ch05 dynamics of circular motion angular

Chapter 5
Dynamics of Uniform
Circular Motion

5.1 Uniform Circular Motion
DEFINITION OF UNIFORM CIRCULAR MOTION
Uniform circular motion is the motion of an object
traveling at a constant speed on a circular path.

Let T be the time it takes for the object to
travel once around the circle.
v
r
T

2
r

Example 1: A Tire-Balancing Machine
The wheel of a car has a radius of 0.29m and it being rotated
at 830 revolutions per minute on a tire-balancing machine.
Determine the speed at which the outer edge of the wheel is
moving.
revolution
min
10
2
.
1
min
s
revolution
830
1 3



s
072
.
0
min
10
2
.
1 3


 
T
  s
m
25
s
072
.
0
m
0.29
2
2





T
r
v

5.2 Centripetal Acceleration
In uniform circular motion, the speed is constant, but the
direction of the velocity vector is not constant.

90

 


90




 

r
t
v
v
v 


r
v
t
v 2



r
v
ac
2


The direction of the centripetal acceleration is towards the
center of the circle; in the same direction as the change in
velocity.
r
v
ac
2


Conceptual Example 2: Which Way Will the Object Go?
An object is in uniform circular
motion. At point O it is released
from its circular path. Does the
object move along the straight
path between O and A or along
the circular arc between points
O and P ?

Example 3: The Effect of Radius on Centripetal Acceleration
The bobsled track contains turns
with radii of 33 m and 24 m.
Find the centripetal acceleration
at each turn for a speed of
34 m/s. Express answers as
multiples of .
s
m
8
.
9 2

g

r
v
ac
2

  g
ac 6
.
3
s
m
35
m
33
s
m
34 2
2



  g
ac 9
.
4
s
m
48
m
24
s
m
34 2
2




5.3 Centripetal Force
  a
F


m
m


F
a


Recall Newton’s Second Law
When a net external force acts on an object
of mass m, the acceleration that results is
directly proportional to the net force and has
a magnitude that is inversely proportional to
the mass. The direction of the acceleration is
the same as the direction of the net force.

Thus, in uniform circular motion there must be a net
force to produce the centripetal acceleration.
The centripetal force is the name given to the net force
required to keep an object moving on a circular path.
The direction of the centripetal force always points toward
the center of the circle and continually changes direction
as the object moves.
r
v
m
ma
F c
c
2



Example 5: The Effect of Speed on Centripetal Force
The model airplane has a mass of 0.90 kg and moves at
constant speed on a circle that is parallel to the ground.
The path of the airplane and the guideline lie in the same
horizontal plane because the weight of the plane is balanced
by the lift generated by its wings. Find the tension in the 17 m
guideline for a speed of 19 m/s.
r
v
m
T
Fc
2


   N
19
m
17
s
m
19
kg
90
.
0
2


T

Conceptual Example 6: A Trapeze Act
In a circus, a man hangs upside down from a trapeze, legs
bent over and arms downward, holding his partner. Is it harder
for the man to hold his partner when the partner hangs
straight down and is stationary of when the partner is swinging
through the straight-down position?

5.4 Banked Curves
On an unbanked curve, the static frictional force
provides the centripetal force.

5.4 Banked Curves
On a frictionless banked curve, the centripetal force is the
horizontal component of the normal force. The vertical
component of the normal force balances the car’s weight.

5.4 Banked Curves
r
v
m
F
F N
c
2
sin 
  mg
FN 

cos

5.4 Banked Curves
r
v
m
FN
2
sin 

mg
FN 

cos
rg
v2
tan 


5.4 Banked Curves
Example 8: The Daytona 500
The turns at the Daytona International Speedway have a
maximum radius of 316 m and are steely banked at 31
degrees. Suppose these turns were frictionless. As what
speed would the cars have to travel around them?
rg
v2
tan 
 
tan
rg
v 
    
mph
96
s
m
43
31
tan
s
m
8
.
9
m
316 2

 
v

5.5 Satellites in Circular Orbits
There is only one speed that a satellite can have if the
satellite is to remain in an orbit with a fixed radius.

r
v
m
r
mM
G
F E
c
2
2


r
GM
v E


Example 9: Orbital Speed of the Hubble Space Telescope
Determine the speed of the Hubble Space Telescope orbiting
at a height of 598 km above the earth’s surface.
  
 
h
mi
16900
s
m
10
56
.
7
m
10
598
m
10
38
.
6
kg
10
98
.
5
kg
m
N
10
67
.
6
3
3
6
24
2
2
11










v

T
r
r
GM
v E 
2


E
GM
r
T
2
3
2


Global Positioning System
hours
24

T
E
GM
r
T
2
3
2


5.6 Apparent Weightlessness and Artificial Gravity
Conceptual Example 12: Apparent Weightlessness and
Free Fall
In each case, what is the weight recorded by the scale?

5.6 Apparent Weightlessness and Artificial Gravity
Example 13: Artificial Gravity
At what speed must the surface of the space station move
so that the astronaut experiences a push on his feet equal to
his weight on earth? The radius is 1700 m.
mg
r
v
m
Fc 

2
  
2
s
m
80
.
9
m
1700

 rg
v

5.7 Vertical Circular Motion
r
v
m
mg
FN
2
1
1 

r
v
m
mg
FN
2
3
3 

r
v
m
FN
2
2
2 
r
v
m
FN
2
4
4 

ch05 dynamics of circular motion angular

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