Algebra
Basic Laws
Roots of a quadratic equation
Linear equations
Areas
Volumes
Logarithms
Algebra Formulae
Inequalities
Logarithms
Algebra Formulae
Inequalities
Mensuration of Surfaces
Mensuration of Solids
Trigonometry
Trigonometric Identities
Co-Function Identities
Pythagorean Identities
Negative Identities
Sum/Difference Formulas
Power Reducing
Rules of Sign
Sum To Product
Reduction Formulae
Product To Sum
alf !ngle
"ther Trigonometry Identities
Dou#le !ngle
To$ Trigonometric Identities
To$ !ythagorean Identities
To$ Sum " #ifference Formulas
To$ Rules of Sign $uadrant
sin cosec
cos sec
tan cot
I II III I&
% % -
% %
% % -
To$ Reduction Formulae Angle"Function -' ()*- ' ()*% ' +,)*- ' +,)*% ' .)*% ' .)*% ' 0)*% '
sin -sin ' cos ' cos ' sin ' -sin ' -cos ' -cos ' -sin '
cos cos ' sin ' -sin ' -cos ' -cos ' -sin ' sin ' cos '
tan -tan ' cot ' -cot ' -tan ' tan ' cot ' -cot ' -tan '
To$ %alf Angle
To$ #ouble Angle
To$ &o'function Identities
To$ (egati)e Angle Identities
To$ !ower Reducing
To$ Sum To !roduct
To$ !roduct To Sum
To$ *ther Trigonometry Identities
Limits
Definition of limit The Intermediate &alue Theorem #efinition of limit
The Sandwich theorem
To$ The Intermediate Value Theorem
To$ The Sandwich Theorem
eri)ati)es
Definition of Derivative
Rolle1s theorem
Increment
2ean &alue theorem
Differential
Newton1s method
Derivatives
Derivatives of !lge#raic Functions
Derivatives of Trigonometric Functions
Derivatives of Inverse Trigonometric Functions
Derivatives of y$er#olic Functions
Derivatives of Inverse y$er#olic Functions
#efinition of #eri)ati)e
Increment
#ifferential
#eri)ati)es
Rolle+s Theorem
Mean Value Theorem
(ewton+s Method
#eri)ati)es of Algebraic functions Let u,-. be differential function with res/ect to - and n0 Let a and 1 be constant0
#efinition of Integral
Basic Integrals
!ower Trigonometric Functions
To$
To$
More Trigonometric Functions
%y/erbolic Functions
n)erse Trig Functions
To$
2-/onential and (atural Log
Integrals In)ol)ing Linear Factors
ntegrals in)ol)ing Trigonometric Functions
#efinition of Integral
Basic Integrals
n)erse Trig Functions
To$
2-/onential and (atural Log
Integrals In)ol)ing a3b-
4.
5.
6.
7.
8.
For ar#itrary $ositive integers m3 n the following reduction formula a$$lies
For m4n-+ the a#ove reduction formula can #e re$laced with
9.
:.
;.
Integrals In)ol)ing 4.
5.
6.
7.
8.
9.
:.
;.
<.
4=.
Integrals in)ol)ing /ower of -
&alculus The Fundamental Theorem of &alculus
#istance Formula
Riemann Sum
#efinite Integral
Mean Value Theorem for #efinite Integral
%y/erbolic Functions
If f is continuous on 5a3 #6 and a regular $artition of 5a3 #6 is determined #y a=x0 , x1 , x2 , .…., xn=b then Tra/e>oidal Rule
Sim/son+s Rule ,n must be an e)en number.
Mid/oint Rule
#is1 Method
Shell Method
Arc Length
#efinition of #ouble Integral
To$ Volume of the solid that lies under z=f(x,y) under z=f(x,y)
To$ !olar
To$
Surface Area
To$ #efinition of Tri/le Integral
7et T #e a lamina that has sha$e R in the 8y-$lane3 and the density is ρ(x,y) Mass of T
To$
Moments of T with res/ect to the a-es
To$
&enter of Mass
of T
To$
Moments of Inertia of T
To$
9!olar Moment of Inertia:
7et & #e a solid that has sha$e ; and the density is ρ(x,y,z) Mass
To$ Moments
To$ &enter of Mass
To$
Moments of Inertia
First /artial deri)ati)e of f with res/ect to x and y
Increment
To$
#eri)ati)e
To$ The &hain Rule
To$ #irectional #eri)ati)e
To$ ?radient of f =grad f
To$ Test for 2-trema
To$ La?range+s Theorem
Vectors
To$ &auchy'Schwar> Inequality
To$ Triangle Inequality
To$ #ot ,scalar. !roduct
To$ &ross ,)ector. !roduct
Statistics #efinition
To$ The 8'(umber Summary
2in ;+ 2edian ; 2a8 Inter +?@I;R or "utliers A ;+ % +?@I;R To$ @'Score ,standardi>ed )alue of -.
