Your Comments
Very nicely done, jolly good work old chap, pip pop pip chiperoo! I say.
You should have made the doughnut problem more difficult by adding sprinkles.
Whenever I read notes or textbooks for any class, I hear it narrated by Gary Gladding in my head.
MAKE IT STOP!
In the prelecture, It's so ignorant of the astronaut to throw the wrench in an unknown direction.
Since there is no air resistance, this wrench may accelerate enough to damage satellites, UFOs or
even small asteroids. Let's not encourage vulgar behavior in space!
"No eternal reward will forgive us now for wasting the dawn." --Jim Morrison That is my favorite
quotation, and after listening to "The WASP (Texas Radio and the Big Beat)" in lecture, I remark
that The Doors were the theme to most of my conscious life…
I don't know what the course policy is on things like this, but can we take a class trip into space with
a bag of wrenches and just fly around wherever we want? That would just be great.
This prelecture made me happy. I think I understand everything :)
We need to go over EVERYTHING.
I wish I weighed 10 times the mass of a wrench.
i think you should be able to say things like "this makes sense" and "you guys did good“
If I slow down and think about it it isn't too difficult to comprehend it just seems like its going to be a
lot of work
That moon problem in the homework is hilarious. "You don't have a traditional rocket". What, are we
being shot by a giant cannon like in Jules Verne's novel "From Earth to the Moon"?
Is there any possible way we could receive bonus for knowing the artists at the beginning of each
lecture? I love hearing such wonderful music walking in to class, but i would also like to be
rewarded for my brilliant knowledge. Thank you.
Mechanics Lecture 10, Slide 1
Exam Reminder:
Next Wednesday (Feb 22) at 7pm
Covers Lectures 1-6 only
(No “work & energy” on the exam)
We offer a conflict exam from 5:15 – 6:45 on the night of the
exam. If you need to sign up for this in your grade-book TODAY.
If you need a conflict-conflict (which requires a real excuse)
contact Dr. Paco Jain (rdjain1@illinois.edu) TODAY.
Exam review in class next Tuesday – we will work through the first
hour exam from Fall 2010
Mechanics Lecture 10, Slide 2
Physics 211
Lecture 10
Today’s Concept:
Center of Mass


Finding it
Using it
Mechanics Lecture 10, Slide 3
Center of MAss
“F=ma again! It's so wonderful when complicated physics
can be simplified so brilliantly.”
r
r
F

MA

cm
Not just for “point” particles, but for ANYTHING!
(if you use center of mass)
Mechanics Lecture 10, Slide 4
Center of Mass
Mechanics Lecture 10, Slide 5
Since you asked..
“Is the center of mass also the center of gravity?”
Center of mass is the average position of stuff weighted by mass:
X CM
m1 x1  m2 x2  ...

m1  m2  ...
Center of gravity is the average position of stuff weighed by its weight:
X CG
w1 x1  w2 x2  ...

w1  w2  ...
If weight can be written like w1=m1g then they are the same.
If the strength of gravity varies across the object they can be different.
We don’t consider Center of Gravity in Physics 211
Mechanics Lecture 10, Slide 6
Center of Mass Discrete Particles
“those equations for the center of mass make me want to VOMIT! all the sum of mi
and sum of the vector of r makes no sense! can we please use words first and then
put the symbols in!!”
Mechanics Lecture 10, Slide 7
Since you asked…
m1r1  m2 r2
m1
m2
m2
m2

r1 
r2 
r1 
r1
m1  m2
m1  m2
m1  m2
m1  m2
m1  m2

m1  m2
m2
r1 
(r2  r1 )
m1  m2
m1  m2
Mechanics Lecture 10, Slide 8
Clicker Question
Three masses are placed on a horizontal frictionless surface at the
corners of an equilateral triangle. Where is the center of mass?
XCM
A)
B)
C)
D)
E)
0
0
0
H/4
H/4
YCM
0
H/2
H/3
H/4
0
y
H
x
Mechanics Lecture 10, Slide 9
Three tiny equal-mass magnets are placed on a horizontal
frictionless surface at the corners of an equilateral triangle. When
the magnets are released, they attract and quickly slide to a single
point. What are the coordinates of that point?
y
XCM
C) 0
X CM  0
YCM
H/3
H
(symmetry)
x
YCM
m1 y1  m2 y2  m3 y3
m0  m0  m H  H


3
m1  m2  m3
3m
Mechanics Lecture 10, Slide 10
Homework Problem
m1 x1  m2 x2  m3 x3  m4 x4
 0.975 m
xcm 
m1  m2  m3  m4
ycm
m1 y1  m2 y2  m3 y3  m4 y4

