Chapter 7 Systems of
particles
7-1 The motion of a complex object

7-2 Two-particle system
7-3 Many-particle system
7-4 Center of mass of solid objects
7-5 Conservation of momentum in a
system of particles
When can the object studied be regarded as a
mass point?
Doing only translational motion
Translational motion
Rotational motion
Translational +
Rotational motion
7-1 The motion of a complex object
When we projectile a rigid body, the
motion of the body looks very complicated.
We can consider the motion of the rigid
body to be a combination of a parabolic
trajectory of a “center of mass” (rotational
motion is not considered) plus a rotation
about “center of mass” (translational
motion is not considered).
How to find the center of mass (CM) of
a particle system?
7-2 Two-particle system
Do an experiment to find CM in a two-particle
system.
As an example, we consider a baton consists
of two particles m 1 , m 2 at its ends A and B,
connected by a thin rigid rod of fixed length and
negligible mass.
A
B
m1
m 2  2m1
m2
We give the rod a push along the frictionless
horizontal surface and examine its motion.
Snapshots of the locations of points A and
B at successive intervals of time.
Clearly both m 1 and m 2 are accelerated, however,
one point in the rod (point C) moves with constant
velocity.
If point c is regarded as a reference, A and B
points rotate with a constant rotational speed.
View the motion
from the reference
of point C.
So point c is actually the center of mass.
Fig 7-5
By building a Cartesian
coordinate, position of
point c is found at :

rcm 


m 1 r1  m 2 r2
m1  m 2
y cm 

r1 
rcm
C

r2
 '
rcm
C’
m2
O

m1
(7-1)
or written as:
x cm 
y
x
m 1 x1  m 2 x 2
m1  m 2
m1 y1  m 2 y 2
m1  m 2
A
(7-2)
C
B
m2
m1
From Eq. (7-1), the velocity and acceleration of
the CM are:


v cm 
d rcm


dt


a cm 
d v cm

m 1 v1  m 2 v 2
m1  m 2


dt

m1 a1  m 2 a 2
m1  m 2
(7-4)
(7-6)

Can we also find a cm is zero from Eq. (7-6)?




m1 a1  m 2 a 2  F1 r  F2 r

ac  0


F 2 r   F1 r
How about the motion if the system has net
external forces?
Suppose there is an external force on each
particle in above expt., then


m1 a1  m 2 a 2 


F
1


F

2
 F1 r  F 2 r  0 , and if write


m 1 a1  m 2 a 2 

F
ext


F ext

 ( m 1  m 2 ) a cm




 F1 ext  F1 r  F 2 ext  F 2 r

F
1 ext

F

2 ext


F



m a  m2a2
 a cm  1 1

m1  m 2

ext




Newton’s second law for
systems of particles
This looks very like a particle of mass m 1  m 2 located
at the center of mass.
7-3 Many-particle system
Consider a system consisting of N
particles of masses m 1 m 2 …… m N . The total
mass is
M   mn
(7-10)
Each particle
can be represented by its 

location rn , velocity v n and its acceleration a n .
The CM of the system can be defined by
logical extension of Eq(7-1):


rcm 


m 1 r1  m 2 r2        m N rN
m1  m 2        m N

1
M

(7-11)
 m n rn
In terms of components, Eq(7-11) can be
written as
(7-12)
1
1
1
y

m
y
x cm 
m
x
 n n , z cm   m n z n .
 n n , cm
M
M
M
Taking the derivative of Eq(7-11)

1
v cm 
M
m

(7-13)
vn
n
Differentiating once again:

a cm 
Or

M a cm 
m
1
M

n
an 
m


F
(7-14)
an
n
1


F
2

        F N (7-15)
By Newton’s third law, in Eq(7-15) the vector
sum of all the internal forces is cancelled, and
Eq(7-15) reduces
 to

 Fext  M a cm (7-16)
Eq(7-16) is just the Newton’s second law for the
system of N particles treated as a single particle

of mass M located
 at the center of mass( rcm ),
experiencing a cm.

rcm 

a cm 
1
M
1
M

 m n rn
m

n
an
We can summarize this important result
as follow:
“The overall translational motion of a
system of particles can be analyze using
Newton’s law as if the mass were
concentrated at the center of mass and
the total external force were applied at
that point.”
These are general results that apply
equally well to a solid object.
Sample problem 7-3
A projectile(射弹) of mass 9.8kg is launched
from the ground with
initial velocity of 12.4m/s at

54
an angle of
above the horizontal (Fig 7-11).
At some time after its launch, an explosion splits
the projectile into two pieces. One piece of mass
6.5kg, is observed at 1.42s after the launch at a
height of 5.9m and a horizontal distance of
13.6m from the launch point. Find the location of
the second fragment at that same time.

