of Centre Mass
Definition
Total momentum of a system of particles
Motion of the centre of mass
Serway and Jewett 9.6 - 9.7
Review: Newtons Second Law
For a particle:
(Net external force) = ma
For a particle or a system of particles:
(Net external force) = F = dp/dt
(Net external impulse) = I = p
Apply Newtons Laws to objects that are not particles:
e.g.,
F
F
or
How will an extended body move (accelerate) when a force is
applied at an arbitrary location?
The motion of the centre of mass is simple; in addition, various
parts of the object move around the centre of mass.
Centre of Mass
Recall: Definition:
rCM
mi ri mi ri
=
=
mi
M
m1
CM
x
or,
MrCM = mi ri
m2
rCM
m3
For continuous objects,
rCM
1
=
r dm
M
(Recall the position vector r
has components x, y, z.)
Dynamics of a system of particles
CM definition:
MrCM = mi ri
Differentiate with respect to time:
Mv CM = mi v i = p total
The total momentum of any collection of particles is equal to the
total mass times the velocity of the CM pointamazing but true!
So Newtons second law gives (differentiate above):
(external forces) = M
total
a CM
This is remarkable. The motion of the centre of mass is the same as
if the object were a single particle at the CM, with all external forces
applied directly to it.
Example
v0
CM
x
A neutron (mass m) travels at speed v0 towards a stationary
deuteron (mass 2m). What is the initial velocity of the CM of the
system (neutron plus deuteron)?
Since:
Mv CM = mi v i
v CM
m v
=
i
M
i
v0
CM
1
/3v0
/3v A 0
1
Examples:
1) springboard diver does a triple reverse dive with one and a
half twists. Her CM follows a smooth parabola (external force is
gravity).
2) A paddler in a stationary canoe (floating on the water, no friction)
walks from the rear seat to the front seat. The CM of the canoe
plus paddler moves relative to the canoe, but not relative to the
land (the canoe moves backwards). Here there is no external
force.
3) Pendulum cart (demonstration).
MOON384,000km
Quiz
A space station consists of a 1000-kg
sphere and a 4000-kg sphere joined by
a light cylinder. A rocket is fired briefly
to provide a 100-N force for 10 seconds.
Compare the velocity (CM) change if
the rocket motor is
A) at the small sphere
B) at the large sphere
C) same for either
F
F
In which case will the space station get to the moon faster?
MOON384,000km
Quiz
A space station consists of a 1000-kg
sphere and a 4000-kg sphere joined by
a light cylinder. A rocket is fired briefly
to provide a 100-N force for 10 seconds.
In which case will the space station
rotate faster?
A) at the small sphere
B) at the large sphere
C) same for either
F
F
Still more amazing CM theorems:
1)
Kinetic Energy = Mv 2CM + (K.E. relative to CM)
(e.g., rigid body: K = Mv 2CM + ICM 2 )
about CM) = ICM , even for an
accelerated body
2) (Torque
alpha
(angular acceleration)
Summary
ptotal = M vCM
(net external force) = M aCM
For practice: Chapter 9
Problems 39, 41, 58
(6th ed)
Problems 41, 43, 57, 69