Chapter 9 Lecture Essential University Physics Richard Wolfson 2 nd Edition © 2012 Pearson Education, Inc. Slide 9-1 Systems of Particles
Chapter 9 Lecture
Essential University PhysicsRichard Wolfson
2nd Edition
© 2012 Pearson Education, Inc. Slide 9-1
Systems of Particles
© 2012 Pearson Education, Inc. Slide 9-2
In this lecture you’ll learn
• How to find the center of mass
of a system of particles
• The principle of conservation
of momentum, and how it
applies to systems of
particles
• How to analyze collisions
– Inelastic collisions
– Elastic collisions
© 2012 Pearson Education, Inc. Slide 9-3
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,
– For a continuous distribution of matter,
– In both cases, M is the system’s total mass.
cm
i im rr
M
cm
r dmr
M
© 2012 Pearson Education, Inc. Slide 9-4
Finding the Center of Mass
• Example: A system of three particles in an equilateral triangle
• Example: A system of continuous matter
• Express the mass element dm in terms
of the geometrical variable x:
• Evaluate the integral:
so
xcm mx1 mx3
4mm(x1 x1)
4m 0
ycm my1 my3
4m
2my1
4m
1
2y1
3
4L 0.43L
dm
M
(w/L)x dx1
2 wL
2x dx
L2
xcm 1
Mx dm
1
M 0
L
x2Mx
L2dx
2
L20
L
x2 dx
xcm 2
L2 0
L
x2 dx 2
L2x3
3 0
L
2L3
3L2
2
3L
© 2012 Pearson Education, Inc. Slide 9-5
More on Center of Mass
• The center of mass of a composite
object can be found from the CMs
of its individual parts.
• An object’s center of mass
need not lie within the object!
– Which point is the CM?• The high jumper clears the
bar, but his CM doesn’t.
© 2012 Pearson Education, Inc. Slide 9-6
Motion of the Center of Mass
• The center of mass obeys Newton’s second law:
net external cmF Ma
• Here most parts of the skier’s body undergo complex
motions, but his center of mass describes the parabolic
trajectory of a projectile:
© 2012 Pearson Education, Inc. Slide 9-7
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.
• Example: Jumbo, a 4.8-t elephant, walks 19 m toward one
end of the car, but the CM of the 15-t rail car plus elephant
doesn’t move. This allows us to find the car’s final position:
J Jf c cf J Ji cf c cfcm
J
J c
(19 m)(4.8 t)
( ) (15 t 4.8 t)
m x m x m x x m xx
M M
m
m m
© 2012 Pearson Education, Inc. Slide 9-8
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
© 2012 Pearson Education, Inc. Slide 9-9
• 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
• 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:
© 2012 Pearson Education, Inc. Slide 9-10
Collisions
• A collision is a brief, intense interaction between objects.
– Examples: balls on a pool table, a tennis ball and racket,
baseball and bat, football and foot, an asteroid colliding with
a planet.
– The collision time is short compared with the timescale of the
objects’ overall motion.
– Internal forces of the collision are so large that we can
neglect any external forces acting on the system during the
brief collision time.
– Therefore linear momentum is essentially conserved during
collisions.
© 2012 Pearson Education, Inc. Slide 9-11
Impulse
• If is the average force acting on one object during a
collision that lasts for time Dt, then Newton’s second law
reads , or
• The product of average force and time that appears in this
equation is called impulse.
F
/ ,F p t D D
p F tD D
• The force in a collision usually
isn’t constant and can fluctuate
wildly, so
( )p J F t dtD
© 2012 Pearson Education, Inc. Slide 9-12
Elastic and Inelastic Collisions
• In an elastic collision, the internal forces of the
collision are conservative.
– Therefore an elastic collision conserves kinetic energy as
well as linear momentum.
• In an inelastic collision, the forces are not
conservative and mechanical energy is lost.
– In a totally inelastic collision, the colliding objects stick
together to form a single composite object.
– But if a collision is totally inelastic, that doesn’t necessarily
mean that all kinetic energy is lost.
© 2012 Pearson Education, Inc. Slide 9-13
Totally Inelastic Collisions
• Totally inelastic collisions are governed entirely by
conservation of momentum.
– Since the colliding objects join to form a single composite
object, there’s only one final velocity:
• Therefore conservation of momentum reads
Before collision After collision
1 1 2 2 1 2 fm v m v m m v
© 2012 Pearson Education, Inc. Slide 9-14
Elastic Collisions
• Elastic collisions conserve both momentum and kinetic
energy:
• Therefore the conservation laws read
Before collision After collision
1 1i 2 2i 1 1f 2 2f
2 2 2 21 1 1 11 1i 2 2i 1 1f 2 2f2 2 2 2
m v m v m v m v
m v m v m v m v
© 2012 Pearson Education, Inc. Slide 9-15
Elastic Collisions in One Dimension
• In general, the conservation laws don’t determine the
outcome of an elastic collision.
– Other information is needed, such as the direction of one
of the outgoing particles.
• But for one-dimensional collisions, when particles collide
head-on, then the initial velocities determine the outcome:
• Solving both conservation laws in this case gives
1 2 21f 1i 2i
1 2 1 2
1 2 12f 1i 2i
1 2 1 2
2
2
m m mv v v
m m m m
m m mv v v
m m m m
© 2012 Pearson Education, Inc. Slide 9-16
Special Cases: 1-D Elastic Collisions;
m2 Initially at Rest
1) m1 << m2:
Incident object rebounds with
essentially its incident velocity
2) m1 = m2:
Incident object stops; struck
object moves away with initial
speed of incident object
3) m1 >> m2:
Incident object continues with
essentially its initial velocity;
struck object moves away with
twice that velocity
© 2012 Pearson Education, Inc. Slide 9-17
Summary
• A composite system behaves as though its mass is
concentrated at the center of mass:
• The center of mass obeys Newton’s laws, so
cm
i im rr
M
cm
r dmr
M
net external cm net external or, equivalently, dP
F Ma Fdt
• In the absence of a net external force, a system’s linear
momentum is conserved, regardless of what happens internally
to the system.
• Collisions are brief, intense interactions that conserve
momentum.
– Elastic collisions also conserve kinetic energy.
– Totally inelastic collisions occur when colliding objects join to make
a single composite object.
(discrete particles) (continuous matter)