To$
&orrelation
M*TI*( Motion
The motion is said to #e one dimensional if it taBes $lace on a straight line $ath and can #e descri#ed with the hel$ of only one s$ace coordinate? The s/eed is defined as the distance covered #y the #ody $er second or the rate of change of distance with time? It is a scalar
A)erage s/eed
The ratio of the total distance covered and the total time taBen? The unit of s$eed in C??S? system is centimeter $er second and meter $er second in 2??S? or S?I? system? Thus3
A)erage Velocity A)erage )elocity is defined as the total dis$lacement divided #y total time in a $articular direction? Its unit in C??S? system is centimeter $er second and metre $er second in 2??S? or S?I? system3 Thus3
Acceleration Acceleration is defined as the rate of change of velocity with time? It is a vector
The acceleration is uniform if its velocity changes #y e
Acceleration due to ?ra)ity ,g.
The acceleration of a freely falling #ody due to earthEs attraction is called acceleration due to gra)ity0
Motion under ?ra)ity
If a #ody falls under gravity then all the e
The ma8imum height attained #y a
ii?
The time taBen to reach the highest $oint 4 The time taBen to reach the $oint from where it is $roJected
The total time taBen #y a #ody in going to the ma8imum height and falling #acB to the $roJected $oint
iv? Ghen a #ody striBes the same $oint from where it was $roJected3 the velocity #ecomes the same as that of the velocity of $roJection? Velocity
The rate of change of dis$lacement with regard to time of an o#Ject is called the )elocity of the o#Ject? It may also #e defined as the s$eed of the o#Ject in a $articular direction3 it is a vector
2nergy
The ca$acity to do worB is called energy? Knergy is neither created nor destroyed? It is converted from one form to another? Knergy 4 mc
where m 4 mass c 4 s$eed of light
inetic 2nergy inetic energy is the energy $ossessed #y a #ody #y virtue of its motion? It is generally denoted #y KB ? Its S?I? unit is L and its dimension is 527 T-6?
ence3
!otential 2nergy ,!020. Potential energy is the energy $ossessed #y a condition or virtue of its $osition or state or configuration? It is generally denoted #y K $? Its S?I? unit is L and its dimension is 527 T-6
ence3
!ower
The time rate at which worB is done is called $ower or worB done $er second is called $ower? It is generally denoted #y P? Its S?I? unit is watt and its dimension is 527 T-6? ence3
or1
GorB is said to #e done only when a force $roduces motion? GorB done in moving a #ody is e
Cnits of Measurement
Acceleration
It is defined as the rate of change of velocity 0
To$ Acceleration due to gra)ity
To$ Angular #is/lacement
To$ Angular momentum or moment of momentum L
To$ Angular )elocity
To$ &oefficient of Friction
To$ &oefficient of Thermal conducti)ity 1 =
To$ &oefficient of )iscosity ,D.
To$ Force
Force 4massMM acceleration 4 m M a F 4 92: M97T -: 4 927T -:
So3 dimension of mass is + and that of length is %+and that of time is > in force? To$
?ra)itational constant
!ccording to Newton universal law of gravitation?
To$ %eat
eat is a form of energy? ; 4 527T-6 To$
Im/ulse
To$ inetic 2nergy ,020.
To$ Latent %eat
eat a#sor#ed $er unit mass during changed of state?
To$ Momentum
To$ Moment of a force of torque of moment of a cou/le
To$ Moment of Inertia
2oment of inertia 4 mass 9length: 4 5276 I 4 5276 To$ !lanc1+s constant
To$ !ower
To$ !otential 2nergy ,!020.
To$ !ressure
To$ S/ecific %eat
Thermal ca$acity for unit mass of the #ody?
To$ S/eed
So3 dimension of length is %+ and of time is >+ in velocity and s$eed?
To$ Stress E
To$ Surface Tension
To$ Thermal &a/acity
The amount of heat energy re
To$ Velocity
To$ or1 of energy
GorB 4 force dis$lacement 4 F s G 4927T -: 97: 4 927 T-: To$ oung modulus ,.
