Basic Formulae Mechanical

7/24/2019 Basic Formulae Mechanical

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Boyles Law Latent Heat of Fusion and Vaporization

Charles Law Black Body Radiation

Gas Law Newtons Law of Cooling

nthalpy of an !deal Gas "inetics #heory of Gas

$pecific Volu%e $pecific !%pulse

ntropy of $tea% $olar Radiation

Flow nergy Con&ecti&e Heat #ransfer

Law of #her%odyna%ics Radiation Heat #ransfer

#her%al 'pansion of $olids Conduction Heat #ransfer#her%al Conducti&ity

Boyles Law:

!f the te%perature (#) re%ain constant* the &olu%e (V) of a gi&en %ass of gas is in&erselyproportional to the pressure (+)

or +V, constant at a gi&en te%perature

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Charles Law
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(a) !f the pressure (+) is held constant* the &olu%e (V) of a gi&en %ass of gas &ariesdirectly as the a-solute te%perature (#)

(a) !f the &olu%e (V) is held constant* the pressure (+) of a gi&en %ass of gas &ariesdirectly as the a-solute te%perature (#)

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Gas Law:

+V , R#

R is uni&ersal gas constant

R , ./0120 % kgf3kg %ol 4"

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Enthalpy of an Ideal Gas:

#he property of enthalpy* H is defined as5

where

6,internal energy of an ideal gas

+,pressure

V,total &olu%e

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Specific Volume:
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or

where*

&,specific &olu%e* %73kg

',8uality of the %i'ture

9:',%oisture fraction of the %i'ture

,specific Volu%e of dry and saturated stea% at a particular pressure* %73kg

,specific &olu%e of saturated water* %73kg

, change in specific &olu%e during e&aporation

,

#op

Entropy of Steam

Change in entropy during heating of water and its e&aporation into stea% is gi&en -y

where*

,change in entropy

C,specific heat

,! for stea%

#9,initial te%perature

#;,final te%perature
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#s,stea% te%perature

721? kcal3kg

#op

Flow Energy:

Flow energy,work done -y a syste%

, p&

where*

p,pressure

&,@Ad for unit %ass

@,area of piston

Ad,displace%ent

!nternal energy ,u (for unit %ass flow)

#op

Law of hermodynamics

First Law5 hen a syste% undergoes a ther%odyna%ics cycle then the net heat () suppliedto the syste% fro% its surrounding is e8ual to the net work() done -y the syste% on itssurrounding1

!n sy%-ols*

Second Law5 !t is i%possi-le for a heat engine to produce net work () in a co%plete cycleif it e'changes heat only with -odies at a single fi'ed te%perature1
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Net heat supplied , Net work done

hermal efficiency

!t can -e seen that the second law i%plies that the ther%al efficiency of a heat engine %ustalways -e less than 9==D1

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hermal E!pansion of Solids:

#op

hermal Conducti"ity:

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Latent #eat of Fusion and Vapori$ation:

where* H , 8uantity of heat re8uired or li-erated* cal

% , %ass of a gi&en su-stance to -e fused or solidified* g

Lf , latent heat of fusion* cal3g

L&, latent heat of &aporization* cal3g

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Blac% Body &adiation:

where* , energy radiated per second -y a -ody* cal3c%;1s

# , a-solute te%perature* 4"

" , proportionality constant

, difference in energy radiation

#9, a-solute te%perature of cold -ody* 4"

#;, a-solute te%perature of hot -ody* 4"

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'ewtons Law of Cooling:

By e'panding the 8uantity in parenthesis and neglecting for s%all te%perature difference* wefind

(inetics heory of Gas:

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Specific Impulse:
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Solar &adiation:

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Con"ecti"e #eat ransfer:

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&adiation #eat ransfer:

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Conduction #eat ransfer:
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#op

)lge*ra

Algebra

Roots of a 8uadratice8uation

@reas

Linear e8uations Volu%e

Logarith%s @lge-ra For%ulae

!ne8ualitiesEensuration of

$olids

Eensuration of $urfaces

Basic Laws
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Roots of a quadratic equation

