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Centre of Mass
• Definition
• Total momentum of a system of particles
• Motion of the centre of mass
Serway and Jewett 9.6 - 9.7
Physics 1D03 - Lecture 16
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Review: Newton’s 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 = Dp
Physics 1D03 - Lecture 16
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Apply Newton’s 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.
Physics 1D03 - Lecture 16
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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.)
Physics 1D03 - Lecture 16
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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 point—amazing but true!
So Newton’s 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.
Physics 1D03 - Lecture 16
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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
mv


i
i
M
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v0
CM
1/ v
3 0
1/ v
3 0
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Examples:
1) A 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).
Physics 1D03 - Lecture 16
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MOON
Quiz
384,000 km
F
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
In which case will the space station get to the moon faster?
Physics 1D03 - Lecture 16
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MOON
Quiz
384,000 km
F
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
Physics 1D03 - Lecture 16
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Still more amazing CM theorems:
1)
Kinetic Energy = ½ Mv 2CM + (K.E. relative to CM)
(e.g., rigid body: K = ½ Mv 2CM + ½ ICM w 2 )
about CM) = ICM a , even for an
accelerated body
2) (Torque
“alpha”
(angular acceleration)
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Summary
• ptotal = M vCM
• (net external force) = M aCM
For practice: Chapter 9
Problems 39, 41, 58
(6th ed)
Problems 41, 43, 57, 69
Physics 1D03 - Lecture 16
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