M2ASCR2M2(TS C(ITS ' MASS Mass ,1ilogram: The Bilogram is the unit of mass? It is e
2ass is denoted #y 1g For all $ractical $ur$oses Conventional 2ass is related to 2ass via the following e
where • •
•
2c is the conventional mass value 2 is the mass value is the density of the test weight material in Bgm -
Cnits of mass
There have historically #een four different Knglish systems of mass Tower weight3 Troy weight3 avoirdu$ois weight3 and a$othecaries weight? Tower weight fell out of use 9due to legal $rohi#ition: centuries ago3 and was never used in the nited States? Troy weight is still used to weigh $recious metals? !$othecaries weight3 once used in $harmacy3 has #een largely re$laced #y metric measurements? !voirdu$ois weight is the $rimary system of mass in the ?S? customary system? The !voirdu$ois units are legally defined as measures of mass3 #ut the names of these units are sometimes a$$lied to measures of force? For instance3 in most conte8ts3 the $ound avoirdu$ois is used as a unit of mass3 #ut in the realm of $hysics3 the term $ound can re$resent $ound-force 9a unit of force $ro$erly a##reviated as l#f:? 2nglish Standards of weight A)oirdu/ois weight
.?UU grains
4
+ dram
+0 drams
4
U. V grains
+0 ounces
4
.))) grains
+U $ounds
4
+ stone
, $ounds
4
+
U
4
++ $ounds
) hundredweights
4
U) $ounds
$ennies
4
@ half$ennies
U grains
4
+ carat
U grains
4
+ $ennyweight
) $ennyweights
4
U,) grains
+ troy ounces
4
@.0) grains
The ton and hundredweight a#ove are referred to as the short ton3 and the short hundredweight 3 to distinguish them from the Oritish Im$erial ton and hundredweight3 which are larger and hence are referred to as the long ton and long hundredweight ? The long ton has limited use in the nited States? + long hundredweight 4 ++ l# W @)?,) Bg + long ton 4 ) long cwt 4 U) l# W +)+0?)U. Bg W +?)+0 t
• •
A/othecariesG weight
The grain has the same definition as for avoirdu$ois weight? + scru$le 9s a$: 4 ) gr W +?(0 g + dram a$othecaries 9dr a$: 4 s a$ W ?,,, g + ounce a$othecaries 9oH a$: 4 + oH t 4 , dr a$ 4 U,) gr W +?+) g + $ound a$othecaries 9l# a$: 4 + l# t 4 + oH a$ 4 @.0) gr W .?U g
• • • •
The $ound and ounce a$othecaries are identical to the $ound and ounce troy?
Troy weight
The grain has the same definition as for !voirdu$ois weight? + $ennyweight 9dwt: 4 U gr W +?@@@ g + ounce troy 9oH t: 4 ) dwt 4 U,) gr W +?+) g + $ound troy 9l# t: 4 + oH t 4 @.0) gr W .?U g
• • •
Metric Standards of weight eight
+ grain 9avoirdu$ois and troy:
4
)?)0U, grammes
+ $ennyweight
4
+?@@@ grammes
+ dram 9avoirdu$ois:
4
+?.. grammes
+ ounce 9+0 drams:
4
,?@) grammes
+ ounce 9troy and a$othecary:
4
+?+)@ grammes
+ $ound 9+0 ounces:
4
)?U@@( Bilogram
+ hundredweight
4
++ $ounds
+ Oritish ton
4
U) $ounds
+ !merican ton
4
))) $ounds
+ milligram
4
)?)+@ grain
+ gramme
4
+@?U grains
+ Bilogram
4
?)U0 $ounds
+
4
+)) Bilograms
+ tone
4
)?(,U Oritish ton
+ gallon of water at 0XF
4
+) l#
+ cu#ic foot water at 0)XF
4
0?. l#
+ cu#ic inch water at 0)XF
4
)?)0+ l#
U hours
4
+UU) min
+ hour
4
.)
+ metre
4
ft in?
#istincti)e Masses of &ertain *bHects *bHect Klectron Proton ranium atom Red #lood cell Dust Particles Rain Dro$ 2os
Mass ,g. +)-) +)-. +)-@ +)-+ +)-( +)-0 +)-0 +)- +) +) +)0 +) +)@ +)) +)U+ +)@@
M2ASCR2M2(TS C(ITS ' M2T2R Meter ' 2eter is +30@)3 .0?. times the wavelength of the orange light in vacuum emitted #y0 ,0 9ry$ton: in the transition $ +) to @d@? The meter is length of the $ath traveled #y the light in vacuum during a time interval of +/((3.(3U@, of a second? 9+(,:? It is denoted #y m0 Cnits of Meter SI
+ meter 9m: 4 +)) centimeters 9cm: + meter 4 +))) millimeters 9mm: + meter 4 +) decimeters 9dm: +) meter 4 + decameter 9dm: +) decameter 4 + hectometer 9hm: +) hectometers 4 + Bilometer 9Bm: + Bm 9Bilometer: 4 +)))m4 +)Ym + hm 9hectometer:4 +))m4 +)Z m + cm 9centimeter: 4 +)mm 4 +)-[m + fermi 4 +f 4 +) -+@ m
+ angstrom 4 +!) 4 +)-+) m + astronomical nit 4 + ! 9distance of the sun from the Karth 4+?U(0+)- ++ m + light year 4 (?U0+) +@ m 9distance that light travels with velocity of +) , m s-+ in + year: + $arsec 4 ?),+)+0 m 9distance corres$onding to an annual $aralla8 of one second:?
