r. field 10/10/2013 university of florida phy 2053page 1 center-of-mass system of particles:...
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R. Field 10/10/2013 University of Florida
PHY 2053 Page 1
Center-of-Mass
System of Particles: Center-of-Mass
• Acceleration of the Center-of-Mass:
• Location of the Center-of-Mass:
NN
N
iitotcmtot vmvmvmvmpPvM
332211
1
cmtottot
net aMdt
• Velocity of the Center-of-Mass:
(net force acting on the system of particles)
(total mass of the system)
NN
N
iiicmtot rmrmrmrmrmrM
332211
1
tot
NNN
iii
totcm M
rmrmrmrmrm
Mr
332211
1
1
N
N
iitot mmmmmM
3211
tot
NNN
ii
tottot
totcm M
vmvmvmvmp
MM
Pv
332211
1
1
NN
N
iiicmtotnet amamamamamaMF
332211
1
(total momentum of the system)
The center-of-mass of a system of particles is the point that moves as though (1) all of the systems mass were concentrated there and (2) all external forces were applied there.
R. Field 10/10/2013 University of Florida
PHY 2053 Page 2
Center-of-Mass: Superposition
BA
BcmB
AcmABA
cm MM
rMrMr
What is the x-coordinate of the center-of-mass of the circular disk of radius 2R with a circular hole of radius R as shown in the figure? Assume the disk has a uniform mass density .
cm
2R
R
hole
Uniform density
Disk A
Let object A+B be a uniform disk of radius 2R, height h, and center at x=y=0.Let object A be the disk with the hole in it.Let object B be a uniform disk with radius R, height h, and center at y=0, x=-R.
• Superposition:
2R
R
Disk B
Disk A+B
3))2((
)(22
2 RR
hRR
hRx
M
Mx
M
Mx
M
MMx B
cmA
BBcm
A
BBAcm
A
BAAcm
If object A has mass MA and its center-of-mass is located at and object B has mass MB and its center-of-mass is located at then the center of mass of the system A+B is located at
Acmr
BA
BcmB
AcmABA
cm MM
xMxMx
0
Bcmr
• Example:
Answer: R/3
0
R. Field 10/10/2013 University of Florida
PHY 2053 Page 3
Center-of-Mass: Example
The figure shows a cubical box with each side consisting of a uniform metal plate of negligible thickness. Each of the four sides have mass, M, and the bottom has mass Mbottom. The box is open at the top (at z = L) and has edge length L. If the the z-coordinate of the center-of-mass is at zCM = L/3, what is Mbottom?
• Fall 2011 Exam 2 Problem 22:
Answer: 2M% Right: 24%
bottombottombottom
cmbottombottom
cmcmcmcm
cm MM
ML
MM
LM
MMMMM
zMzMzMzMzMz
4
2
4
)2/(4
4321
44332211
z
y
x
MLMMz bottomcm 2)4(
MML
MLM
z
MLM
cmbottom 24
)3/(
24
2
0
R. Field 10/10/2013 University of Florida
PHY 2053 Page 4
Collisions in 1 Dimension: Elastic
v2 =0 v1
x-axis M1 M2
Lab Frame Before
121
12
2' v
MM
Mv
• Elastic Collision:An elastic collision is one in which the kinetic energy is conserved (i.e. the initial total kinetic energy is equal to the final total kinetic energy). If a projectile with mass M1 and speed v1 traveling to the right along the x-axis collides with a target particle at rest with mass M2, what are the velocities of the two particles after they undergo an elastic collision?
22112211 '' vMvMpvMvMp finalinitial
v'2 v'1
x-axis M1 M2
Lab Frame After
22111 ')'( vMvvM
2222
12112
12222
12112
1 )'()'( vMvMKEvMvMKE finalinitial 2
2221
11111212
12112
1 )'()')('())'(( vMvvvvMvvM
(momentum conservation)
211 '' vvv
(energy conservation)
(1)
(2)
(3)
121
211' v
MM
MMv
Divide eq. 2 by eq. 1 and multiply by 2
Note: a2 - b2 = (a-b)(a+b)
Multiply eq. 3 by M1 and add it to eq. 1
0
0
121 '' vvv (4)
(5)Insert eq. 5 into eq.4
R. Field 10/10/2013 University of Florida
PHY 2053 Page 5
1d Collisions: Completely Inelastic
v2 v1
x-axis M1 M2
CM Frame Before
V
x-axis M1+M2
Lab Frame After
• Completely Inelastic Collision (Lab Frame):A projectile with mass M1 and speed v1 traveling to the right along the x-axis collides with a target particle at rest with mass M2. If the two particles stick together to form a single particle of mass M1+M2, what is its velocity? What is the velocity of the center-of-mass of the two particles system before the collision? What is the change in the kinetic energy before and after the collision?
21
21
211
21
1
2
1v
MM
MMKEKEKEv
MM
MVV ifcm
VMMpvMvMp finalinitial )( 212211 (momentum conservation)
0
v2 =0 v1
x-axis M1 M2
Lab Frame Before
V = 0
x-axis M1+M2
CM Frame After
0
• Completely Inelastic Collision (CM Frame):
In the CM frame the initial two particles have equal and opposite momentum.They stick together and are at rest with zero final momentum and zero final kinetic energy. This corresponds to the maximal loss of kinetic energy consistent with momentum conservation.
CMCMcmtotcmtot ppPvM 210
CMCM pp 21
0)( 2121 VMMpppP CMf
CMCMcmtot