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Uniform Circular Motion
Part I
http://en.wikipedia.org/wiki/G-force
http://www.youtube.com/watch?v=EXQKl-28zn0
Uniform Circular Motion
• Uniform circular motion: motion in a circle
of constant radius at constant speed
• Instantaneous velocity is always tangent to
circle (perpendicular to the radius)
Note that the magnitude of v is
constant but not the direction


v1  v2  v
f is the frequency
 
v1  v2
T is the period
2r
v
 r  2fr
T
 is angular velocity

v1

v1

v

v

v3

v2

v2
  
v  v2  v1


 v2  (v1 )

v

v2

 v1
Looking at the change in velocity in the limit that the time interval becomes

infinitesimally small, we see that
v1
s
ac

ac

| v | s

v
r
 vs
| v |
r

| v | vs

t
r.t
v2
ac 
r

v2

v


 v1

v2
Centripetal Acceleration
• an object that moves on a circular path of radius r
with constant speed v has an acceleration a.
• The direction of the acceleration vector always
points towards the center of rotation C (thus the
name centripetal) Its magnitude is constant
Centripetal Force
• If we apply Newton’s law to analyze uniform circular
motion we conclude that the net force in the direction that
points towards C must have magnitude:
mv 2
Fc 
r
• This force is known as “centripetal force”
• Centripetal force is not a new kind of force. It is simply
the net force that points from the rotating body to the
rotation center C. Depending on the situation the
centripetal force can be friction, the normal force or
gravity or combination of forces.
What about Centrifugal Force?
• There is no centrifugal force pointing outward;
what happens is that the natural tendency of the
object to move in a straight line must be
overcome.
• If the centripetal force vanishes, the object flies
off tangent to the circle.
Problem Solving Strategies
• Draw the force diagram for the object
• Choose one of the coordinate axes (the yaxis in this diagram) to point towards the
orbit center C along the radius.
• Determine Fnet
• Set Fnet
mv 2

r
Example
• A hockey puck moves around a circle at
constant speed v on a horizontal ice surface.
The puck is tied to a string looped around a
peg at point C. In this case the net force
along the y-axis is the tension T of the
string. Tension T is the centripetal force.
Example
Example
•
The Rotor is a large hollow cylinder of radius R that is rotated rapidly around
its central axis with a speed v. A rider of mass m stands on the Rotor floor with
his/her back against the Rotor wall. Cylinder and rider begin to turn. When the
speed v reaches some predetermined value, the Rotor floor abruptly falls
away. The rider does not fall but instead remains pinned against the Rotor
wall. The coefficient of static friction μs between the Rotor wall and the rider
is given.
We draw a free body diagram
for the rider using the axes
shown in the figure.
The normal reaction FN is the
centripetal force.
Fx ,net
Fy ,net
mv 2
 FN  ma =
(eqs.1) ,
R
 f s  mg  0 , f s   s FN  mg   s FN (eqs.2)
mv 2
Rg
If we combine eqs.1 and eqs.2 we get: mg   s
 v2 
 vmin 
R
s
Rg
s
Example
• In a 1901 circus performance Allo Diavolo introduced the stunt of
riding a bicycle in a looping-the-loop. The loop sis a circle of radius R.
What is the minimum speed v that Diavolo should have at the top of
the loop and not fall We draw a free body diagram for Diavolo when
he is at the top of the loop. Two forces are acting: Gravitational force
Fg and the normal reaction FN from the loop. When Diavolo has the
minimum speed v he has just lost contact with the loop and thus FN =
0. The only force acting on Diavolo is Fg The gravitational force Fg is
the centripetal force.
The skid marks show straight
lines. Why?
Banked Roads
• As long as the tires do not slip, the friction is
static. If the tires do start to slip, the friction is
kinetic, which is bad in two ways:
1. The kinetic frictional force is smaller than the
static.
2. The static frictional force can point towards the
center of the circle, but the kinetic frictional force
opposes the direction of motion, making it very
difficult to regain control of the car and continue
around the curve.
Nonuniform Circular Motion
If an object is moving in a circular path but at
varying speeds, it must have a tangential
component to its acceleration as well as the radial
one.
Nonuniform Circular Motion
This concept can be used for an object moving along any curved path, as a
small segment of the path will be approximately circular.
A child on a sled comes flying over the crest of a small hill. His sled does not leave the
ground, but he feels the normal force between his chest and the sled decreases as he goes
over the hill. Explain?
A 0.150 kg ball on the end of a 1.10 m long cord is
swung in a vertical circle.
(a) determine the minimum speed the ball must have
at the top of its arc so that the ball continues moving
in a circle
(b) Calculate the tension in the cord at the bottom of
the arc, assuming the ball is moving twice the speed
of part(a)
What is the apparent weight at the top and the bottom of the track?
What must be the rotational speed of a space station in zero gravity so that the
occupants experience an artificial g?
•
https://en.wikipedia.org/wiki/Artificial_gravity