2nglish Standards of length ,Im/erial. Cnits of Length
+ inch
4
Diameter of half$enny
.?( inches
4
+ linB
+ inches
4
+ foot
feet
4
+ yard
0 feet
4
+ fathom
@ V yards
4
@ linBs
+)) linBs
4
00 feet
) yards
4
U) $oles
, furlongs 0),) feet $er hour
4 4
,) chains + !dmiralty Bnot
0),) feet
4
+ nautical mile 9Oritish:
Surface
+UU s
4
+ s
( s
4
+ s
) \ s
4
+ s
U) s
4
+ rood
U roods
4
U,U) s
0U) acres
4
+ s
Metric Standards and 2qui)alents
The following e
+ inch
4
?@U centimeters
+ foot
4
)?U.(( centimeters
+ yard
4
)?(+U(( meter
+ mile
4
+?0)( Bilometers 4 @,) feet
+ chain
4
)?++0, meters 4 yards
+ centimeter
4
)?(.)++ inch
+ meter
4
(?.)++ inches 4 ?,),U feet 4 +?)(0+U yards?
+ Bilometer
4
)?0+. mile
+ radian
4
@.?(0 degrees 9angular measure:
Square Measure
+ s
4
0?U@+0 s
+ s
4
(?() s
+ s
4
)?,0+0 s
+ rood
4
+)?++. acres 4 U) $erches
+ acre
4
)?U)U0, hectare
+ s
4
@( hectares
+ s
4
)?+@@ s
+ s
4
+)?.0( s
+ are
4
++(?0 s
+ hectare
4
?U.++ acres 4 +)3))) s
+ cu#ic inch
4
+0?,. cu#ic centimeters?
+ cu#ic foot
4
)?),+. cu#ic meter 4 ,?+. litres
+ cu#ic yard
4
)?.0U@@ cu#ic meter 4 .0U?@@ litres
+ gallon
4
U?@U@(0 liters 4 )?+0)@ cu#ic foot
+ ?S?!? gallon
4
+ cu#ic inches 4 )?,0. im$erial gallon
+ liter
4
+?.@(, $ints 4 )?)) gallon
+ cu#ic centimeter
4
)?)0+ cu#ic inch?
+ cu#ic decimeter
4
0+?)U cu#ic inches?
+))) cu#ic centimeters
4
+ liter 4 )?).@ #ushel
+ cu#ic meter
4
@?+U, cu#ic feet 4 +?).(@U cu#ic yards
&ubic Measure
Thermodynamics
First Law
Second Law
inetics
Strength of Materials
Stress
where3
]4normal stress3 or tensile stress3 $ a P4force a$$lied3 N !4cross-sectional area of the #ar3 m 4shearing stress3 Pa !s4total area in shear3 m To$
Strain
where3 4tensile or com$ressive strain3 m/m 4total elongation in a #ar3 m 4original length of the #ar3 m To$
%oo1eGs Law
Stress is $ro$ortional to strain
where3 K4$ro$ortionality constant called the elastic modulus or modulus of elasticity or ^oungEs modulus3 Pa
To$ !iossonGs Ratio
where3 v4PoissonEs ratio 4lateral strain 4a8ial strain To$
Cnit Volume &hange
where3 4change in volume
4original volume
4strain 4PoissonEs ratio To$
2longation due to its weight
where3 4total elongation in a material which hangs vertically under its own weight G4weight of the material To$
Thin Rings
where3 4Circumferential or hoo$ Stress S4Circumferential or hoo$ tension !4Cross-sectional area 4Circumferential strain K4^oungEs modulus To$
Strain 2nergy
where3 4total energy stored in the #ar or strain energy P4tensile load 4total elongation in the #ar 74original length of the #ar !4cross-sectional area of the #ar K4^oungEs modulus 4strain energy $er unit volume
To$
Thin alled !ressure )essels
where3 4normal or circumferential or hoo$ stress in cylindrical vessel3 P a 4normal or circumferential or hoo$ stress in s$herical vessel3 P a and longitudinal stress around the circumference P4internal $ressure of cylinder3 P a r4internal radius3 m t4thicBness of wall3 m To$
MohrGs &ircle for Bia-ial Stress
To$
!ure Shear
where3 4Shearing Stress3 P a 4Shearing Strain or angular deformation 4Shear modulus3 Pa K4^oungEs modulus3 Pa &4PoissonEs ratio
To$
Torsion formula for Thin walled tubes
where3 4ma8imum shearing stress3 Pa 4Shearing stress at any $oint a distance 8 from the centre of a r4radius of the section3 m
section
d4diameter of a solid circular shaft3 m 4$olar moment of inertia of a cross-sectional area3 m U T4resisting tor
and
To$
Torsion formula for &ircular Shafts
where3 4I $3 $olar moment of inertia for thin-walled tu#es r4mean radius t4wall thicBness To$
Fle-ure Formula
where3
4Stress on any $oint of cross-section at distance y from the
neutral a8is
4stress at outer fi#re of the #eam c4distance measured from the neutral a8is to the most remote fi#re of the #eam I4moment of inertia of the cross-sectional area a#out the centroidal a8is To$
Shear Stress In Bending
where3 F4Shear force ;4statistical moment a#out the neutral a8is of the cross-section
#4width I4moment of inertia of the cross-sectional area a#out the Centroidal a8is? To$
Thin'alled %ollow Members ,Tubes.
