Physics 218
Lecture 15
Dr. David Toback
Physics 218, Lecture XV
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Today’s Lecture: Rest of Chap 8
• Center of Mass
• Center of Mass and
Translational Motion
• Collision & Explosion
Problems using:
–Conservation of Momentum
–Center of Mass
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Center of Mass (CM)
What is the “Center of Mass?”
• More importantly “Why do we
care?”
• This is a special point in space
where “it’s as if the object could
be replaced by all the mass at
that one little point”
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Center of Mass (CM) Cont…
Examples where this is useful:
• We have a spherical cow that weighs
two tons. We can model defenestrating
her as if she were a single point
• We can model the earth moving
around the sun as a single point at “the
center of the earth”
At some level we’ve been assuming these
things for doing problems all semester
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Center of Mass (CM) Cont…
Yet another example: there
are only a couple of points
on a ruler that you can put
your finger under and hold
it up
–Your finger provides the
normal force
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Visual Examples
The center of mass has the same trajectory as a
point since both have the same acceleration and
initial velocity
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How do you calculate CM?
1. Pick an origin
2. Look at each “piece of mass” and figure
out how much mass it has and how far it is
(vector displacement) from the origin. Take
mass times position
3. Add them all up and divide out by the sum
of the masses
The center of mass is a displacement vector
“relative to some origin”
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Spelling out the math:



m1x1  m 2 x 2
X cm for 2 particles 
m1  m 2




m1x1  m 2 x 2  m3x 3
X cm for 3 particles 
m1  m 2  m3
etc...

Note that x is the 3 - D vector displaceme nt
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2-D Example
Three balls with
masses m1, m2
and m3 are are
located at the
points given to
the right.
Where is the
center of mass?
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D
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So what?
2 ways to solve
collision/explosion
problems:
1. Conservation of
Momentum in all
directions
2. Watching the Center of
Mass
Need to be able to do both
– Pick easier method
– Physics is the same
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Two balls in outer space
Two balls are moving in outer space. They
have known masses 2M and 3M and speeds
4V and 2V, respectively, and they collide at
the origin. The directions are as shown in the
figure. After the collision, the two balls stick
together and form a blob.
What is the final velocity of the blob?
Y
2M
(ignore gravity)
Speed = 4v
X
Speed = 2v
3M
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Toy Rocket Problem
Your friend fires a toy rocket into the air with an unknown
velocity. You observe that at the peak of its trajectory it has
traveled a distance d in the x-direction and that it breaks
into two equal mass pieces. Part I falls straight down with
no initial velocity.
Where does the 2nd half of the toy end up?
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Two Balls in Two Dimensions
Before a collision,
ball 1 moves with
speed v1 in the x
direction, while ball 2
is at rest. Both have
the same mass. After
the collision, the balls
go off at angles Q and
–Q. What are the
velocities, v’1 and v’2,
after the collision?
Q
-Q
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Next Week
• Homework 7: Due Monday
• Reading for Tuesday:
– Chapter 9: Rotational Motion
– Reading Questions: Q9.1 & Q9.7
• Exam 2: Thursday, October 26th
– Mini-practice exam for Exam 2
– Bonus points available!!!
• Reading & questions for Chap 10
– Tuesday, afterPhysics
exam
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Collisions & Explosions
Momentum before
collision is equal to
momentum after
collision.


m1v1  m 2 v 2 
'
'
m1v1  m 2 v 2
True in both X and Y
directions separately
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So what?
• Now we can
show that it’s as
if the entire
body moves as if
it’s a single
point
• Derivation
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2D Example
Three balls with
masses m1, m2 and
m3 are are located at
the points given
below.
Where is the center of
mass?
What is the center of
mass if all the
masses are equal?
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Y1
X1
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Exam II
• Mean without bonus points is 69%.
• Approximate curve
–
–
–
–
–
>100 A+
>90 A
>80 B
>55? C (but could go higher…)
Below this are the D’s and F’s
• If you aren’t caught up with the HW and
didn’t do well on the exam, either catch up
quickly or consider dropping
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Collisions in Two Dimensions
• There is nothing new here
• Simply use the same vector
techniques
–break things up in the X and Y
directions!
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General Example
Two Balls:
Ball 1: m1 with velocity v1
Ball 2: m2 with velocity 0.
Choose axis so v1 points in
X direction.
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Simple Example
We are given two balls with masses m1 and m2
and an origin. The balls are placed at a
distance x1 and x2 from the origin.
Where is the
center of mass
if:
1. In general
2. m2 = m1 ?
3. m1 = 0
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CM and Integrals…
A rod of length L and
mass M has a
uniform density.
Calculate the center of
mass.
What if the density
varied linearly from
0 to some value at
the end?
Hint: This requires an
integral
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Three guys on a raft
Three guys are hanging out on a raft at the
locations given below. The origin is at the
left.
Where is the center of mass?
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