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Chapter 14
Chapter 14
Static Equilibrium III
Static Equilibrium III
Chapter 15
Chapter 15
Oscillatory Motion II
Oscillatory Motion II
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View Full Document Example: Slipping Sphere
Example: Slipping Sphere
A uniform sphere is supported by a rope. The point
where the rope is attached to the sphere is located
so a continuation of the rope would intersect a
horizontal line through the sphere’s center a
distance
R
/
2
beyond the center, as shown. What is
the smallest value for
m
s
between the wall and the
sphere?
Taking torques about the contact between the sphere
and the wall, assuming a mass
m
, yields:
3
2
RT
cos30
∘
mgR
Summing forces in both the
x
and
y
directions:
N
T
sin30
∘
and
s
N
T
cos30
∘
mg
Example: Slipping Sphere
Example: Slipping Sphere
A uniform sphere is supported by a rope. The point
where the rope is attached to the sphere is located
so a continuation of the rope would intersect a
horizontal line through the sphere’s center a
distance
R
/
2
beyond the center, as shown. What is
the smallest value for
m
s
between the wall and the
sphere?
There are three unknowns,
m
s
,
T
, and
N
. Substituting
into the force equation in the
y
direction
T
from the
torque equation and
N
from the
x
force equation:
mg
s
sin30
∘
cos30
∘
2
mg
3cos30
∘
3
2
s
tan30
∘
1
→
s
cos30
∘
2sin30
∘
s
cos30
∘
3
/2
.866
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View Full Document Example: Double Welled Potential
Example: Double Welled Potential
Consider the potential given by
U
x
−
ax
2
2
bx
4
4
Find the equilibrium points and determine
if they are stable.
The equilibrium points are found from:
dU
dx
−
ax
bx
3
0
→
x
0,
a
/
b
Stability is determined by the second derivative of the potential at equilibrium.
d
2
U
dx
2
0
−
a
3
bx
2

0
−
a
0
unstable
d
2
U
dx
2
a
/
b
−
a
3
bx
2

a
/
b
−
a
3
a
2
a
0
stable
Simple Harmonic Motion
Simple Harmonic Motion
Mathematically such a force is described as:
The is the force exerted by an ideal spring of spring constant
k
.
From Newton’s 2
nd
we can write:
Simple harmonic motion results when an
object is subject to a linear restoring force and
is called
simple harmonic motion
, SHO.
F
−
kx
F
m
d
2
x
dt
2
−
kx
An object experiencing such a force means that when it is displaced from
equilibrium there is a force proportional to the distance from equilibrium that
accelerates the object back towards its equilibrium position.
How do we describe such motion?
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View Full Document Simple Harmonic Motion
Simple Harmonic Motion
From Newton’s 2
nd
:
Simple harmonic motion results when an
object is subject to a linear restoring force and
is called
simple harmonic motion
, SHO.
F
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This note was uploaded on 04/16/2008 for the course PHYS PHYS2A taught by Professor Hicks during the Spring '08 term at UCSD.
 Spring '08
 Hicks
 Static Equilibrium

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