Lecture 10: Center of Mass
Physics 2210
Fall Semester 2014
Today's Concepts:
a) Finding the Center of Mass
b) Using the Center of Mass
Mechanics Lecture 10, Slide 2
Unit 10: Prelecture Feedback
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Open homeworks earlier.
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As of now, all homeworks until the end of
the semester are 'open'.
Make checkpoint answers available the
same day.
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They're available at 11 AM the same
day. Flexibility for people with
emergencies.
Checkpoint question with men pulling pole
Homework-type questions
Calculus practice, especially multivariable
integration
Mechanics Lecture 10, Slide 3
Center of Mass
Mechanics Lecture 10, Slide 4
Example
A system consists of three particles located as
shown in the Figure. Find the center of mass of
the system. m1 = m2 = 1 kg and m3 = 2 kg.
y(m)
2
m3
1
m1
m2
1
2
x(m)
Example
Find the center of mass of the triangle
shown below. Assume uniform mass per
unit area , i.e. M = ·(½ b h).
y
h
b
x
Center of Mass for System of Objects
Mechanics Lecture 10, Slide 7
Flashcard 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
C) same
x
x
CM
Case 1
Case 2
Mechanics Lecture 10, Slide 8
Kinematic Quantities
If the total
force is zero,
the CM won’t
accelerate
Mechanics Lecture 10, Slide 9
Flashcard Question
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?
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 10
Flashcard Question
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?
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 11
Interesting Application:
Exoplanet Discovery
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Extrasolar planets
usually too small to be
seen directly.
Need to detect indirectly
by observing changes to
the star.
“astrometry”: observe
“wobble” as star moves
in order to to keep
c.o.m. fixed!
Homework Problem
m1 x1 + m2 x2 + m3 x3 + m4 x4
xcm =
m1 + m2 + m3 + m4
m1 y1 + m2 y2 + m3 y3 + m4 y4
ycm =
m1 + m2 + m3 + m4
Mechanics Lecture 10, Slide 13
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
=
m1 + m2 + m3 + m4
2
2
vcm = vcm
+
v
,x
cm , y
Mechanics Lecture 10, Slide 14
A
B
C
D
m5
CM moves toward m5
cm
Mechanics Lecture 10, Slide 15
A
B
C
m5
CM moves toward m5
cm
Mechanics Lecture 10, Slide 16
Flashcard 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
1)
M
T
C) Same
F = 2T
2)
M
T
Mechanics Lecture 10, Slide 17
Flashcard 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?
a1
C) Same
1)
T
M
F = 2T
a2
2)
M

aCM
T

FNet ,External
=
M Total
SAME
Mechanics Lecture 10, Slide 18
Checkpoint
Checkpoint
Checkpoint
Flashcard Question
Now, if the guy on the left has mass 2M and the guy on the right 1M,
where would they meet?
a) -3 m
b) -1 m
c) 0 m
d) +1 m
Flashcard Question
Now, if the guy on the left has mass 2M and the guy on the right 1M,
where would they meet?
a) -3 m
b) -1 m
c) 0 m
d) +1 m