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Chapter 10: Circular Motion1
Rotation vs. Revolution
Two types of circular motion
Definition of terms:
Axis: straight line around which
rotation takes place
Rotation: an object turns about an
internal axis (the axis within the
body of the object)
Revolution: an object turns about
an external axis (axis is located
outside the object).
Earth rotates around its
internal axis through its
geographical poles once every
24 hours
Earth revolves around the sun
once every 365 ¼ days.
Ferris wheel rotates around
its axis
o But the riders revolve
around the Ferris
wheel axis.
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All parts of the turntable
rotate at the same
rotational speed
Rotational Speed
Types of speed:
Linear speed: distance per unit of time (speed in a
straight line)
Tangential speed: speed of an object moving along a
circular path
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The direction of motion is always tangent to
the circle
For circular motion, linear speed and
tangential speed can be used interchangeably.
Rotational speed (also called angular speed):
number of rotations per unit of time
All parts of a rigid merry-go-round
or turn-table rotate about their axis
in the same amount of time.
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All parts have the same rate
of rotation or the same
number of rotations per unit
of time
Rotational speed is
sometimes called
revolutions per minute
(RPM)
Chapter 10: Circular Motion2
Tangential and Rotational Speed
Relating tangential and rotational speed
Tangential speed and rotational speed are
related.
Tangential speed is directly proportional
to radial distance x rotational speed
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The faster the object turns, the faster
the tangential speed is.
Tangential speed is directly
proportional to rotational speed and the
distance from the center (axis) of
rotation
o The farther you are from the axis
of rotation, the faster you move.
o Mrs. Hardy’s favorite horse on a
merry-go-round is the outside;
unfortunately, it doesn’t go up and
down!
o Time for a demonstration!
v = rω
ω (Greek letter omega) represents
angular (rotational) speed.
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Units for ω are radian/s
Radian is not a “true” unit, but is
a placeholder
Units for v are m/s.
Where do these units come from?
Interpreting the equation:
v = rω
Railroad Train Wheels

How do train wheels stay on the track?
The flanges at the edge of the wheel are only
used in emergency situations or when they
follow slots to switch tracks.
BLN assignment: Discover lab, pg. 173 and
Figure 10.5. Fasten a pair of cups together
both wide-side together and narrow-side
together. Which orientation rolls more
efficiently? Which stays on the track better?
Which rolls faster? Use 2 meter sticks are
your railroad tracks.
See the adjacent figure – the wheels of trains
are tapered; this tapered shape is essential on
the curves in the track.

On a curve, outer wheels travel faster than
inner wheels
o Train wheels are connected, so each
has the same RPM at any time.
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You move faster if the rotation
rate is faster.
You move faster if you are farther
from the axis (bigger r)
If you’re at the center (axis of
rotation), you have no tangential
speed because r = 0.
o You will have rotational
speed because you rotate in
one place.
o Rotational speed is the
same for all points on a
rigidly rotating system.
Tangential speed depends on rotational
speed and the distance from the axis of
rotation.
Chapter 10: Circular Motion3
Due to the tapered wheel, when rounding a curve, wheels on the outer track
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ride on the wider part of the wheel (traveling a greater distance)
the opposite wheel then rides on the narrower parts (covering a smaller distance in the
same time period).
Wheels have different linear speeds for the same rotational speed, which is v = rω in
action.
Centripetal Force
An object in circular motion, even at constant speed, undergoes
acceleration because of the changing direction.
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Net force causes the change in direction (otherwise the
object would travel in a straight line)
Centripetal acceleration: an acceleration directed toward
the center of the circle.
o “Centripetal” means “center seeking”
Centripetal force: a force that is directed toward a fixed
center that causes an object to follow a circular path
o If the centripetal force ceased to exist, the object
would move off in a straight line (tangent line), as
predicted by Newton’s law of inertia.
Motion is circular, at a constant speed, centripetal force acts at
right angles (perpendicular) to the path of the moving object.
Examples of centripetal force:
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Whirling a yo-yo on the end of its string
Moon orbiting the Earth
Orbiting electrons around the nucleus of an atom
Anything that moves in a circular path is acted on by a
centripetal force.
Centripetal force is not
a basic force of nature.

Name given to any
force (string tension,
gravity, electrical
force) that is
directed toward a
fixed center.
Chapter 10: Circular Motion4
Everyday examples of centripetal force

Car rounding a curve: friction between the tires and the road provides the
centripetal force that keeps the car on the road.
o If the road is slick, the car slides sideways and fails to follow the curve
o Tangential skid off the road

