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PreClass Notes: Chapter 9, Sections 9.1, 9.2
• From Essential University Physics 3rd Edition
• by Richard Wolfson, Middlebury College
• ©2016 by Pearson Education, Inc.
• Narration and extra little notes by Jason Harlow,
University of Toronto
• This video is meant for University of Toronto
students taking PHY131.
Outline
“Most parts of the dancer’s body
undergo complex motions during
this jump, yet one special point
follows the parabolic trajectory
of a projectile. What is that
point, and why is it special?” –
R.Wolfson
Systems of Particles:
• 9.1 Centre of Mass
• 9.2 Momentum
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Center of Mass
• The center of mass of a composite object or system of
particles is the point where, from the standpoint of
Newton’s second law, the mass acts as though it were
concentrated.
• The position of the center of mass is a weighted
average of the positions of the individual particles:
– For a system of discrete particles,
– M is the system’s total mass.
cm
i im rr
M
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Center of Mass
For a continuous distribution of matter:
cm
r dmr
M
𝑟
𝑑𝑚
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More on Center of Mass
The center of mass of the
airplane is found by
treating the wing and
fuselage as point particles
located at their respective
centers of mass.
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More on Center of Mass
• An object’s center of mass need not lie within the object!
– Which point is the CM?
More on Center of Mass
• An object’s center of mass need not lie within the object!
– Which point is the CM?
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More on Center of Mass
• The high jumper clears the bar, but his CM doesn’t.
Motion of the Center of Mass
• The center of mass obeys Newton’s second law:
net external cmF Ma
• Here the hammer rotates as it is going through the air,
but its center of mass describes a simple parabolic
trajectory of a projectile:
[Image of flying hammer from http://www.racetomars.ca/mars/ed-module/artificial_gravity/ ]
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Motion of the Center of Mass
In the absence of any external forces on a
system, the center of mass motion remains
unchanged; if it’s at rest, it remains in the same
place—no matter what internal forces may act.
Recall: Momentum
• a property of moving things
• Defined as the mass of an object multiplied by its
velocity
• in equation form:
𝑝 = 𝑚 𝑣
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Momentum and the Center of Mass
• The center of mass obeys Newton’s law, which can
be written or, equivalently,
• where is the total momentum of the system:
• with the velocity of the center of mass, and
net external cmF Ma
net external
dPF
dt
P
cmi iP m v Mv cmv
cmcm
dva
dt
• When the net external force is zero, .
• Therefore the total momentum of the system is
unchanged:
This is the conservation of linear momentum.
0dP dt
constantP
Conservation of Momentum
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Law of conservation of momentum:
In the absence of an external force, the momentum of
a system remains unchanged.
• When a cannon is fired, the force on the cannonball inside the cannon barrel is equal and opposite to the force of the cannonball on the cannon.
• The cannonball gains momentum, while the cannon gains an equal amount of momentum in the opposite direction—the cannon recoils.
Conservation of Momentum
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Conservation of Momentum
Conservation of Momentum
• Example: A system of three billiard balls:
– Initially two are at rest;
all the momentum is in
the left-hand ball:
– Now they’re all moving, but the
total momentum remains the same:
