MATHEMATICAL FORMULA:

Force: force is defined as an external agent which changes or tends to change the state of
rest or of unifor otion of a !od"# Force is a $ector %uantit"#
Characteristics of a force: the %uantities !" which a force is full" re&resented are called
characteristics of the force# The $arious characteristics of a force are
'# Magnitude: the %uantit" of a force is called its agnitude#
(# )irection: the line along which the force acts is called direction or line of action of
the force# It is re&resented !" *a+ indicating the angle that the line of action of the
force a,es with the fixed or reference axes such as O-. O/ or with an" other
hori0ontal or $ertical line# *!+ Indicating the angle that the line of action of the force
a,es with the geogra&hical axes such as 1orth. 2outh. East or 3est#
4# 2ense: the wa" in which the force acts along its line of action is ,nown as sense or
nature of the force. i#e# &ush or &ull#
5# 6oint of a&&lication: the &oint at or through which the force acts on a !od" is called
its &oint of a&&lication#
Equilibrium: an" s"ste of forces which ,ee&s the !od" at rest is said to !e in
e%uili!riu. or when the condition of the !od" is unaffected e$en though a nu!er of
forces acted u&on it. it is said to !e in e%uili!riu#
Laws of Equilibrium: there are two laws of e%uili!riu de&ending u&on the forces or
the cou&le acting on a rigid !od"#
'# Force law or e%uili!riu
(# Moent law of e%uili!riu#
Force law of Equilibrium: for an" s"ste of forces ,ee&ing a !od" in e%uili!riu. the
alge!raic su of forces. in an" direction is 0ero. i#e# 7 F 8 9#
This law is a&&lica!le to all s"stes of co&lanar forces and includes#
*a+Two force principle: if a !od" in e%uili!riu is acted u&on !" two forces. the" ust
!e e%ual. o&&osite and collinear#
*!+ Three force principle: if a !od" in e%uili!riu is acted u&on !" three forces. then
the resultant of an" two forces ust !e e%ual. o&&osite and collinear with the third force#
*c+ Four force principle: if a !od" in e%uili!riu is acted u&on !" four forces. then the
resultant of an" two forces ust !e e%ual. o&&osite and collinear with the resultant of
other two forces#
Moment Law of Equilibrium: For an" s"ste of forces ,ee&ing a !od" in e%uili!riu.
the alge!raic su of the oents of all the forces a!out an" &oint in their &lane is 0ero
i#e# 7 M 89 or 7 F#d 8 9# This law is a&&lica!le onl" to co&lanar. non:concurrent *&arallel
or non:&arallel+ force s"ste#
Lamis theorem: If three co&lanar forces acting on a &oint in a !od" ,ee& it in
e%uili!riu. then each force is &ro&ortional to the sine of the angle !etween the other two
forces# Thus. P/ sin = Q / sin = / sin !#
Trusses: a truss is an articulated structure co&osed of straight e!ers arranged and
connected in such a wa" that the" transit &riaril" axial forces#
'# 6lane truss: if all the e!ers lie in one &lane. the truss is called &lane truss#
(# 2&ace truss: a three diensional truss is called a s&ace truss#
"ssumptions:
'# Each truss is co&osed of rigid e!ers all l"ing in one &lane#
(# 3eight of the e!ers is neglected !ecause the" are sall in co&arison with the
loads#
4# Forces are transitted fro one e!er to another through sooth &ins fitted in
the e!ers#
#or$ %one:
'# 3or, done !" an elastic string during stretching 8 *initial tension ;final
tension+<( #Extension# 8 Mean tension# Extension#
(# 3or, done in resisting a !od" !" a height h 8 gh#
4# 3or, done in rotation 8 Tor%ue# Angle turned through#
5# 3or, done !" a cou&le 8 Moent of the cou&le# Angle in radians turned through#
Ener&': energ" is the ca&acit" to do wor,# Infact. energ" is &ossessed !" a !od".
