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Physics 111: Lecture 13 Today’s Agenda. Potential Energy & Force Systems of Particles Center of mass Velocity and acceleration of the center of mass Dynamics of the center of mass Linear Momentum Example problems. Potential Energy & Force. - PowerPoint PPT Presentation
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Physics 111: Lecture 13, Pg 1
Physics 111: Lecture 13
Today’s Agenda
Potential Energy & Force
Systems of Particles
Center of mass
Velocity and acceleration of the center of mass
Dynamics of the center of mass Linear Momentum
Example problems
Physics 111: Lecture 13, Pg 2
Potential Energy & Force
For a conservative force we define the potential energy function:
Therefore:
Consider some potential energy functions we know, and find the forces:
Spring:
Gravity near earth:
Newton’s Gravity:
CdxFU
dxFWU2
1
x
x
U kx Cx 1
22
U mgy Cy
FdU
dxkxx
FdU
dymgy
UGMm
RCR F
dU
dR
GMm
RR
2
It’s true!!
dxdU
F
Physics 111: Lecture 13, Pg 3
Potential Energy Diagrams
Consider a block sliding on a frictionless surface, attached to an ideal spring.
0x
U
m
xU kxs
1
22
Physics 111: Lecture 13, Pg 4
Potential Energy Diagrams
Consider a block sliding on a frictionless surface, attached to an ideal spring.
F = -dU/dx = -slope
m
x
x 0x
U
F
F
U kxs 1
22
Physics 111: Lecture 13, Pg 5
Potential Energy Diagrams
The potential energy of the block is the same as that of
an object sliding in a frictionless “bowl”:
Ug = mgy = 1/2 kx2 = Us
0x
U
m
2xgm2k
y
is the height of an object in the bowl at position x
Physics 111: Lecture 13, Pg 6
Equilibrium
F = -dU/dx = -slope So F = 0 if slope = 0. This is the case at the
minimum or maximum of U(x).
This is called an equilibrium position.If we place the block at
rest at x = 0, it won’t move.
0x
U
m
x
Physics 111: Lecture 13, Pg 7
Equilibrium
If small displacements from the equilibrium position result in a force that tends to move the system back to its equilibrium position, the equilibrium is said to be stable.
This is the case if U is a minimum at the equilibrium position.
In calculus language, the equilibrium is stable if the curvature (second derivative) is positive.
0x
U
m
x
F
F
Physics 111: Lecture 13, Pg 8
Equilibrium
Suppose U(x) looked like this:
This has two equilibrium positions, one is stable (+ curvature) and one is unstable (- curvature).
Think of a small object sliding on the U(x) surface:If it wants to keep sliding when you give it a little push,
the equilibrium is unstable.If it returns to the equilibrium position when you give it a
little push, the equilibrium is stable.If the curvature is zero (flat line) the equilibrium is
neutral.
U
x0
stable
unstable
neutral
Balancecone
Birds
Physics 111: Lecture 13, Pg 9
System of Particles
Until now, we have considered the behavior of very simple systems (one or two masses).
But real life is usually much more interesting! For example, consider a simple rotating disk.
An extended solid object (like a disk) can be thought of as a collection of parts. The motion of each little part depends on where it is in the object!
Physics 111: Lecture 13, Pg 10
System of Particles: Center of Mass
How do we describe the “position” of a system made up of many parts?
Define the Center of Mass (average position):For a collection of N individual pointlike particles whose
masses and positions we know:
Rr
CM
i ii
N
ii
N
m
m
1
1
y
x
r1
m1
r3
r2
r4m4
m2
m3
RCM
(In this case, N = 4)
Icetable
Physics 111: Lecture 13, Pg 11
System of Particles: Center of Mass
If the system is made up of only two particles:
Rr
r rCM
i ii
N
ii
N
m
m
m m
m m
1
1
1 1 2 2
1 2
y
x
r2r1
m1
m2
RCM
r2 - r1
m m m
m m1 2 1 2 2 1
1 2
r r r
R r r rCMm
M 1
22 1
where M = m1 + m2
So:
Physics 111: Lecture 13, Pg 12
System of Particles: Center of Mass
If the system is made up of only two particles:
y
x
r2r1
m1
m2
RCM
r2 - r1
where M = m1 + m2
+
R r r rCMm
M 1
22 1
If m1 = m2
the CM is halfway betweenthe masses.
R r r rCM 1 2 112
Physics 111: Lecture 13, Pg 13
System of Particles: Center of Mass
If the system is made up of only two particles:
y
x
r2r1
m1
m2
RCM
r2 - r1
where M = m1 + m2
+
If m1 = 3m2
the CM is now closer tothe heavy mass.
