Chapter 12: Rotational Motion1
Ch. 12.3 Rotational Inertia and Rolling
Great Shapes Race
Which shape will win the race the hollow hoop or the solid cylinder? They each have the same
mass and radius. Why?
Solid cylinder
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Smaller moment of inertia (½mr2 instead of mr2 for the hollow hoop).
o Hoop’s moment of inertia means that it takes more time for it to get rolling.
o Inertia is a measure of “laziness.”
Any solid cylinder, regardless of mass or radius, will roll down an incline with greater
acceleration than any hollow cylinder.
o Hollow cylinder has more “laziness per mass” than a solid cylinder.
Objects of the same shape but different sizes accelerate equally when rolled down an
incline.
o Smaller objects rotate more times down the incline than larger objects, but will
reach the bottom in the same time.
o All objects of the same shape have the same “laziness per mass” ratio.
Objects of the same shape but different sizes accelerate equally
when rolled down an incline.
Chapter 12: Rotational Motion2
Ch. 12.4 Angular Momentum
Anything that rotates keeps on rotating until something stops it.
All moving objects have “inertia of motion,” or momentum.
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Linear Momentum (p) = mass x velocity
Angular momentum (L)
o Angular momentum = moment of inertia x rotational velocity (also known as
angular speed)
o L=Iω
o Vector quantity whose magnitude is the rotational speed
o By convention, the rotational velocity vector and the angular momentum
vector have the same direction and lie along the axis of rotation.
Example: gyroscope – low-friction swivels can be turned in any direction without
exerting a torque on the whirling gyroscope.
If an object is small compared with its radial distance to its axis of
rotation, like a rock on a string or a planet orbiting around the sun
in a circle, its angular momentum is just the linear momentum (mv)
multiplied by the radial distance.
o Angular momentum (L) = mvr.
A net torque is required to change the angular momentum of an object.
Newton’s first law of inertia for rotating systems:
An object or system of objects will maintain its angular momentum unless acted upon
by an unbalanced external torque.
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It’s easier to balance on a moving bicycle than on one at rest.
o Spinning wheels have angular momentum
o When our CG is not above a point of support, a torque is produced.
o When the bicycle is at rest, we fall over.
o If the bicycle is moving, the angular momentum of the wheels requires a
greater torque to change direction.
An object or system of objects will maintain its angular momentum unless acted on by
an unbalanced external torque.
Chapter 12: Rotational Motion3
Ch. 12.5 Conservation of Angular Momentum
Just like linear momentum …
Angular momentum is conserved if no net torque acts on the rotating system
Law of conservation of angular momentum:
If no unbalanced external torque acts on a rotating system, the angular momentum of
that system is constant.
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With no net external torque, the product of rotational inertia and rotational
velocity will be the same whatever the situation.
Angular momentum is conserved when no net external torque acts on an
object.
Example of the figure below (or ice skaters)
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When the man has his arms extended, his moment of inertia is high and his
rotational velocity is low.
When he pulls his arms in, the man’s moment of inertia is low, but now his
rotational velocity is high.
Whenever a rotating body contracts, its rotational speed increases.
Rotational speed can be changed by making variations in rotational inertia.
o Moving some part of the body toward or away from the axis of rotation.
Chapter 12: Rotational Motion4
For a classroom demonstration, a student sits on a piano stool
holding a sizable mass in each hand. Initially, the student
holds his arms outstretched and spins about the axis of the
stool with an angular speed of 3.72 rad/s. The moment of
inertia in this case is 5.33 kg▪m2. While still spinning, the
student pulls his arms in to his chest, reducing the moment of
inertia to 1.60 kg▪m2. (a) What is the student’s angular speed
now? (b) Find the initial and final angular momenta of the
student.
A 0.11 kg mouse rides on the edge of a Lazy Susan that has a
mass of 1.3 kg and a radius of .25 m. If the Lazy Susan begins
with an angular speed of 3.0 rad/s, what is its angular speed
after the mouse walks from the edge to a point 0.15 m from
the center?
Cats can perform zeroangular-momentum by
turning one part of the
body against the other.
When finished the cat is
not turning. This
maneuver rotates the
cat’s body through an
angle but doesn’t lead
to continuing rotation.
Conservation of angular
momentum is not
violated.
Chapter 12: Rotational Motion5
Ch. 12.6 Simulated Gravity
Remember the ladybug sitting on the bottom of the can when we studied centrifugal force?
Now, we have a colony of ladybugs living in a bicycle tire.
If thrown into the air or dropped from an airplane, the ladybugs will be “weightless” and will
float while in free fall.
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If the wheel is spun, the ladybugs will feel themselves pressed to the outer part of
the inner surface.
Simulated gravity
If spun at the right speed, the ladybugs will experience the same gravity that they
feel on Earth.
From within a rotating frame of reference, there seems to be an outwardly
directed centrifugal force, which can simulate gravity.
o “Up” is toward the center of the wheel.
o “Down” is radially outward, away from the center of the wheel.
Why is this important?
If people of Earth are to become spacefaring people, they will need the option of living in
space, to put themselves closer to the far off worlds that they wish to visit.
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Simulated gravity is necessary so that people can function normally while living in
space.
Support Force
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In today’s space vehicles, people feel weightless because they lack a support force.
o Not pressed “down” by gravity
o Not spinning, so no centrifugal force to simulate gravity
Future space travelers will live in habitats that will spin and provide a support force,
simulating gravity.
Chapter 12: Rotational Motion6
From outside the space
habitat, the man is seen at
rest.
The floor pushes up on the
man; the man pushes down
on the floor (action-reaction
pair).
Only force exerted on the
man is by the floor.
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Challenges of Simulated Gravity
We’re used to 1g gravity here at Earth’s surface.
In a rotating spaceship the acceleration is the centripetal/
centrifugal acceleration due to rotation.
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The size of the acceleration (and therefore gravity,
is due to the radial distance and the square of the
rotational speed.
For a given RPM, acceleration increases with
increasing radial distance.
o Doubling the distance doubles the
acceleration
o At the axis where radial distance is zero,
there is no acceleration due to rotation.
To simulate Earth’s gravity, a very large space
habitat is required (almost 2 km in diameter).
o Very large compared with space shuttle.
o Economics will probably dictate a much
smaller structure that will probably not
rotate.
 The people living on board will
experience weightlessness.
 Larger habitats that would rotate will
probably come later.
Centrifugal force
Directed toward the
center
Centripetal force
From inside the habitat,
there is the man-floor
interaction plus the
centrifugal force exerted on
the man at his center of
mass.
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Feels like gravity
Has no reaction
counterpart (nothing
that pulls back on the
“gravity”)
Centrifugal force is
not part of an
interaction, but
results from rotation.
o Fictitious force
Assignment: Read the last
paragraph on page 225.
Write a minimum of 2 “good”
paragraphs about what it
would be like to live on a
rotating space habitat, with
variations on the different
gravity levels. What would
you do in that environment?