where3
4shearing stress at any $oint of a #lue t4thicBness of tu#e <4shear flow T4a$$lied tor
Stress &oncentration
To$ &ur)ed Beam in !ure Bending
where3
4normal stress 24#ending moment d!4cross-sectional area of an element r4distance of curved surface from the centre of curvature !4cross-sectional area of #eam R4distance of neutral a8is from the centre of curvature R +4distance of centroidal a8is from the centre of curvature To$
Bending of a Beam ,a. Bending of a Beam Su//orted at Both 2nds
,b. Bending of a Beam Fi-ed at one end
where3
d4 #ending dis$lacement3 m F4force a$$lied3 N I4length of the #eam3 m a4width of #eam3 m #4thicBness of #eam3 m ^4^oungEs modulus3 N/m
2lectrical &ircuits Laws of Resistance
+?
The resistance of a conductor varies directly as its length?
50
The resistance of a conductor varies inversely to its cross section area?
60
The resistance of a conductor de$ends on the material?
70
The resistance of a conductor de$ends on its tem$erature?
The a#ove factors can #e summed u$ mathematically as
where ` is constant re$resenting the nature of material and is Bnown as s$ecific resistance? To$ 2ffect of Tem/erature on Resistance ,a. Resistance of $ure metals and alloys increases with rise in tem$erature? ,b. Resistance of electrolytes3 insulators and semiconductors decreases with rise in tem$erature?
If R o 4 Conductor resistance at )XC R t 4 Conductor resistance at tXC t 4 rise in tem$erature and
o 4 Tem$erature coefficient of resistance at )XC3
Then
R t 4 R o 9+ % ot:
The tem$erature coefficient at )XC is defined at the change in resistance $er ohm for a rise in tem$erature of +XC from )XC? To$ ?rou/ing of &ells
! single cell has an e?m?f? of a#out +?@ volts? If either more voltage is needed or more current is re
Ghen cell are connected in series3 e?m?f? of the #attery is e
where e is e?m?f? of the cell and n refers to the num#er of cells in series? To$ &ells in !arallel
The cells when arranged in $arallel have the same e?m?f?3 #ut internal resistance of the unit is reduced? If n cells are connected in $arallel3 each of e?m?f? K3 then e?m?f? of the #attery 4 e?m?f? of one cell 4 K
where r is the resistance of one cell To$ &ells in Series and in !arallel
n 4 num#er of cells in each row m 4 num#er of $arallel rows? N 4 total num#er of cells 4 mn? 7et e?m?f of one cell
4e
K?2?F? of #attery
4ne volts
Internal resistance of each row 4nr ohms
To$ &ell 2fficiency
The efficiency of a Cell is considered in two ways 9+: !m$ere-hour 9!?h: efficiency? 9: Gatt-hour 9G?h?: efficiency?
To$ #0&0 Motors
D?C? motors are classified according to the method of e8citation? They may #e of the shunt3 series or com$ound ty$es?
Series Motor
The s$eed of a series motor is given #y the relation
Shunt motor
where R a is the armature resistance? Since constant?
is $ractically constant at all loads3 s$eed is there almost
To$ Ty/es of Armature inding
The two main ty$es of winding are +? ?
La/ inding0 It is also Bnown as $arallel winding or multi$le winding? In this ty$e of winding3 the num#ers of $arallel $aths 9!: are e
To$ Sli/
The rotor of induction motor rotates at somewhat lesser s$eed than the synchronous s$eed and actual s$eed is Bnown as sli$?
where Nr is rotor or actual while N is synchronous s$eed?
To$ A& through Resistance and Inductance
In the resistance $art of the circuit the current is in $hase with the voltage3 while in the inductive $art it is ()X out of $hase? ence3 to determine the current3 the effect of resistance and reactance has to #e com#ined which is named as im$edance
where
is the $hase angle #etween the voltage and current and cos
is called the $ower factor? To$
&ircuits &ontaining Resistance Inductance and &a/acitance
To$ Transformer
Transformer is a device for transferring energy from one alternating current circuit to another without any change in fre
To$ Three !hase Transformer
Ghenever the su$$ly is three $hase and it is desired to transform current at another voltage3 then either a single three $hase transformer or three se$arate single $hase transformers can #e used? owever3 in $ractice a single three $hase transformer is used? The three $hase winding of a transformer can #e connected either in star or in delta?
