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AMME2500 Formula Sheet 2012 P age 1 of 3
SOME USEFUL FORMULAE
1. Two Dimensional KinematicsNon-rotation reference attached to
B for 2 points A and B in a rigid body
Equation of velocity: B/ABA vvv +=
Equation of acceleration: B/ABA aaa += or
( ) ( ) ( ) ( ) ( ) ( )tB/AnB/AtBnBtAnA aaaaaa +++=+
where: ( )ra = n)( and ra = t)( are the normal and tangential
accelerations.Rotation reference attached to B for 2 points A and B
in a rigid body
Equation of velocity:relBA vrvv ++=
Equation of acceleration: ( ) relrelBA avrraa ++++= 2 where relv
and rela are relative velocity and relative acceleration
respectively
2. Two Dimensional Kinetics
Equation of motion (G centre of mass): xx amF = yy amF = IMG =
Work Energy principle: eg VVTU ++=21
Work done by external force = rF dU and by external couple dMU =
,22
2
1
2
1IvmT += kinetic energy, hmgVg = gravitational P.E.,
2
2
1kxVe = elastic P.E.
Impulse-Momentum principle:
Linear: 22122
1
vvG-GF mmdtt
t==
=
=
122
1
12
2
1
:
:
yy
t
ty
xx
t
tx
vmvmdtFy
vmvmdtFx
Angular: 12GG 122
1
IIHHdtMt
t G==
3. Steady Mass Flow and Variable MassSteady Mass Flow:
Resultant force: ( )12 vvGF == 'm& Resultant moment about O:
( )1122 vdvdHM == 'mOO &
where 'm is mass flow rate, v1 entering velocity and v2 leaving
velocity.
Variable Mass: uvF mm && = where m is mass of system at
time t, v is velocity of system, u is velocity of the rejected
mass,
F the external force acting on the system.
4. Eulers First Law
FR =Gm &&
where RG is the position vector of mass center and F is the
external force acting on the body
5. Eulers Second Law
H =& where H is angular momentum and = i ii FrM is the
external moment acting on the body
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AMME2500 Formula Sheet 2012 Page 2 of 3
6. Angular Momentum
==
z
y
x
zzyxxz
yxyyxy
xzxyxx
III
III
III
IH
where
( ) ( ) ++== B
n
i
iiixx dmzyzymI22
1
22 == B
n
i
iiixy xydmyxmI1
( ) ( ) ++== B
n
i
iiiyy dmzxzxmI22
1
22 == B
n
i
iiixz xzdmzxmI
1
( ) ( ) ++== B
n
i
iiizz dmyxyxmI22
1
22 == B
n
i
iiiyz yzdmzymI1
7. Parallel Axis Theorem
xxXxXX ImdI +=2
where Xxd is the distance between the these two axes
(XXandxx)
8. Parallel Plane Theorem
xyGGXY IYmXI +=
whereXG, YG, ZG are the coordinate of the center of mass.
9. Kinetic Energy of A Rigid Body
GGGmT HVV += 2
1
2
1T
where VG is the velocity of the center of mass, is the angular
velocity of the body, HG isangular momentum of the body about the
center of mass.
10. Lagranges Equations
( )NCk
kkk
Qq
V
q
T
q
T
dt
d=
+
& r,,k L1=
where Tis the kinetic energy, Vthe potential energy, qk the
generalized coordinates and( )NCkQ the non-conservative generalized
forces.
11. Integration
( ) 11
Cbtalnbbta
dt+=
( ) ( ) ( ) 2111
1 Ctatlnata
dtatln += where C1 and C2 are constant of integration
