MOTION OF A CENTER OF MASS
Consider a system of particles where the position of the CM is given by:
mi ri
rcm
M
Taking the derivative wrt time gives:
drcm
dt
vcm
vi
M
dr
1
1
mi i
mi vi
M
dt M
velocity of i th particle
Mvcm
Psys
mi ri
d
dt
mi vi
Mvcm
pi
Psys
Total Linear Momentum of a System
1. The total linear momentum of the system is equal to the total mass of the system
times the velocity of the center of mass of the system.
2. The total linear momentum of the system is equal to that of a single particle of
mass M located at the center of mass and moving with a velocity equal to that of
the center of mass.
dvcm
dt
d 1
mi vi
dt M
dv
1
1
mi i
M
dt M
mi ai
Fi
acm
Macm
Fi
Fext
mi ai
Fint
Fi = net force on i th particle
Fext = net external force on i th particle
Fint = net internal force on i th particle
Macm
Fext
By N 3L
Fint
Fint
0
Therefore,
Fext
Macm
Fext
Macm
Fext
Macm
M
dvcm
dt
dPsys
dt
d
Mvcm
dt
dPsys
dt
N2L for a System
A system moves as if all the mass was concentrated on a single particle of mass M
located at the center of mass and the net external force were applied to that point.
If
Psys
Pi
Fext
0, then :
MVcm
Pf
constant
Law of Conservation of Linear Momentum for a System
Thus,
If
Fext
0, then
Psys and Vcm are constants in time.