Linear e+uations

)reas


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Volumes

Logarithms

)lge*ra Formulae


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Ine+ualities

rigonometry #rigono%etry +age 9

#rigono%etric !dentities Co:Function !dentities

+ythagorean !dentities Negati&e !dentities

$u%3ifference For%ulas +ower Reducing

Rules of $ign $u% #o +roduct
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Reduction For%ulae +roduct #o $u%

Half @ngle ther #rigono%etry !dentities

ou-le @ngle
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#op

rigonometric Identities:

#op

,ythagorean Identities:

#op
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Sum - .ifference Formulas

#op

&ules of Sign

/uadrant sin cos tan cosec sec cot! !! : :!!! : : !V : :

#op

&eduction Formulae

)ngle-Function sin cos tan

:I :sin I cos I :tan I2=J: I cos I sin I cot I

2=J I cos I :sin I :cot I9.=J: I sin I :cos I :tan I9.=J I :sin I :cos I tan I;0=J I :cos I :sin I cot I;0=J I :cos I sin I :cot I7?=J I :sin I cos I :tan I

#op

#alf )ngle


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#op

.ou*le )ngle

#op

Co0function Identities

#op

'egati"e )ngle Identities


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#op

,ower &educing

#op

Sum o ,roduct

#op

,roduct o Sum


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#op

1ther rigonometry Identities

#op

.efinition of Integral Basic integrals

,ower rigonometric Functions 2ore rigonometric Function

#yper*olic FunctionsIn"ol"ing

In"erse rig Functions E!ponential and 'atural Log

Integrals In"ol"ing !n Integrals In"ol"ing a3*!

Integrals In"ol"ing Linear Factors In"ol"ing a4 5 *4 !4

In"ol"ingIntegrals in"ol"ing power of !

Integrals in"ol"ing rigonometric Functions

.efinition of Integral

Basic Integrals
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Derived SI Units

Energy:

The capacity to do work is called energy. Energy is neither created nor destroyed. It is

converted from one form to another.

Energy = mc2

where

m = mass


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c = speed of light

Kinetic Energy: Kinetic energy is the energy possessed by a body by virtue of its

motion. It is generally denoted by Ek. Its S.I. unit is and its dimension is !"#2T$2%.

&ence'

otential Energy !"E"#: (otential energy is the energy possessed by a condition orvirtue of its position or state or configuration. It is generally denoted by Ep. Its S.I. unit is

and its dimension is !"#2T$2%

&ence'

ower:

The time rate at which work is done is called power or work done per second is calledpower. It is generally denoted by (. Its S.I. unit is watt and its dimension is !"#2T$)%.

&ence'

$or%:

*ork is said to be done only when a force produces motion. *ork done in moving abody is e+ual to the force e,erted on the body and the distance moved by the body in the

direction of force. It is generally denoted by w. Its S.I. unit is and its dimension is.

&ence' * = -orce , istance

= -. , S.


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@cceleration Eo%ent of !nertia

@cceleration due to Gra&ity Eo%ent of Force of #or8ue

@ngular isplace%ent Eo%entu%

@ngular Eo%entu% +lankKs Constant

@ngular Velocity +otential nergy

Coefficient of Friction +ower

Coefficient of #her%al Conducti&ity +ressure

Coefficient of Viscosity $pecific Heat

Force $peed

Fre8uency $tress

Gra&itational Constant $urface #ension

Heat #her%al Capacity

!%pulse Velocity


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"inetic nergy ork

Latent Heat oung Eodulus

Acceleration:

It is defined as the rate of change of velocity "

Top

Acceleration due to gravity:

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Angular Dis&lace'ent:

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Angular 'o'entu' or 'o'ent of 'o'entu' L/

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Angular velocity:
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Top

(oefficient of )riction:

Top

(oefficient of *+er'al conductivity %,:

Top

(oefficient of viscosity: !-#

Top

)orce/

-orce =massacceleration = m a
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- = 0"1 0#T$21 = 0"#T$21

So' dimension of mass is and that of length is 3and that of time is 42 in force.

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.ravitational constant:

5ccording to 6ewton universal law of gravitation.

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/eat:

&eat is a form of energy.