R r r rCMm
M 1
22 1
R r r rCM 1 2 114
Physics 111: Lecture 13, Pg 14
System of Particles: Center of Mass
The center of mass is where the system is balanced!Building a mobile is an exercise in finding centers of
mass.
m1m2
+m1 m2
+
Baton
Physics 111: Lecture 13, Pg 15
System of Particles: Center of Mass
We can consider the components of RCM separately:
( , , ) , ,X Y Zm x
M
m y
M
m z
MCM CM CMi ii i ii i ii
y
x
r1
m1
r3
r2
r4m4
m2
m3
RCM
(In this case, N = 4)
Physics 111: Lecture 13, Pg 16
Example Calculation:
Consider the following mass distribution:
(24,0)
12m4
24m12)m2(0mM
xmX i ii
CM
6m4
0m12)m2(0mM
ymY i ii
CM
(0,0)
(12,12)
m
2m
m
RCM = (12,6)
Physics 111: Lecture 13, Pg 17
Rr r
CMdm
dm
dm
M= =òò
ò
System of Particles: Center of Mass
For a continuous solid, we have to do an integral.
y
x
dm
rwhere dm is an infinitesimal mass element.
Physics 111: Lecture 13, Pg 18
System of Particles: Center of Mass
We find that the Center of Mass is at the “center” of the object.
y
x
RCM
Physics 111: Lecture 13, Pg 19
System of Particles: Center of Mass
The location of the center of mass is an intrinsic property of the object!!(it does not depend on whereyou choose the origin or coordinates whencalculating it).
y
xRCM
We find that the Center of Mass is at the “center” of the object.
Physics 111: Lecture 13, Pg 20
System of Particles: Center of Mass
We can use intuition to find the location of the center of mass for symmetric objects that have uniform density:
It will simply be at the geometrical center !
+CM
+ +
++ +
Physics 111: Lecture 13, Pg 21
System of Particles: Center of Mass
The center of mass for a combination of objects is the average center of mass location of the objects:
+
+y
x
R2
R1
RCM
m1
m2
+
21
2211CM
N
1ii
N
1iii
CM
mm
mm
m
m
RRR
RR
R2 - R1
so if we have two objects:
122
1 M
mRRR
Pisa
Bottle
Physics 111: Lecture 13, Pg 22
Lecture 13, Act 1Center of Mass
The disk shown below (1) clearly has its CM at the center. Suppose the disk is cut in half and the pieces arranged as
shown in (2): Where is the CM of (2) as compared to (1)?
(a) higher (b) lower (c) same
(1) (2)
XCM
Physics 111: Lecture 13, Pg 23
Lecture 13, Act 1Solution
The CM of each half-disk will be closer to the fat end than to the thin end (think of where it would balance).
(1) (2)
X
X
The CM of the compound object will be halfway between theCMs of the two halves.
XXCM
This is higher than the CM of the disk
Physics 111: Lecture 13, Pg 24
System of Particles: Center of Mass
The center of mass (CM) of an object is where we can freely pivot that object.
Gravity acts on the CM of an object (show later)
If we pivot the objectsomewhere else, it willorient itself so that theCM is directly below the pivot.
This fact can be used to findthe CM of odd-shaped objects.
+CM
pivot
+ CM
pivot
+
pivot
CM
mg
Doublecone
Physics 111: Lecture 13, Pg 25
System of Particles: Center of Mass
Hang the object from several pivots and see where the vertical lines through each pivot intersect!
pivot
pivotpivot
+CM
The intersection point must be at the CM.
Oddshapes
Physics 111: Lecture 13, Pg 26
Lecture 13, Act 2Center of Mass
An object with three prongs of equal mass is balanced on a wire (equal angles between prongs). What kind of equilibrium is this position?
a) stable
b) neutral
c) unstable
3 prongedobject
Fork, spoon,and match
Physics 111: Lecture 13, Pg 27
Lecture 13, Act 2Solution
The center of mass of the object is at its center and is initially directly over the wire
If the object is pushed slightly to the left or right, its center of mass will not be above the wire and gravity will make the object fall off
+ CM + CM
mgmg
(front view)
Physics 111: Lecture 13, Pg 28
Consider also the case in which the two lower prongs have balls of equal mass attached to them:
+ CM
mg
Lecture 13, Act 2Solution
In this case, the center of mass of the object is below the wire
mg
+ CM
When the object is pushed slightly, gravity provides a restoring force, creating a stable equilibrium
Physics 111: Lecture 13, Pg 29
Velocity and Accelerationof the Center of Mass
If its particles are moving, the CM of a system can also move. Suppose we know the position ri of every particle in the system
as a function of time.
