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Circular Motion
Unit 5
An axis is the straight line around
which rotation takes place. When
an object turns about an internal
axis- that is, an axis located within
the body of the object- the motion
is called rotation.
When an object turns about an
external axis, the motion is called
revolution.
Earth undergoes both types of
rotational motion. It revolves
around the sun once every
365 ¼ days and it rotates
around an axis passing through
its geographical poles once
every 24 hours.
Linear speed
Linear speed is the distance moved per unit
of time. A point on the outer edge of a
merry-go-round moves a greater distance
in one complete rotation that a point near
the center. The linear speed is greater on
the outer edge of rotating objects than it is
closer to its axis. The speed of something
moving along a circular path can be called
tangential speed because the direction of
motion is always tangent to the circle.
Rotational speed
Rotational speed is the number of rotations
per unit time. All parts of the merry-goround rotate about their axis in the same
amount of time. Thus, all parts have the
same rate of rotation. Rotational speed is
expressed in revolutions per minute
(RPM).
Tangential speed and rotational speed are related.
The faster it turns, the faster your tangential speed
is.
Tangential speed is directly proportional to
rotational speed and the radial distance from the
axis of rotation. At the center of a rotating platform,
right at its axis, you have no tangential speed at
all, but you do have rotational speed. As you move
away from the center, you move faster and fasteryour tangential speed increases while your
rotational speed stays the same.
Centripetal force

Any force that causes an object to follow a
circular path
 Means “center seeking” or “towards the center”
 When an automobile rounds a corner, the
sideways-acting friction between the tires and
the road provides the centripetal force that holds
the car on a curved path. If the force of friction is
not great enough, the car fails to make the curve
and the tires slide sideways. The car skids.
Centrifugal force
 Outward
force
 Means “center fleeing” or “away from the
center”
Suppose you are whirling a can. If the string on the whirling
can breaks, it if often wrongly stated that centrifugal force
pulls the can from its circular path. But in fact, when the
string breaks the can goes off in a tangential straight-line
path because no force acts on it.
Now suppose there is a ladybug inside the whirling can.
The can provides the centripetal force, not the centrifugal
force necessary to hold the ladybug in a circular path. The
“centrifugal-force effect” is attributed not to any real force
but to inertia- the tendency of the moving body to follow a
straight-line path
Our view of nature depends on the frame of reference from
which we view it. For instance, when sitting on a fastmoving train, we have no speed at all relative to the train,
but we have an appreciable speed relative to the reference
frame of the ground outside.
From a stationary frame outside the whirling can, we see
there is no centrifugal force acting on the ladybug. We do
see centripetal force acting on the can. In the rotating
frame of reference of the whirling can, both centripetal
force and centrifugal force act on the ladybug. However,
centrifugal force is an effect of rotation. It is not part of an
interaction so it cannot be a true force.
Gravity is simulated by centrifugal force. If
the spinning can freely falls, the ladybug
inside will experience a centrifugal force that
feels like gravity when the can spins at an
appropriate rate.
Today we live on the outer surface of a spherical planet,
held here by gravity. In the years ahead many people will
likely live in huge, lazily rotating space stations where
centrifugal force simulates gravity. The simulated gravity
will be provided so the people can function normally.
Occupants in today’s space shuttles feel weightless
because they lack a support force. They’re not pressed
against a supporting floor by gravity, nor do they
experience a centrifugal force due to spinning. But future
space travelers need not be subject to weightlessness.
Their space habitats will likely spin effectively supplying a
support force and nicely simulating gravity.
The comfortable 1g we
experience at Earth’s surface is
due to gravity. Inside a rotating
spaceship the acceleration
experienced is the centripetal/
centrifugal acceleration due to
rotation.
Formulas

Period: the time it takes for one full rotation or
revolution of an object


Unit is seconds
Frequency: the number of rotations or
revolutions per unit time

Unit is Hertz
T= 1
f
Formulas
When an object spins in a circle, the
distance it travels in one revolution is the
circumference of a circle. The time it takes
is the period.
speed= 2Πr
T
formulas
An object can move around in a circle with a
constant speed yet still be accelerating
because its direction is constantly
changing. This is centripetal acceleration.
centripetal acc= (linear speed)2
radius
Formulas
If the mass is being accelerated towards the
center of a circle, it must be acted upon by
an unbalanced force that gives it this
acceleration.
centripetal force= mv2
r
After closing a deal with a client,
Kent leans back in his swivel
chair and spins around with a
frequency of 0.5Hz. What is
Kent’s period of spin?
Curtis’ favorite disco record has
a scratch 12cm from the center
that makes the record skip 45
times each minute. What is the
linear speed of the scratch as it
turns?
Missy’s favorite ride at the fair is
the rotor, which has a radius of 4m.
The ride takes 2s to make one full
revolution.
What is Missy’s linear speed on the
rotor?
What is Missy’s centripetal
acceleration on the rotor?
Captain Chip, the pilot of a
60500kg jet plane, is told that he
must remain in a holding pattern
over the airport until it is his turn to
land. If Captain Chip flies his plane
in a circle whose radius is 50km
once every 30 min, what centripetal
force must the air exert against the
wings to keep the plane moving in
a circle?
Formula
 Torque:
a measurement of the tendency of
a force to produce a rotation about an axis
torque= perpendicular force x lever arm
The lever arm is the distance from the pivot point, or
fulcrum, to the point where the component of force
perpendicular to the lever arm is being exerted. The
longer the lever arm, the greater the torque.
Keep in mind that when an object is balanced, all torques
must also balance.
Ned tightens a bolt in his car
engine by exerting 12N of force
on his wrench at a distance of
0.40m from the fulcrum. How
much torque must Ned produce
to turn the bolt?
Mabel and Maude are seesawing
on the school playground and
decide to see if they can move to
the correct location to make the
seesaw balance. Mabel weighs
400N and she sits 2m from the
fulcrum of the seesaw. Where
should 450N Maude sit to balance
the seesaw?
Moment of inertia:
the resistance of an object to changes in its
rotational motion
Hoop rotating about its center: I= mr2
Hoop rotating about its diameter: I= (1/2)mr2
Solid cylinder: I= (1/2)mr2
Stick rotating about its center of gravity: I= (1/12)ml2
Stick rotating about its end: I= (1/3)ml2
Solid sphere rotating about its center of gravity: I= (2/5)mr2
Formula
Angular momentum: the measure of how
difficult it is to stop a rotating object
Angular momentum (L)= (mass)(velocity)(radius)
On the Wheel of Fortune game
show, a contestant spins that
15kg wheel that has a radius of
1.40m. What is the moment of
inertia of this disk-shaped
wheel?
Trish is twirling her 0.60m
majorette’s baton that has a
mass of 0.40kg. What is the
moment of inertia of the baton
as it spins about its center of
gravity?
Jupiter orbits the sun with a
speed of 2079m/s at an average
distance of 71,398,000m. If
Jupiter has a mass of
27
1.90x10 kg, what is its angular
momentum as it orbits?