Upload
damian-brooks
View
216
Download
2
Embed Size (px)
Citation preview
Chapter 7 Systems of particles
7-5 Conservation of momentum in a system of particles
7-4 Center of mass of solid objects
7-3 Many-particle system
7-2 Two-particle system
7-1 The motion of a complex object
When can the object studied be regarded as a mass point? Doing only translational motion
Translational + Rotational motion
Translational motion 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
As an example, we consider a baton consists of two particles , at its ends A and B, connected by a thin rigid rod of fixed length and negligible mass.
12 2mm
1m 2m
1m
2m
A
B
Do an experiment to find CM in a two-particle system.
Clearly both and are accelerated, however, one point in the rod (point C) moves with constant velocity.
2m1m
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.
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
1m
2m1r
x
y
O21
2211
mm
rmrmrcm
21
2211
mm
xmxmxcm
21
2211
mm
ymymycm
cmr
2r
'cmr
C C’
By building a Cartesian coordinate, position of point c is found at :
(7-2)
(7-1)
or written as:
1m
2m
A
BC
From Eq. (7-1), the velocity and acceleration of the CM are: (7-4) (7-6)
21
2211
mm
vmvm
dt
rdv cmcm
21
2211
mm
amam
dt
vda cmcm
Can we also find is zero from Eq. (7-6)?cma
rr FFamam 212211
0
ca
rr FF 12
How about the motion if the system has net external forces?
Suppose there is an external force on each particle in above expt., then
This looks very like a particle of mass located at the center of mass.
rextrext FFFFFFamam 2211212211
021
rr FF
extextext FFF 21, and if write
cmext ammF )( 21
extFamam
2211
21
2211
mm
amamacm
21 mm
Newton’s second law for systems of particles
7-3 Many-particle system
Consider a system consisting of N particles of masses …… . The total mass is
(7-10) Each particle can be represented by its
location , velocity and its acceleration . The CM of the system can be defined by
logical extension of Eq(7-1):
1m 2m Nm
nmM
nvnr
na
In terms of components, Eq(7-11) can be written as
Taking the derivative of Eq(7-11) (7-13)
Differentiating once again: (7-14)
nncm xmM
x1 nncm ym
My
1 nncm zm
Mz
1
nncm vmM
v1
nncm amM
a1
Nnncm FFFamaM 21Or (7-15)
nnN
NNcm rm
Mmmm
rmrmrmr
1
21
2211
(7-11)
(7-12), , .
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
(7-16)
cmext aMF
cma
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( ), experiencing .
cmr
nncm amM
a1
nncm rmM
r1
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 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.
54
0v
0
CM
1m
2m
Solution: If the projectile had not exploded, the
location of the projectile at t=1.42s should have been
It is the location of the CM.
mssmtvx x 4.1042.1)/3.7(0
m
ssmssm
gttvy y
3.4
)42.1()/80.9(2
1)42.1()/0.10(
2
1
22
20
21
2211
mm
xmxmxcm
21
2211
mm
ymymycm
? ?
By Eq(7-12)
Fig 7-11
mm
ymMyy cm 9.0
2
112
m
kg
mkgmkg
m
xmMxx cm
7.3
1.3
)6.13()5.6()4.10()6.9(2
112
0v
0
Cm
1m
2m
7-4 Center of mass of solid objects 1) If an object has symmetry, the CM must lie at the geom
etrical symmetrical center of the object. Suppose the mass is uniformly distributed.
cm
2) If the object has no symmetry, sometimes it is also easy to find its cm position:
(a) (b)
cm
cm
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
x
y
C xDR
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.
RRRR
Rx
m
mx D
x
Dx 3
1)(
)2()(
22
2
xD
xxDDc mm
xmxmx
0 CM for the big circular plate
?
3) If we encounter solid irregular objects, we can divide infinite small elements. And the sums of Eqs(7-12) transform into integrals:
xdmM
mxM
x nnm
cm
1lim
10
ydmM
myM
y nnm
cm
1lim
10
zdmM
mzM
z nnm
cm
1lim
10
dmrM
rcm1 (7-19)In vector form are
(7-18)
x
O
y
z
Sample problem 7-5 A thin strip of
material is bent into the shape of a semicircle of radius R. Find its center of mass.
Fig 7-14
x
y
0
d
Mdm cmy
Solution: The strip has symmetry about the y axis.
So
( the small element of mass dm subtends an angle . The location of the dm is )
0cx
RR
dR
dM
RM
ydmM
ycm
637.02
sin
sin11
0
0
dsinRy
7-5 Conservation of momentum in a system of particles 1) For a system containing N particles, the total
momentum is (7-21) Here (7-22)
If the net external force acting on a system is zero, then and so the total linear momentum of the system remains constant.
cm
N
n
nnN
nnn
N
nn vM
M
vmMvmPP
111
extcmcm FaMdt
vdM
dt
Pd
P
)( imM
0dtPd
P
2) If we view the system from the cm frame, the velocity of a particle in this frame is
(7-24) Then in this cm frame, the total momentum is
(7-25)
0
''111
cmcm
cm
N
nn
N
nnn
N
nnn
vMvM
vmvmvmP
)(' cmnn vvv
'nv
Sample problem 7-8 As Fig 7-17 shows a cannon whose mass M is 1
300kg 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?
bcv
MvcEm
Fig 7-17
0
fxix PP
bcv
MvcEm
Fig 7-17
Solution:
0 bEcEf mvMvP
cEbEbc vvv
Momentum in horizontal direction is conserved.