&urrent and )oltage in star
The e?m?f? #etween any line and the neutral gives the $hase voltage while the e?m?f? #etween two outer terminals is Bnown as line voltage? Current in each line is the same as $hase current i?e? 7ine current I 74Phase current IP
Line Voltage and &urrent in #elta
*hmGs Law
where3 I 4 current3 am$ & 4 $otential difference3 volts R 4 resistance3 ohms To$ Resisti)ity
where3 ` 4 resistivity or s$ecific resistance3 ohm-m 7 4 length of wire3 m ! 4 cross-sectional area of wire3 m To$ &onductance ,?.
where3 b 4 conductivity or s$ecific conductance3 mhos/m To$ 2lectric !ower
where3 P 4 electric $ower3 watts To$ 2lectric 2nergy
where3
K 4 electric energy3 Loules & 4 $otential difference3 &olts I 4 Current3 !m$eres R 4 resistance3 ohms t 4 time3 seconds K 4 )?UIRt3 cal
For Oattery
where
To$ irchhoff+s Law ,i.
The alge#raic sum of all the currents directed towards a Junction $oint is Hero?
The alge#raic sum of all the voltage rise taBen in a s$ecified direction around a closed circuit is Hero?
,ii.
Series Circuits
Parallel Circuits
&oulomb+s Law
The force acting #etween two charged #odies < + and < in air is $ro$ortional to the $roduct of charges and inversely $ro$ortional to the s
Machines
! machine is a device #y which a force a$$lied to some $oint in certain direction is made availa#le at some $oint and in some other direction? The force P a$$lied on the machine is called the effort or the $ower while the resistance G overcome #y the machine is called weight or load? •
Mechanical Ad)antage
The ratio3 load/effort is called mechanical advantage of a machine? The mechanical advantage should #e greater than one? If3 in a machine3 the ratio is less than one3 it would #e more accurate to call is mechanical disadvantage?
•
Velocity Ratio
The ratio
is called the velocity ratio of a machine? The two distances are moved in the same interval of time3 so they are $ro$ortional to the velocities of the effort and load? Pd+ 4 Gd in a $erfect machine? or
where d+ and are d are the distances moved #y effort P and load G res$ectively? ence3 in a $erfect machine the mechanical advantage is e
In all machines some worB is always wasted friction? The result of it is that the worB done #y the effort in a given time3 called total worB or worB in$ut 94 $ d +: is always greater than the worB done on the load 94 G d: called useful worB or worB out$ut? The difference of the latter from the former 4 lost worB 9Pd +-Gd:?
The ratio is called efficiency of machine? It is also defined as the ratio
In any actual machine the efficiency is always less than one #ut in a $erfect or ideal machine in where there is no friction at all the efficiency is e
"r Mechanical ad)antage 4 2fficiency J )elocity ratio i?e? 2?!? 4&elocity Ratio To$ The !rinci/le of or1
In any actual machine3 the useful worB o#tained is always less than total worB done #y the effort? This is #ecause 9i: worB has to #e done in lifting its $arts which have weight3 and 9ii: #ecause there is always some internal friction which has to #e overcome? ! $erfect or ideal machine is one which has no weight and the efficiency of the machine is unity? Princi$le of the worB is the $rinci$le of conservation of energy i?e?3 the worB done #y a machine is e
If G increase3 then d decrease in the same ratio? ence in a machine3 whatever is gained in $ower is lost in s$eed or distance? Cses of a machine
9i: This ena#les one to lift weight or overcome resistances much greater than one could do unaided as in the case of a $ulley-system3 a wheel and a8le3 a crow #ar3 a sim$le screw JacB3 etc? 9ii: This ena#le one to convert a slow motion at some $oint into a more ra$id motion at some other desired $oint3 viH?3 a #icycle3? ! sewing machine etc? !n o$$osite effect may also #e arranged in $ractice when necessary 9iii: This ena#le one to use a force acting at a $oint to #e a$$lied at a more convenient $oint3 as in the case of a $oBer for stirring u$ fire3 or to use a force acting at a $oint in a more convenient manner3 e?g?3 lifting of a mortar #ucBet to the to$ floor #y means of a ro$e $assing over a $ulley fi8ed at the to$ of the #uilding3 the other end of the ro$e #eing $ulled down #y an agent remaining on the ground? 9iv: This ena#les one to convert a rotatory motion into a linear motion or vice versa3 as in the case of a racB and $inion3 etc 9v: This ena#les one to convert a to and fro motion into a rotatory motion or vice versa3 e?g?3 a cranB used in the heat engine? To$ Ty/es of Sim/ly Machines
The following si8 sim$le machines re$resent the ty$es of $rinci$les used in maBing $ractical machines? 9+: Pulley? 9: Inclined $lane? 9: 7ever? 9U: Gheel and a8le? 9@: Screw and? 90: Gedge? To$
,4. !ulley
! $ulley is a sim$le machine consists of a grooved wheel3 called sheave3 over which a string can $ass? The wheel is ca$a#le of turning freely a#out an a8le $assing through its centre? The a8le is fi8ed to a frame worB3 called the #locB? The $ulley is fi8ed or mova#le accordingly as its #locB is fi8ed or mova#le?