 1 m
m1  m2  m3  m4
Mechanics Lecture 10, Slide 11
Homework Problem
vcm, x 
vcm, y 
m1v1, x  m2v2, x  m3v3, x  m4v4, x
m1  m2  m3  m4
m1v1, y  m2v2, y  m3v3, y  m4v4, y
2
2
vcm  vcm

v
,x
cm , y
m1  m2  m3  m4
Mechanics Lecture 10, Slide 12
A
B
C
D
m5
CM moves toward m5
cm
Mechanics Lecture 10, Slide 13
Center of Mass (Continuous)
“Lots of proofs but not too bad. Not to sure about continuous objects though.”
Mechanics Lecture 10, Slide 14
Example Uniform Stick
A uniform stick of length L and mass M.
dm
x=L
x=0
dx M
dm  M

dx
L
L
xcm
1

M
1
 xdm  M

L
0
M
1 L
x dx   xdx
L
L 0
L
11 2
11 2
L
2
L 0  
  x  

L  2 0 L 2
2
Mechanics Lecture 10, Slide 15
CheckPoint
A yummy glazed doughnut is shown below. Where is the center
of mass of this fantastic culinary delight?
x
A) In the center of the hole.
B) Anywhere along the blue
dashed line going through
the solid part of the dough.
C) The center of mass is not defined in
cases where there is missing mass.
“It's not ok that you featured a checkpoint
about a glazed doughnut, when there is no
Krispy Kreme doughnuts shop in Chambana. So
here's a deal: I get an A on this exam, you
give me a dozen Krispy Kreme Doughnuts. :)”
Mechanics Lecture 10, Slide 16
Cube Example
“TRIPLE INTEGRALS!?!? Do you have no mercy?.”
Mechanics Lecture 10, Slide 17
Cube Example: Using Volume Mass Density
“How would we calculate the center
of mass if density is a function (not
constant)?”
Mechanics Lecture 10, Slide 18
Cube Example: CM Calculation
MTotal
L

2
“Could you go over why it is 3 integrals when calculating x,y, and z for the box example in the
prelecture? It makes sense if combining all of the different dimensions to have three
integrals but when broken up into individual dimensions it seemed like the three integrals were
still there for all three dimensions !”
Mechanics Lecture 10, Slide 19
Center of Mass for System of Objects
Mechanics Lecture 10, Slide 20
Clicker Question
The disk shown in Case 1 clearly has its CM at the center.
Suppose the disk is cut in half and the pieces arranged as
shown in Case 2
In which case is the center of mass highest?
A) Case 1
B) Case 2
x CM
Case 1
C) same
xx
x CM
x
Case 2
Mechanics Lecture 10, Slide 21
F=MA for Center of Mass!
If the total force
is zero, the CM
won’t accelerate
demo
Mechanics Lecture 10, Slide 22
CheckPoint
Two objects, one having twice the mass of the other, are initially at
rest. Two forces, one twice as big as the other, act on the objects in
opposite directions as shown.
2F
2M
M
F
aCM 
FNet , External
M Total
Which of the following statements about the acceleration of the
center of mass of the system is true:
A) aCM = F/M to the right
B) aCM = F/(3M) to the right
C) aCM = 0
D) aCM = F/(3M) to the left
E) aCM = F/M to the left
Mechanics Lecture 10, Slide 23
Clicker Question
Two pucks of equal mass, on a frictionless table, are being pulled at
different points with equal forces. Which one gets to the end of the
table first?
A) Puck 1
B) Puck 2
C) Same
1)
M
T
F = 2T
2)
M
T
Mechanics Lecture 10, Slide 24
Clicker Question
Two pucks of equal mass, on a frictionless table, are being pulled at
different points with equal forces. Which one gets to the end of the
table first?
a
1
C) Same
1)
M
T
F = 2T
a2
2)
M
T
aCM 
FNet , External
M Total
SAME
Mechanics Lecture 10, Slide 25
CheckPoint
2F
2M
M
F
aCM 
Which of the following statements about the
acceleration of the center of mass of the system is true:
FNet , External
M Total
C) aCM = 0
D) aCM = F/(3M) to the left
C) “The acceleration of the masses is the same so the
center of mass will not move.”
D) “a = fnet/mtotal. F net is just F pointing to the left,
and total mass is 3M.”
Mechanics Lecture 10, Slide 26
CheckPoint
Two guys who weigh the same are holding onto a massless pole while
standing on horizontal frictionless ice. If the guy on the left starts to
pull on the pole, where do they meet?
“the center of mass will not change,
so they will meet in the middle”
3 m
A) 3 m
0m
B) 0 m
3m
C) 3 m
“Two guys "weight" the same? Someone's been spending too much time in the
lab and not enough time around the English language.”
demo
Mechanics Lecture 10, Slide 27
Clicker Question
A large guy with mass 2M and a smaller guy with mass M are holding onto
a massless pole while standing on frictionless ice, as shown below. If the
little guy pulls himself toward the big guy, where would they meet?
A) 3 m
B) 1 m
C) 0 m
D) 1 m
E) 3 m
M
2M
3 m
0m
3m
Mechanics Lecture 10, Slide 28
A large skinny guy with mass 2M and a smaller guy with mass M are
holding onto a massless pole while standing on frictionless ice, as shown
below. If the little guy pulls himself toward the big guy, where would they
meet?
A) 3 m
B) 1 m
C) 0 m
D) 1 m
E) 3 m
X CM
M
2M
3 m
0m
3m
m  M  (3m)
2M   3m
m1 x1  m2 x2
 1 m


m1  m2
3M
Mechanics Lecture 10, Slide 29