whereas wor, is done !" a force#
(ifferent forms of Ener&': Mechanical energ". Electrical energ". Heat energ". Light
energ". 2ound energ". 6ressure energ" etc#
Mechanical energ": The energ" &ossessed !" a !od" due to its &osition or otion is
called echanical energ"# Mechanical energ" is of two ,inds#
'# )inetic ener&': the energ" &ossessed !" a !od" !" $irtue of its otion is called
=inetic energ"# It is easured !" the aount of wor, which the !od" can do due
to its otion# )inetic ener&' of translation: the energ" due to rectilinear otion
of a !od" is called ,inetic energ" of translation# )E = m*
+
/ +# )inetic ener&' of
rotation: the ,inetic energ" due to rotar" otion of the !od" is called ,inetic
energ" of rotation# )E of otation = , -
+
/+ #
(# Potential ener&': the energ" &ossessed !" a !od" !" $irtue of its &osition is called
&otential energ"# This is gi$en !" gh#
4# Law of conser*ation of ener&': is a &article is acted u&on !" a conser$ati$e force
s"ste> the su of the ,inetic energ" and &otential energ" is constant#
Energ" stored in a s&ring: the energ" stored in the s&ring is &otential energ" and is gi$en
!" $ s
+
/+# 3here. , 8 s&ring constant. s 8change in the length of the s&ring#
I&ulse: if a constant force 6 acts for a tie t on a !od". then i&ulse of the force 8 6#t
Moentu: &roduct of ass and the $elocit" is the linear oentu# Thus. Linear
momentum = m*.
"n&ular momentum: it is gi$en !" Tor%ue#tie where tor%ue 8 T = , .
Relati$e $elocit": two !odies o$ing in the sae direction will ha$e their relati$e
$elocit" e%ual to the difference of their $elocities# Thus. relati$e $elocit" will !e u : $#
Two !odies o$ing in o&&osite directions: the relati$e $elocit" in this case will !e e%ual
to ? u ;$#
Angular $elocit" *@+: it is defined as the angle in radians turned through !" a &article in
unit tie# Thus. - = d /dt. If 1 is the re$olutions &er inute of a !od" then - = + / 0 /
12. Tangential or linear $elocit" of a &article undergoing cur$ilinear otion and ha$ing
angular $elocit" @ is gi$en !" $ 8 @r# And A 8 @t angular acceleration 8 = v
(
< r#
Thick cylinder
Economic Order Quantity:
B 8C( A2<I
6rocureent cost<"ear 8 A2<B
In$entor" carr"ing cost<"ear 8 I B<(
A$erage %uantit" held in in$entor" 8 B<(#
1o of orders &laced &er "ear 8 Annual usage < EOB#
Gear drive:
Minimum 0o of teeth on pinion 34
p
5
8 '5 for (9
9
stu! teeth#
84' for '5#D
9
full de&th#
8 'E

for (9
9
full de&th#
6
s
= 7er*ice factor:
8 ' for stead" load#
8 '#(D for light shoc,#
8 '#D for ediu shoc,#
8 '#F for hea$" shoc,#
Tooth form factor 3'5:
' = .8+9 :.1;9 / < for '5#D
9
. $ G(9 <s#
" 8 #'D5 : #H'(<0 for (9
9
full de&th#
" 8 #'ED : #HD<0 for (9
9
stu! teeth#
Tangential tooth load. Ft 8 I
d
C
$
! J " #
T
&
< r
&
8T
g
< r
g
. where T
&
8 6#K9# '999#C
s
< ( J 1
&#
= %=%esi&n stress
8 D9 :F9 1<( for cast iron#
8'59 ? ('9 1<( for Cast steel#
8('9 1<( for Allo" steel#
>elocit' factor 36
*
5:
8 4 < 4;L for ordinar" gear $ M F <s#
8 5#D < 5#D ;L for carefull" cut gear. $ 8 F to '4 <s#
8 K < K;L for &recision gear $ 8 '4 to (9 <s#
8 D#DD < *D#DD ; L+ for high $elocit" gear $G (9 <s#
!8face width. generall" ta,en as H#D to '(#D #
#ear an% Lubrication factor 36
w
5:
8 '#'D for good lu!rication#
8 '#(D for noral lu!rication#
8 '#4D for &oor lu!rication#
Mathematical Formulae:
6ircumference of a circle 8 + r#
(iameter of circle 8 + r#
"rea of circle 8 r
+
= (
+
/9#
7in
+
? @cos
+
? = 8#
7ec
+
? =8 @ tan
+
?#
6osec
+
? = 8 @ cot
+
?#
7in 3:?5 = : sin ?#
6os 3:?5 = cos?#
7in 3A2 : ?5 = sin ?#
6os 3A2 : ?5 = sin ?#
7in 3A2@ ?5 = 6os ?#
6os 3A2 @ ?5 = : sin ?#
7in 38;2 : ?5 = sin ?