7 = !"#2T$2%

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I'&ulse:

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Kinetic Energy !K"E"#:
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Top

Latent /eat:

&eat absorbed per unit mass during changed of state.

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0o'entu':

#op

0o'ent of a force of torque of 'o'ent of a cou&le/

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0o'ent of Inertia:

"oment of inertia = mass 8 0length12= !"#2%

I = !"#2%


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Top

lanc%1s constant:

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ower:

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otential Energy !"E"#:

Top

ressure/

Top
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S&ecific /eat:

Thermal capacity for unit mass of the body.

Top

S&eed

So' dimension of length is 3 and of time is 4 in velocity and speed.

Top

Stress : 2

Top

Surface *ension/

Top


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*+er'al (a&acity:

The amount of heat energy re+uired by a body for unit rise of temperature.

Top

3elocity:

Top

$or% of energy/

*ork = force 8 displacement = - 8 s

* =0"#T$21 8 0#1 = 0"#2T$21

Top

4oung 'odulus !4#:

0EASURE0E5*S SI U5I*S 6 *i'e


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*i'e !second#: 6 The second is the duration of 9'92':)';;


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Boiling ,oints of Li+uids

Li+uid .eg6 F@lcohol 907

Linseed il ?==Eercury ?/.+araffin >7?

+etroleu% 79?#urpentine 79>

ater (pure) ;9;

ater (sea) ;97ater (saturated

NetwonKs Law Li8uid +ressureEo%ent of !nertia +ascalKs +rinciple


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$pring Constant $urface #ensionBouncing Capillary

Laws of Conser&ation of Eo%entu% Vi-rationsensity

'ewton7s Laws of 2otion

First Law5 @ -ody at rest or in unifor% %otion will re%ain at rest or in unifor% %otion unless so%ee'ternal force is applied to it

Second Law of 2otion5 hen a -ody is acted upon -y a constant force* its resulting acceleration isproportional to the force and in&ersely proportional to the %ass*

where'

a=acceleration' ms2

-=force' 6

E,%ass of a -ody* kg

hird Law of 2otion5 !t states that to e&ery action force there is an e8ual and opposite reactionforce1

"otion /

isplace%ent

>elocity

5cceleration
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where'

S=distancecoveredby a moving body in time t' m

>=>elocity of a moving body' ms

5 =acceleration of a moving body' ms2

>


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K=spring constant

0b1 -= @k,

where'

-=force e,erted by spring against *. The minus sign indicates that , and - are in opposite directions.

3ibrations

. Simple &armonic "otion

where'

T=period of a vibration' s

n=fre+uency or vibration per unit time' s

#op

2. Spring (endulum

where'

T=period' s

"=mass of pendulum

K=spring

#op

). Simple (endulum
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where'

l=length of the pendulum

g=acceleration due to gravity

#op

A. *avelength

where'

>=total distance traveled in one second

B=length of one wave

C=number of waves per second

#op

>1 $peed of sound

where'

>=speed of sound at temperature tcD' ms

>o=speed at


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where'

6=beat fre+uency' i.e.' number of beats per second

6' n2=fre+uencies of two sources producing the sound' vibrationss

#op

01 oppler ffect

where'

6o=fre+uency heard by the observer

ns=fre+uency of the source

>=velocity of sound

>s=velocity of source

>o=velocity of the observer

#op

.1 !ntensity of sound

where'

E=intensity of sound at any distance d' microwattscm2or decibels

Eo=intensity of sound at unit distance' decibels

#op

9. >ibrating Strings
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where'

>=velocity of sound' ms

6=fre+uency or number of waves passing by per second

B=length of one wave or wavelength

-=tension in a rope or string' 6

"=mass of string per unit length' kgm

L,distance -etween two consecuti&e nodes* %

#op

9=1 $ound a&e #hrough Gas

where'

>=wave velocity' cms

(=gas pressure' dynescm2

G=gas density' gcm)

K=proportionality constant

$tress +ure $hear$train #orsion For%ula for #hin alled #u-es

HookeKs Law #orsion For%ula for Circular $haft+iossonKs Ratio Fle'ure For%ula6nit Volu%e Change $hear $tress in Bending

longation due to its eight #hin alled Hollow Ee%-ers (#u-es)#hin Rings $tress Concentration