CM
v0
0

m
2

m1
Solution:
If the projectile had not exploded, the
location of the projectile at t=1.42s should
have been
x  v 0 x t  ( 7 . 3 m / s )  1 . 42 s  10 . 4 m
y  v0 y t 
1
gt
2
2
 (10 . 0 m / s )  (1 . 42 s ) 
1
( 9 . 80 m / s )  (1 . 42 s )
2
2
2
 4 .3 m
It is the location of the CM.
?
x cm 
m 1 x1  m 2 x 2
m1  m 2
?
y cm 
m1 y1  m 2 y 2
m1  m 2
By Eq(7-12)
x2 

Mx cm  m 1 x 1
Fig 7-11
m2
( 9 . 6 kg )  (10 . 4 m )  ( 6 . 5 kg )  (13 . 6 m )
3 . 1kg
 3 .7 m
y2 
My

Cm
v0
cm
 m1 y1
m2
 0 .9 m
0

 m2

m1
7-4 Center of mass of solid objects
1) If an object has symmetry, the CM must lie at
the geometrical symmetrical center of the object.
Suppose the mass is uniformly distributed.
cm
cm 

2) If the object has no symmetry, sometimes it is
also easy to find its cm position:
cm
(a)
(b)

Sample problem 7-4
Fig 7-13 shows a
circular metal plate
of radius 2R from
which a disk of
radius R has been
removed. Find the
cm (x) of the plate.
Fig 7-13
y
R
D
C x
x
Solution:
Due to the mirror symmetry about the x axis,
the cm must lie along the x axis.
If the hole is filled with a disk of the same
material of radius R, the cm of composite
disk is at the origin of the coordinate system.
xc  0 
x x  (
?
m D xD  m x xx
mD
mx
CM for the big
circular plate
mD  mx
)xD 
 R
2
 (2 R )  R
2
2
 ( R ) 
1
3
R
3) If we encounter solid irregular objects, we
can divide infinite small elements. And the
sums of Eqs(7-12) transform into integrals:
x
z
O
x cm 
y cm 
z cm 
y
1
lim
M
1
m  0
M
1
m  0
M
lim
lim
m  0
In vector form are
x

m n 
y n m n 
z

n
n
m n 
rcm 
1
M

1
M
1
M
1
M
 xdm
 ydm
(7-18)
 zdm

r dm
(7-19)
Sample problem 7-5
Fig 7-14
A thin strip of
material is bent
into the shape of a
semicircle of radius
R. Find its center
of mass.
y

y cm

0
dm 
M

d
x
Solution:
The strip has symmetry about the y axis.
So x c  0
y cm 

R

1
M
 ydm

 sin  d 
0



1
M
2R

 R sin 
0
M

d
 0 . 637 R
( the small element of mass dm subtends
an angle d  . The location of the dm
is y  R sin  )
7-5 Conservation of momentum in a
system of particles
1) For a system containing
N particles,

the total momentum P is ( M   m i )
N 
N
N

m n vn


(7-21)
P   Pn   m n v n  M 
 M v cm
n 1
Here

dP
dt
n 1
M
n 1

d v cm
dt
M

 M a cm 


Fext
(7-22)
If the net external force
 acting on a
system is zero, then d P  0 and so the
dt
total linear momentum P of the system
remains constant.
2) If we view the system from the cm frame,
the velocity v n ' of a particle in this frame is



(7-24)
v '  (v  v )
n
n
cm
Then in this cm frame, the total momentum is

P'
N

n 1

m nvn ' 
N

n 1

m nvn 


 M v cm  M v cm  0
N

n 1

m n v cm (7-25)
Sample problem 7-8
As Fig 7-17 shows a cannon whose mass
M is 1300kg fire a 72kg ball in a horizontal
direction with a speed vbc of 55m/s
relative to the cannon. The cannon is
mounted and can recoil (后退) freely.
(a) what is the velocity vcE of the recoiling
cannon with respected to the Earth?
(b) what is the initial velocity vbE of the
ball with respected to the Earth?
vcE
M
m
Fig 7-17

v bc
Solution:
Momentum in horizontal direction is conserved.


Pix  P fx  0
P f  Mv cE  mv bE  0
v bc  v bE  v cE
vcE
M

m
Fig 7-17

v bc