6os 38;2 : ?5 = :cos?#
7in 38;2 @ ?5 = : sin ?
6os 38;2 @ ?5 = : cos?
7in 3"@B5 = 7in" 6osB @ 6os" 7inB#
6os 3"@B5 =6os" 6osB : 7in" 7inB#
7in 3":B5 = 7in" 6osB : 6os" 7inB#
6os 3":B5 = 6os" 6osB @7in" 7inB#
7in6 @7in( = + 7in 36@(5/+ .6os 36:(5/+.
7in6 C 7in( = + 6os 36@(5/+. 7in36:(5/+.
6os6 @6os( = + cos36@(5/+ .6os 36:(5/+.
6os6 C 6os( = + sin36@(5/+. 7in 3(:65/+.
+ sin" cosB = sin 3"@B5 @sin 3":B5.
+ 6os" 7inB = 7in 3"@B5 : 7in 3":B5.
+6os" 6osB = 6os 3"@B5 @ 6os 3":B5.
+ 7in" 7inB = 6os 3":B5 : 6os 3":B5.
tan 3"@B5 = 3tan" @ tanB5 / 38: tan"tanB5#
tan 3":B5 = 3tan" : tanB5 / 38@tan"tanB5#
7in +" = + 7in" 6os"#
6os+" = 6os
+
" C 7in
+
" = 8 C +7in
+
" = +6os
+
" C 8.
7in+" = + tan" / 38@tan
+
"5.
6os+" = 38 C tan
+
"5 / 38 @ tan
+
"5#
tan+" = + tan" / 38: tan
+
"5#
7inD" = D 7in" C 97in
D
".
6osD" = 96os
D
" C D6os"#
tanD" = 3Dtan": tan
D
"5 / 38 C Dtan+"5
7um of certain series:
8. 8@+@D@9@ EE. @n = n 3n@+5 / +#
+. 8
+
@+
+
@D
+
@9
+
@ F
+
@ EE..@n
+
= n 3n@85 3+n@85 / 1.
D. 8
D
@ +
D
@D
D
@9
D
@ EEE..@ n
D
= Gn 3n@85/+H
+
.
9. 8
9
@+
9
@ D
9
@ 9
9
@EEE. @ n
9
= n 3n@85 3+n@853Dn+ @Dn :85/D2.
F. I 3+r:85 = 8@D @F @EEE @ 3+n:85 = n
+
#
1. I 3+r:85
+
=n 39n
+
:85/D#
J. I 3+r:85
D
= n
+
3+ n
+
:85#
N O ; M 8 G P Q R S T U 7 V W
X Y Z [ J \ I ] ^ _ ` a C
b c d e A f g h i j k
l m n o p @ @ @ @ J [ Z Y
X A o n m m m l k W q U 7 @ I
r s t
{}
.
9 9 9 9 9 9
P q g Q
n
A
8'<E *:Z +