Your Comments
frictionless ice, weightless ropes, these kinds of things don't
even exist and I wish we'd learn how to do things as
if they happened in real life.
I'm honestly lost. It is completely different from how I solved this
in highschool and I would like a much more detailed explanation
of how all the formulas are applied into actual problems and how to solve them.
Is it okay if I have no idea what's going on when the pre-lecture
goes through the derivation of concepts?
gravity can not be held responsible if someone falls in love on valentines day
That doughnut was a krispy kreme right?
how do I sign up for the conflict exam?
Stuff is pretty cool. Though... how WOULD you get the center of mass of a
completely un-uniform solid mass like a chess piece? I can read the formula,
but applying it to such an object is sort of fuzzy for me.
This one was much easier than the last few prelectures.
i tried to drown myself in the toilet but my neck is too big to allow my face to
reach the water…
As an aside, do you know any good (or terrible) physics jokes?
Mechanics Lecture 10, Slide 1
1st Exam Wednesday February20:
90 Minutes. 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.
You can sign up for this in your grade-book before Feb 16.
If you have a serious conflict with both exams, please email Prof.
Bezryadin (bezryadi@illinois.edu)
Exam review in class next Tuesday – we will work through the first
hour exam from Fall/2010
The old exams are a fantastic way to practice. We have put hundreds
of great problems online (with formula sheet)…
There is no lab next week. You will have discussion but no quiz.
(always look at the syllabus – its all there)
Mechanics Lecture 9, 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

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
Center of Mass Discrete Particles
“I hate sigmas. I only want to see them on frat houses, and not in problems.”
Mechanics Lecture 10, Slide 7
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
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)
“The calculus was also some new stuff, like triple integrals and integrating in 3-D”
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
1
1
xcm 
xdm 

M
M

L
0
M
1 L
x dx   xdx
L
L 0
L
11 2 2
11 2
L
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.
“1. The center of mass is geometrical, and
need not be contained within an object.
2. It doesn't really matter, since this is a
university and somebody is bound to eat it
long before someone else has the time to
consider its physical and mathematical
properties in detail.”
Mechanics Lecture 10, Slide 16
Cube Example
“can you run over the triple intergral deriviation please.”
Mechanics Lecture 10, Slide 17
Cube Example: Using Volume Mass Density
Mechanics Lecture 10, Slide 18
Cube Example: CM Calculation
MTotal
L

2
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!
The CM moves like
a point particle !
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.
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
aCM 
FNet , External
Which of the following statements about the
acceleration of the center of mass of the system is true:
M Total
C) aCM = 0
D) aCM = F/(3M) to the left
C) “Because both masses have the same acceleration but
are moving in opposite directions, the center mass for
acceleration is zero.”
D) “The net force on the center of mass is 1F to the left.
Acceleration is a=F/m The mass of the system is 3m, so
the acceleration of the center of mass is F/3m to the
left.”
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?
-3 m
A) -3 m
0m
B) 0 m
3m
C) 3 m
“They would meet at the initial center of mass because there is no external force to move
the initial center of mass? This question confuses me because there are no external
forces so how could they change their positions by pulling on a pole. It seems to be
analogous to holding a fan behind the sail of a sailboat; there would be no net force and
demo
therefore no movement since all forces remain in the system...”
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)
m1 x1  m2 x2 2M   -3m
 -1 m


m1  m2
3M
Mechanics Lecture 10, Slide 29