The Single Fi-ed !ulley
Shows a fi8ed $ulley in which the #locB of the $ulley is fi8ed to a rigid su$$ort? The load G is attached to one end of the string $assing round the groove of the $ulley and effort P is a$$lied at the other of the $ulley and effort P is a$$lied at the other end? Gith a $erfectly smooth $ulley and a weightless string3 the tension of the string will #e the same throughout? ence the distance through which the load is raised3 is e
P !" 4 G "O !" 4 "O
2echanical advantage
In $ractice3 $ulleys are not $erfectly frictions3 and G is always less than P3 that is3 the mechanical advantage is always less than +? Out in s$ite of this3 the arrangement is useful as the o$erator can use the weight of this #ody for raising the land? It is generally used for raising weights3 drawing curtains etc?
Single Mo)able /ulley
ere one end of the string $asses round the $ulley ! and is attached to a fi8ed su$$ort 2 as shown in? The effort P is a$$lied at the other end of the ro$e $assing over a fi8ed $ulley O? The load w to #e raised is attached to the #locB of mova#le $ulley !? The fi8ed $ulley is used only to a$$ly the $ower in the downward direction? It is assumed that the $ulleys are frictionless and the tension in the string is the same at all $oints and is e
Thus a given effort can raise twice its weight? If the weight of the $ulley is w3 then G%w4P
"r 2echanical advantages 4 The single mova#le $ulley is much used in cranes? Sails in #oats and flags are raised and lowered with the hel$ of mova#le $ulleys?
&ombination of !ulleys
! com#ination of $ulleys is very often used to secure a mechanical advantages greater than two? Different systemEs having different mechanical advantages are used for different $ur$oses3 #ut the most im$ortant com#ination3 which is in general use3 is given here?
!ulley Bloc1
This system consists of two #locBs3 each containing or $ulleys? The u$$er #locB is fi8ed to a su$$ort and the lower one is mova#le to which the load G is attached? The string is attached to the u$$er or to the lower #locB3 and is then $assed round a mova#le and a fi8ed $ulley in turn finally $assing over a fi8ed $ulley3 the effort $ #eing a$$lied at the free end? It should #e noted that when the string is attached to the u$$er #locB3 the num#ers of wheels in the two #locBs must #e the same3 #ut when it is attached to the lower #locB3 the num#er of wheels in it will #e one less than that in the u$$er one? The tension everywhere round the string is the same and is e
To$ ,5. The Inclined !lane
!n inclined $lane is a smooth rigid flat surface inclined at an angle to the horiHontal? It is used to facilitate the rising of a heavy #ody to a certain height #y the a$$lication of a force which is less than the weight of the #ody? The fair-case and the roads on the hills are ty$ical e8am$les? 7et !O #e a $lane inclined at an angle ' to the horiHontal line !C and OC is the height of the $lane? The #ody $laced on the $lane is acted u$on #y three forces 9i: G3 its weight acting vertically downwards 9G4mg:3 9ii: $3 the force or effort3 and 9iii: R3 the reaction of the $lane? &ase0 I0 7et the force P act u$ward along the $lane?
In order that the #ody may #e in e
ence mg sin ' > P 4 ) mg cos ' > R 4 ) P 4 mg sin ' R 4 mg cos '?
that is3 a #ody of weight G 4 mg can #e su$$orted #y a force P 4 G/ acting u$ the $lane? To$ ,6. The Le)er
! lever is a sim$le machine and consists the rest of the lever can turn? This fi8ed $oint is called the fulcrum? The $er$endicular distance #etween the fulcrum and the $oint of a$$lication of the $ower is called the $ower arm the $er$endicular distance #etween the fulcrum and the $oint of a$$lication of the weight arm of the lever?
Mechanical ad)antage of le)er
7et !O #e a lever with the $ower P is a$$lied at ! and weight G to #e lifted is sus$ended at O? !F and Of are the $ower and weight arms res$ectively? The forces act $er$endicular to the arms and Bee$ the lever in e
This is Bnown as the $rinci$le of lever? The lever is used for $ulling weights or overcoming resistance #y the a$$lication of force at a suita#le $oint? From the a#ove relation it is clear that #y increasing the length of the $ower arm3 we can lift a greater weight with the lever?