R rCM i ii
N
Mm
1
1
VR r
vCMCM
ii
i
N
i ii
Nd
dt Mm
d
dt Mm
1 1
1 1So:
AV v
aCMCM
ii
i
N
i ii
Nd
dt Mm
d
dt Mm
1 1
1 1And:
The velocity and acceleration of the CM is just the weighted average velocity and acceleration of all the particles.
N
1iimM
Physics 111: Lecture 13, Pg 30
Linear Momentum:
Definition: For a single particle, the momentum p is defined as:
Newton’s 2nd Law:
F = ma
dtm
p = mv(p is a vector since v is a
vector).
So px = mvx etc.
dv )m(dtd v
dtdF
p
Units of linear momentum are kg m/s.
Physics 111: Lecture 13, Pg 31
Linear Momentum:
P p v ii
N
i ii
Nm
1 1
For a system of particles the total momentum P is the vector sum of the individual particle momenta:
m Mi ii
N
CMv V
1But we just showed that
P VM CMSo
N
1iiiCM m
M1
vV
Physics 111: Lecture 13, Pg 32
Linear Momentum:
So the total momentum of a system of particles is just the total mass times the velocity of the center of mass.
Observe:
We are interested in so we need to figure out
ddt
Md
dtM mCM
CM i ii
i neti
P VA a F ,
ddtP Fi net
i,
P VM CM
Physics 111: Lecture 13, Pg 33
Linear Momentum:
Suppose we have a system of three particles as shown. Each particle interacts with every other, and in addition there is an external force pushing on particle 1.
m1
m3
m2
F13
F31 F32
F23
F21F12
F1,EXT
EXT1
3231
2321
EXT11213i
NETi
,
,,
F
FF
FF
FFFF
(since the other forces cancel in pairs...Newton’s 3rd Law)
All of the “internal” forces cancel !!Only the “external” force matters !!
Physics 111: Lecture 13, Pg 34
Linear Momentum:
Only the total external force matters!
m1
m3
m2
F1,EXT
EXTNETi
EXTidt
d,,
FFP
CMEXTNETM
dt
dA
PF
,
Which is the same as:
Newton’s 2nd law applied to systems!
Physics 111: Lecture 13, Pg 35
Center of Mass Motion: Recap
We have the following law for CM motion:
This has several interesting implications:
It tells us that the CM of an extended object behaves like a simple point mass under the influence of external forces: We can use it to relate F and A like we are used to doing.
It tells us that if FEXT = 0, the total momentum of the system can not change.The total momentum of a system is conserved if there
are no external forces acting.
FP
AEXT CMddt
M
Pork chop
Pendulum
Physics 111: Lecture 13, Pg 36
Example: Astronauts & Rope
Two astronauts at rest in outer space are connected by a light rope. They begin to pull towards each other. Where do they meet?
M = 1.5m m
Physics 111: Lecture 13, Pg 37
Example: Astronauts & Rope...
They start at rest, so VCM = 0. VCM remains zero because
there are no external forces. So, the CM does not move! They will meet at the CM.
Finding the CM:
M = 1.5m m
CM
L
If we take the astronaut on the left to be at x = 0:
xM m(L
M m
m(L
mLcm
( ) ) )
.
0
2 5
2
5
x=0 x=L
Physics 111: Lecture 13, Pg 38
Lecture 13, Act 3Center of Mass Motion
A man weighs exactly as much as his 20 foot long canoe. Initially he stands in the center of the motionless canoe, a
distance of 20 feet from shore. Next he walks toward the shore until he gets to the end of the canoe.What is his new distance from the shore.
(There no horizontal force on the canoe by the water).
20 ft
? ft
20 ft
(a) 10 ft (b) 15 ft
(c) 16.7 ft
before
after
Physics 111: Lecture 13, Pg 39
Lecture 13, Act 3Solution
X
X
Since the man and the canoe have the same mass, the CM of the man-canoe system will be halfway between the CM of the man and the CM of the canoe.
x 20 ft
Initially the CM of the system is 20 ft from shore.
CM of system
Physics 111: Lecture 13, Pg 40
Lecture 13, Act 3Solution
Since there is no force acting on the canoe in the x-direction, thelocation of the CM of the system can’t change!
20 ftx
X
X
CM of system
Therefore, the man ends up 5 ft to the left of the system CM, and the center of the canoe ends up 5 ft to the right.
10 ft
5 ft
15 ft
He ends up moving 5 ft toward the shore (15 ft away).
Physics 111: Lecture 13, Pg 41
Recap of today’s lecture
Systems of particles (Text: 8-1)
Center of mass (Text: 8-1 & 12-6)
Velocity and acceleration of the center of mass (Text:8-3)
Dynamics of the center of mass (Text: 8-3 to 8-4)Linear Momentum
Example problems
Look at textbook problems Chapter 8: # 3, 7, 17, 29, 35, 77, 111