite Side

tanθ = Adjacent Side

sidesinhypotenuse

oppositeθ =hypotenuse

djacent sideaθ

Make Sure that the djace t s decos

hypotenuseaθ = triangle is

right-angled

sidetandj t id

oppositeθ =

Vikasana – Bridge Cource 2012

djacent sidea

Hypotenues= cosec θ (i e cosecant of θ)OppositeSide = cosec θ (i.e. cosecant of θ).

= sec θ (i e secant of θ)Hypotenues = sec θ (i.e. secant of θ)AdjacentSide

= cot θ (i.e. cotanent of θ)AdjacentSideO it Sid cot θ (i.e. cotanent of θ)OppositeSide

21

21

23

23

232

121

21

31

33

Angle θ

→

trig-ratio

↓

0O 30O 45O 60O 90O 120O 180O

sin θ 0 1 0

cos θ 1 0 − −1

tan θ 0 1 ∞ − 0tan θ 0 1 ∞ 0

(i) sin (90O+θ) = cos θ

(ii) cos (90O+θ) = sin θ

(iii) tan (90O+θ) = −cot θ

21

cos 120O = cos (90O + 30O) = − sin 30O = −

23

sin 120O = sin (90O + 30O) = cos 30O = 2

Some important Trigonometric FormulaeSome important Trigonometric Formulae

sin (A+B) = sin A cos B + cos A sin Bsin (A+B) sin A cos B + cos A sin B

cos (A+B) = cos A cos B − sin A sin Bcos (A+B) cos A cos B sin A sin B

. tan (A+B) =BABA

tantan1tantan

−+

BA tantan1