Washing machine: during the spin cycle, the wash tub rotates at high speed and the
tub wall produces a centripetal force on the wet clothes, forcing them into a circular
path.
o The holes in the wash tub prevent the tub from exerting the same force on
the water.
o The water escapes tangentially out of the holes.
http://bmx.transworld.net/1000142227/videos/the-monday-edit-spot-bowl-session/
http://www.youtube.com/watch?v=TM_jORDiEz8
Calculating Centripetal Forces
Centripetal force on an object depends on
the object’s tangential speed, mass, and
radius of its circular path.
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Greater speed and greater mass
require greater centripetal force
Traveling in a circular path with a
smaller radius requires greater
centripetal force.
Centripetal force = mass x speed2
radius of curvature
2
Fc = mv
r
Centripetal force is measured in
newtons (N) when mass is in kg,
velocity is in m/s, and radius is in m.
Spin Out is a carnival ride consisting of a
large open cylinder. Riders stand inside
with their backs against the cylinder wall.
What is the centripetal force acting on
you in the ride if the radius of the
cylinder is 3.5 m, your mass is 50 kg, and
you are traveling at 5 m/s?
Chapter 10: Circular Motion5
Freddie swings a 2-kg stone at the end of a
thin rope of length 1.2 m. He tugs mightily,
swinging the stone so fast that the rope is
almost horizontal. If the string tension is
200 N, show that the stone moves at 11
m/s.
Definitions and another way to solve for
tangential speed
Period: the time it takes for one full
rotation or revolution of an object (T).
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Unit: s
Frequency: The number of rotations or
revolutions per unit of time (generally 1
second) (f).
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Unit: 1/s (s-1), also known as Hertz
(Hz)
Period and Frequency are reciprocals of
Centripetal acceleration
An object moving around a circle with
constant speed is still accelerating due to
the constantly changing direction.
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Acceleration is always directed
toward the center of the circle
Centripetal acceleration = linear
speed2/radius
ac =vt2
r
Unit: m/s2
Missy’s favorite ride at the Topsfield Fair is
the rotor, which has a radius of 4.0 m. The
ride takes 2.0 s to make one full revolution.
A) What is Missy’s linear speed on the Rotor?
B) What is Missy’s centripetal acceleration
on the Rotor?
each other: T =
f=
Speed of an object traveling in a circle is
the distance it travels in 1 revolution
(circumference of the circle), divided by
the time (time for 1 revolution = the
period, (T).
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v = 2πr
T
Unit: m/s
Captain Chip, the pilot of a 60,500-kg jet
plane, is told that he must remain in a
holding pattern until it is his turn to land. If
Captain Chip flies his plane in a circle whose
radius is 50.0 km once every 30.0 min, what
centripetal force must the air exert against
the wings to keep the plane moving in a
circle?
Chapter 10: Circular Motion6
Adding Force Vectors
Figure 10.11 – conical pendulum  string attached to a
bob sweeps out a cone
Two forces act on the bob:
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Gravity = weight (mg)
Tension (string)
Both forces are vectors
o Ty (vertical) is exactly equal and
opposite to gravity (the bob is not
accelerating vertically, so the net force
= 0)
o Tx (horizontal) causes the net force on
the bob  the centripetal force.
o Centripetal force lies along the radius of
the circle that is swept out.
Vehicle rounding a banked curve
At the appropriate speed, the car doesn’t slide up or down the curve.
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Friction plays no role in keeping the car on the track
o Angle of the curve is chosen for zero friction at the chosen speed
Two forces act on the car
o Gravity
o Normal force – resolved into x and y components.
o Ny is equal and opposite gravity – no net force vertically
o Nx is the centripetal force that keeps the car in its circular path
Centripetal force is always the net force that acts exactly along the radial direction,
toward the center of the circular path.
Chapter 10: Circular Motion7
The 3 balls demo
Mass of object, radius of circle, tangential speed
http://www.mindbit
es.com/lesson/4618physics-in-actionthe-three-balls-demo
Centripetal and Centrifugal Forces
Centripetal force is a center-seeking force.
Centrifugal force is a “center-fleeing” or “away from the center” force.
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Apparent outward force on a rotating or revolving body
Misconception is that a centrifugal force pulls outward on a body in circular
motion, like a yo-yo being whirled on a string. If the string breaks, it is often
wrongly stated that the centrifugal force pulls the yo-yo from its circular path.
Reality: When the string breaks, the yo-yo goes off in a tangential straight-line
path due to the absence of force acting on it.
Similar situations:
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In a car and not wearing a seat belt. When the car stops abruptly, you hit the dashboard.
o Absence of the force of the seat belt … you followed Newton’s 1st law of
motion
In a car and not wearing a seat belt. Car makes a sharp left-hand corner, you slide
outward (to the right).
o Not due to “centrifugal force,” but because of the absence of a centripetal
force. Again, Newton’s 1st law of motion (you continued on in the direction
you were heading before the car turned left).
No force is
pulling the can
outward.
Only the force
from the string is
pulling the can
inward.
Chapter 10: Circular Motion8
Ladybug inside the whirling can …
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The can presses against the bug’s feet and provides the
centripetal force that holds the ladybug in a circular
path.
Neglecting gravity, the only force on the bug is the force
of the can on its feet.
The “centrifugal-force effect” is not attributed to any real force, but is a consequence
of inertia – the tendency of a moving body to follow a straight-line path.
Inertia
Centrifugal Force in a Rotating Reference Frame
Our view of nature depends on the frame of reference from
which we view it.
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Inside a fast-moving car, you have no speed relative to
the car
o You’re moving at the same speed of the car
relative to the stationary ground outside
Ladybug in the can … From a stationary frame of
reference outside the can, you don’t see a centrifugal
force.
o You do see a centripetal force acting on the can
and the ladybug, producing a circular motion.
o From inside the rotating system, the ladybug
sees both the centripetal force (by the can) and
the centrifugal force acting on the ladybug.
o As long as the ladybug is rotating, she feels the
centrifugal force.
o Once the rotation stops, there is no longer any
centrifugal force felt.
Fundamental difference
between “gravity-like”
centrifugal force and
actual gravitational force
…
Gravitational force is
always due to an
interaction between one
mass and another.
In a rotating reference
frame, the centrifugal
force has no mass.
There is no interaction
counterpart.
Centrifugal force is an effect of rotation. It is not part of an interaction and, as a result,
cannot be a true force. Physicists call it a fictitious force, unlike gravitational,
electromagnetic, and nuclear forces.
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However, to observers in a rotating system, centrifugal force is very real.