Uniform Circular Motion
Physics 1
DEHS 2008-09
• The direction from the
edge of the circle towards
the center of the circle is
called the radial direction
– Sign convention is that the
+r direction is toward the
center of the circle
• The direction
perpendicular to the radial
direction is called the
tangential direction
tangential
Anatomy of a Circle
radial
Circular Velocity & Acceleration
• Because the direction that the object is
moving in is constantly changes, velocity
constantly changes
– An object’s instantaneous velocity is directed in
the tangential direction
• Because velocity constantly changes, there is a
constant acceleration
– The acceleration is always directed radially inward
(in the negative rˆ direction)
Uniform Circular Motion
• Uniform Circular Motion (UCM) occurs when
an object follows a circular path at a constant
speed
– The object experiences an acceleration toward the
center of the path called centripetal acceleration
– The object must be kept in circular motion by at
least one force, the total of which we call the
centripetal force
– Comes from the Latin “centrum” = center and
“petus” = seeking
Periodic Motion
• Any repeated cycle, such as UCM, can be
described in terms of its period and its frequency
– Period (T) is the time that it takes for one cycle to be
completed, usually measured in seconds
– Frequency (f) is the number of cycles completed per
unit time, usually measured in Hz
• 1 Hz = 1/s or s-1
1
T
f
1
f 
T
Converting units for f
• Our standard units for frequency is Hz, but another
common unit is revolutions/minute or rpm
• Example: Convert 45 rpm to Hz
Centripetal Acceleration
• Always directed radially inward
2
v
acp 
r
• v is the object’s speed measured in m/s and r
is the radius measured in m
• Because the rˆ is constantly changing, the
direction of ac is always changing

Example 11-1
To test the effects of high accelerations on the human body,
NASA has constructed a large centrifuge at the Manned
Spacecraft Center in Houston. In this device, astronauts are
place in a capsule that moves in a circular path with a
radius of 15 m. If the astronauts in this centrifuge
experience 9.0 g of acceleration, what is the linear speed of
the capsule?
Example 11-2
The moon’s nearly circular orbit about the Earth has a radius
of about 384,000 km and a period of 27.3 days. Determine
the acceleration of the Moon toward the Earth
Since v is equal to the distance traveled (the
circumference of the circle) divided by the
time that it takes to travel around the circle
(the period), we have
C
v
T

and
C  2r
2r
v
 2rf
T
so
Two different ways to express ac
• ac as a function of T
4 r
acp  2
T
2
• ac as a function of f
acp  4 rf
2
2
Centripetal Force
• The condition necessary for an object to move in
UCM is that the net force in the radial direction
must be equivalent to the centripetal force where
r
Fr macp
• If this condition is NOT met, then the motion of the
object will not be a circle (it’ll become straight line
motion, a parabola, etc.
The “Centrifugal” Force
• An extremely common misconception is that there
is a force, the so called centrifugal (“center-fleeing”)
force that acts radially outward on an object moving
in a circular path – this is incorrect, there is NO
outward force!!!!
• This perceived force is due to the object’s tendency
to follow a straight-line path (inertia)
– Example: ball on a string… why does it pull your hand
outward?
Example 11-2
A puck attached to a string undergoes circular motion
on an air table. If the string breaks at the point
indicated in the picture, which path will the puck
take? Why does it take this path?
Example 11-3 (conceptual)
A car is driven with constant speed around a circular
track.
a) Is the car’s velocity constant?
b) Is its speed constant?
c) Is the magnitude of its acceleration constant?
d) Is the direction of its acceleration constant?
Horizontal UCM Problems
1. Draw a free body diagram (LABEL YOUR
DIRECTIONS! – especially the +r direction)
•
Common radial forces: Tension, friction, normal force
2. Write an equation that satisfies the condition
for UCM (ΣFr = macp)
1. Many times you’ll have to apply some other
condition (usually that f ≤ fs,max)
Example 11-4
A 1200 kg car rounds a corner of radius r = 45 m. If the coefficient of
static friction between the tires and the road is μs = 0.82, what is
the greatest speed the car can have in the corner without
skidding?
Example 11-5
When you take your 1300 kg car out for a spin, you go around a
corner of radius 59 m with a speed of 16 m/s. You have a 110 g
fuzzy dice hanging from your rearview mirror. You notice that it
hangs at an angle to the vertical.
a) Assuming your car doesn’t skid, what is the force exerted
on
it by static friction?
b) Calculate the angle that your dice make with the vertical.
c) Calculate the tension in the string holding up the dice.
d) You notice that your keys (which are heavier than your dice)
are also hanging off to the side. Compare the angle of you
keys to the angle of your dice.
Example 11-6
You may find it surprising that the rotation of the Earth in fact makes
us feel lighter. To illustrate this fact calculate what a bathroom
scale would report for the weight of a 100 kg person (a) at the
North Pole (where there is no rotation) and (b) at the equator.
Earth’s equatorial radius is 6,378.1 km. (c) Does the weight your
scale measures increase or decrease as your latitude increases?
WHY?
Example 11-7
You place a coin 15 cm from the center of a record on a turntable.
You can adjust the “speed” of the turntable, measured in rpm.
a) What is the largest value for the “speed” so that the coin does
not slip off the record?
b) Calculate the linear speed of coin when the record spins at
this rate.
Example 11-8
A puck of mass m = 1.5 kg slides in a circle of radius
r = 20 cm of a frictionless table while attached to
a hanging cylinder of mass M = 2.5 kg by a cord
through a hole in the table. What speed does the
puck need to slide to keep the cylinder at rest?
Example 11-9
The Rotor, an amusement park ride, is
essentially a large hollow cylinder that is
rotated rapidly through its central axis.
The rider enters through a door, leans up
against a canvas-covered wall and the
cylinder begins to spin. When the
cylinder’s frequency reaches a
predetermined value, the floor will fall
away. If you are the designer of this ride,
what minimum frequency could you
safely have the the floor fall away so that
the riders remain pinned the the wall and
not fall? The coefficient of static friction
between a rider’s clothes and canvas is
0.40 and the cylinder’s radius is 2.1 m.
Vertical UCM Problems
1. Draw a free body diagram (LABEL YOUR
DIRECTIONS! – especially the +r direction)
•
•
Common radial forces: Tension, normal force, gravity
VERY IMPORTANT!! The +r direction will change based on where
the object is in its motion
2. Write an equation that satisfies the condition for
UCM (ΣFr = macp)
1. Many times you’ll have to solve for some min or
max value (remember that T and FN cannot be
negative!)
Example 11-10
As part of a circus act, a person
drives a motorcycle with a
constant speed v around the
inside of a vertical track of radius r
= 5 m. The combined mass of the
rider and motorcycle is m = 350
kg.
a) Calculate the minimum value for
v so that the motorcycle’s wheels
maintain contact with the track.
b) Calculate the normal force
exerted on the motorcycle at
points A, B, and C.
Example 11-11
As you ride on a Ferris wheel, you notice that your apparent weight is
different at the top than at the bottom. Consider a rider of mass 55 kg
riding a Ferris wheel of radius 7.2 m that completes one revolution
every 28 s.
a) What is the rider’s apparent weight at the top of the Ferris wheel?
b) What is the rider’s apparent weight at the bottom of the Ferris
wheel?
c) What is the minimum period such that a rider wouldn’t lose contact
with his/her seat?