Transcript File

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.
5.1 Uniform Circular Motion
Let T be the time it takes for the object to
travel once around the circle.
r
2 r
v
T
5.2 Centripetal Acceleration
In uniform circular motion, the speed is constant, but the
direction of the velocity vector is not constant. Therefore, this
means that an acceleration is occurring. It is given by
2
v
ac 
r
The direction of the centripetal acceleration is towards the
center of the circle;
5.3 Centripetal Force
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.

a


F
m



F  ma
5.3 Centripetal 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.
2
v
Fc  mac  m
r
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
2
v
Fc  FN sin   m
r
FN cos   mg
5.4 Banked Curves
2
v
FN sin   m
r
FN cos   mg
2
v
tan  
rg
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. At what
speed would the cars have to travel around them?
2
v
tan  
rg
v
316 m 9.8 m
v  rg tan 

s 2 tan 31  43 m s 96 mph 
Problems to be solved
• 5.10, 5.23, 5.30, 5.41, 5.50, 5.60
• B5.1 A woman runs on a circular track of
radius 100m at a speed of 8m/s. What is
her acceleration? Ans: 0.64m/s2
• B5.2 A racing car rounds a turn at 60m/s.
If the force needed to provide the
centripetal acceleration is equal to the
weight of the car, what is the radius of the
turn? Ans: 367.35m
• B5.3 A car is travelling around a flat curve
of radius 0.25km. The coefficient of static
friction between the tires and the road is
0.4. At what speed will the car begin to
skid? Ans: 31.30m/s
• B5.4 A racing car travels around a curve
of radius 1000m. If the frictional force is
zero and the speed is 50m/s, at what
angle is the curve banked? Ans: 14.310