The Straight Le)ers
a:
#:
c:
Ghen the lever is straight and the effort and the weights act $er$endicular to the lever3 the following three distinct classes of levers are found in $ractice according to the relative $ositions of !3 O and Fthe $oints of a$$lication of the effort3 the weight and the fulcrum res$ectively? In each of these cases3 three $arallel forces are acting on the #ody? They3 therefore3 are in the e
A le)er of the first order
In this case the effort P and the weight G act on the o$$osite sides of the fulcrum F? ere reaction R is the middle force3 therefore3 P and G act in the same direction and R in the o$$osite direction? To lift the weight3 the effort must #e a$$lied downwards3 and the reaction acts u$wards so that lever $resses downwards on the fulcrum? TaBing moments a#out F3 we have
!F may #e either greater3 e
Second *rder
In this case3 the weight is $laced #etween the effort and the fulcrum? ence G is the middle force? Therefore3 P and R act in the same direction and G in the o$$osite direction? To lift the weight3 the effort must #e a$$lied in the direction? The reaction of the fulcrum also acts u$wards? TaBing moments a#out the fulcrum F3 we have3
Out in this class of levers !K is greater than Of3 there force3 mechanical advantage G/P is greater than unity? !lso
G 4 P % R
"r
R 4 G- P
The e8am$les of this class are a crow #ar3 a wheel #arrow3 a tin o$ener3 foot #ellows and lifting of a lid of #o8? ! $air of ordinary nut-cracBers and corB s
Third *rder
ere the effort P is $laced #etween the weight and the fulcrum as shown in? Since is the middle force3 therefore3 G and R act in the same direction and P in the o$$osite direction? !lso P 4 R %G "r
R4P-G
Thus to lift weight the effort must #e a$$lied u$wards3 the reaction of fulcrum acts downwards or lever $resses fulcrum u$ward? TaBing moments a#out the fulcrum F3 we have3
Out in this lever !F is less than Of3 therefore3 mechanical advantage is less than unity? This shows a mechanical disadvantage? This arrangement gives G a large movements for a small movement of the effort P3 a fact which is Just o$$osite to what ha$$ens in the other two ty$es of levers? The e8am$les of this ty$e of lever are the forearm3 treadle of a sewing machine in BicBing a foot#all and in using a cricBet #at? ! fire tongs3 a $air of force$s used in a weight #o83 the u$$er and lower Jaws of the mouth are e8am$les of dou#le levers of this class? To$ ,7. The heel and A-le
The wheel and a8le is a modification of the lever? It consists of two cylinders of different diameters ca$a#le of turning a#out a common fi8ed a8is? The larger cylinder is called the wheel and the smaller the a8le? The load G to #e raised is attached to a ro$e coiled around the wheel in the o$$osite direction3 so that when the ro$e round the wheel is un-coiled3 the ro$e round the a8le is coiled u$ and there#y the weight is raised? Shows a section where "O is the radius r of the a8le "! the radius R of the wheel? TaBing moments a#out "3 P "! 4 G "#
A//lication
The windlass #y which water is drawn from a well is the same as the wheel and a8le3 the cranB-handle of which serves the $ur$ose of the wheel? The ca$stan used for lifting an anchor in shi$s3 a coffee grinder3 a s$anner used to wind a nut3 the steering of a motor car3 #icycle $edal etc? are all a$$lications of the wheel and a8le? To$ ,8. Screw
The screw is a rod usually of some hard metal on the surface of which is cut a s$iral groove? The successive turns of these grooves are se$arated #y a s$iral ridge Bnown as thread? The screw worBs in a collar or nut through which a hole is #ored3 having a groove to fit the thread to fit groove of the screw? ! screw is generally $rovided with an arm #y means of which it can #e rotated? Ghen the screw turns in a fi8ed nut3 it moves forward or #acBward in the direction of its length? In each turn of the screw3 the distance moved is e
Mechanical Ad)antage
To find the $itch of screw3 count the num#er of grooves or threads in one centimeter and thus calculate $itch as
when the screw worBs without friction?
Screw Kac1
Screw JacB as shown in is used for lifting heavy loads liBe an automo#ile in garages and worBsho$s for re$air $ur$ose etc? It consists of a strong hollow metallic ta$ering cylinder ! having at the to$ a hole in the form of a nut N with grooves cut on its inner side? ! s
Oy using a system of cog wheel3 the load is raised through a distance e
inch of Lifting &rab
This machine is used for lifting com$aratively heavy loads #y means of a small effort such as can #e e8erted #y hand? It is used when greater velocity ratio and also mechanical advantage is re
Single !urchase &rab
It consists of two stands ! + and !? Connected rigidly together #y the three stays O +3 Oand O and having #earings for the s$indle O and the drum D? "n the s$indle C is Beyed the $inion K3 which gears with and drives the larger s$ur wheel F on the drum D? The ends of C are s