$train nergy Cur&ed Bea% in +ure Bending#hin:walled +ressure &essels Bending of a Bea%

EohrKs Circle for Bia'ial $tress
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Stress

where' H=normal stress' or tensile stress' pa

(=force applied' 6

5=cross$sectional area of the bar' m2

=shearing stress' (a

5s=total area in shear' m2

Top

Strain

where'

=tensile or compressive strain' mm

=total elongation in a bar' m

=original length of the bar' m

Top

#oo%e7s Law

$tress is proportional to strain

where'

E=proportionality constant called the elastic modulus or modulus of elasticity or oungJs modulus' (a

Top
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iosson7s Ratio

where'

v=(oissonJs ratio

=lateral strain

=a,ial strain

Top

Unit 3olu'e (+ange

where'

=change in volume

=original volume

=strain

=(oissonJs ratio

Top

Elongation due to its weig+t

where'

=total elongation in a material which hangs vertically under its own weight

*=weight of the material

Top
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*+in Rings

where'

=ircumferential or hoop Stress

S=ircumferential or hoop tension

5=ross$sectional area

=ircumferential strain

E=oungJs modulus

Top

Strain Energy

where'

=total energy stored in the bar or strain energy

(=tensile load

=total elongation in the bar

#=original length of the bar

5=cross$sectional area of the bar
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E=oungJs modulus

6,strain energy per unit &olu%e

Top

*+in $alled ressure vessels

where'

=normal or circumferential or hoop stress in cylindrical vessel' (a

=normal or circumferential or hoop stress in spherical vessel' (a and longitudinal stress around the circumference

(=internal pressure of cylinder' (a

r=internal radius' m

t=thickness of wall' m

Top

2ohr7s Circle for Bia!ial Stress

Top

,ure Shear
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where'

=Shearing Stress' (a

=Shearing Strain or angular deformation

L=Shear modulus' (a

E=oungJs modulus' (a

>=(oissonJs ratio

Top

orsion formula for hin walled tu*es

where'

=ma,imum shearing stress' (a

=Shearing stress at any point a distance , from the centre of a section

r=radius of the section' m

d=diameter of a solid circular shaft' m

=polar moment of inertia of a cross$sectional area' mA


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T=resisting tor+ue' 6$m

6= rpm of shaft

(=power' k*

=angle of twist' radian

#=length of shaft' m

L=shear modulus' (a

do=outer diameter of hollow shaft' m

di=inner diameter of hollow shaft' m

and

Top

orsion formula for Circular Shafts

where'

=Ip' polar moment of inertia for thin$walled tubes

r=mean radius

t=wall thickness

Top

Fle!ure Formula
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where'

=Stress on any point of cross$section at distance y from the neutral a,is

=stress at outer fibre of the beam

c=distance measured from the neutral a,is to the most remote fibre of the beam

I=moment of inertia of the cross$sectional area about the centroidal a,is

Top

Shear Stress In Bending

where'

-=Shear force

7=statistical moment about the neutral a,is of the cross$section

b=width

I=moment of inertia of the cross$sectional area about the entroidal a,is.

Top

*+in6$alled /ollow 0e'bers !*ubes#

where' =shearing stress at any point of a blue
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t=thickness of tube

+=shear flow

T=applied tor+ue

M=distance between a reference point and segment ds

N=angle of twist of a hollow tube

Top

Stress Concentration

#op

Cur"ed Beam in ,ure Bending

where' =normal stress

"=bending moment

d5=cross$sectional area of an element

r=distance of curved surface from the centre of curvature

5=cross$sectional area of beam

M=distance of neutral a,is from the centre of curvature

M=distance of centroidal a,is from the centre of curvature

Top

Bending of a Bea'

!a# Bending of a Bea' Su&&orted at Bot+ Ends
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8*9 Bending of a Beam Fi!ed at one end

where' d= bending displacement' m

-=force applied' 6

I=length of the beam' m

a=width of beam' m

b=thickness of beam' m

=oungJs modulus' 6m2


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