School

McMaster University
Department

Civil Engineering

Course Code

CIVENG 2Q03

Professor

Dimitrios Konstantinidis

McMaster University Department of Civil Engineering

Winter 2012 CE 2Q03 (D. Konstantinidis)

Useful Formulas for the Dynamics of Systems of Particles and Rigid Bodies

SYSTEMS OF PARTICLES

Position/Velocity/Acceleration Vectors

i

r is the position of mass particle i ( 1, 2, 3, ,in=K)

ii

=vr

&

iii

==avr

&&

&

RIGID BODIES

Position/Velocity/Acceleration Vectors

x is the position of a material point on the body

Between any two points on the body with positions

1

x and 2

x:

12 12

()

âˆ’

=Ã— âˆ’vv Ï‰xx

12 1 2 1 2

12 12

()[()]

âˆ’

=âˆ’=âˆ’=

=Ã— âˆ’ + Ã— Ã— âˆ’

aa vv xx

Î±xx Ï‰Ï‰ xx

&&&&&&

where Ï‰ and Î± are the angular velocity and

angular acceleration respectively.

Center of Mass

r is the position vector of the center of mass,

which is denoted as G.

11

1

nn

ii ii

ii

n

i

i

mm

m

m

==

=

==

âˆ‘âˆ‘

âˆ‘

rr

r, where

1

n

i

i

mm

=

=âˆ‘

=vr

&, ==avr

&&

&

Center of Mass

x is the position vector of the center of mass,

which is denoted as G.

dm dm

m

dm

==

âˆ«

âˆ«

âˆ«

BB

B

xx

x, where mdm=âˆ«

B

=

vx

&,

=

=avx

&&&

Linear Momentum

11

nn

iii

ii

m

==

==

âˆ‘âˆ‘

GG v

m=Gv

Linear Momentum

ddm==

âˆ«

âˆ«

BB

GGv

m

=

Gv

Angular Momentum

The angular moment of the system of particles

relative to a fixed point P is

11

() ()

nn

PiPiiPii

ii

m

==

=âˆ’Ã—=âˆ’Ã—

âˆ‘âˆ‘

HrrGrrv

If P is the origin,

11

nn

Oiiiii

ii

m

==

=Ã—=Ã—

âˆ‘âˆ‘

HrGrv

Angular Momentum

The angular moment of the rigid body relative to a

fixed point P is

() ()

PP P

ddm= âˆ’Ã—= âˆ’Ã—

âˆ«âˆ«

BB

HxxGxxv

If P is the origin,

Oddm=Ã— =Ã—

âˆ«âˆ«

BB

HxGxv

1

r

i

r

r

G

O

system

boundary

x

x

G

O

rigid body B

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