physics - kar · mosos po a s ep,t important step, in understanding logarithms is to realize that...
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PHYSICSPHYSICS
VIKASANA - VIGNANA PATHADEDEGE NIMMA NADIGEBridge Course Program for SSLC Students who want to take up Science in I PUC in 2012
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
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+
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
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.
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”
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
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