Transcript Chapter 10 - galileo.harvard.edu

Chapter 10
Rotational Motion
(rigid object about
a fixed axis)
What is meant by a “rigid object”? and a
“rigid object about a fixed axis”?
Overview: Our approach
• Introduction to thinking about rotation
• Translational – Rotational motion analogy
• Angular/Rotational quantities
– constant angular acceleration motion
•
•
•
•
Torque and Rotational inertia
Rotational dynamics problem solving
Determining moments of inertial
Rotational Kinetic Energy
– Energy Conservation
• “Rolling friction” - comment
Introduction
• The goal
– Describe rotational motion
– Explain rotational motion
• Help along the way
– Analogy between translation and rotation
– Separation of translation and rotation
• The Bonus
– Easier than it looks
– Good review of translational motion
– Encounter “modern” topics
Introduction
• The goal Just like translational motion
– Describe rotational motion kinematics
– Explain rotational motion dynamics
• Help along the way
– Analogy between translation and rotation
– Separation of translation and rotation
• The Bonus
– Easier than it looks
– Good review of translational motion
– Encounter “modern” topics
Introduction
• The goal
– Describe rotational motion
– Explain rotational motion
• Help along the way A fairy tale
– Analogy between translation and rotation
– Separation of translation and rotation
• The Bonus
– Easier than it looks
– Good review of translational motion
– Encounter “modern” topics
Introduction
• The goal
– Describe rotational motion
– Explain rotational motion
• Help along the way
– Analogy between translation and rotation
– Separation of translation and rotation
• The Bonus A puzzle
– Easier than it looks
– Good review of translational motion
– Encounter “modern” topics
• A book is rotated through a point about a
vertical axis by 900 and then through the
same point in the book about a horizontal
axis by 1800. If we start over and perform
the same rotations in the reverse order,
the orientation of the object:
1. will be the same as before.
2. will be different than before.
3. depends on the choice of point.
• A book is rotated through a point about a
vertical axis by 900 and then through the
same point in the book about a horizontal
axis by 1800. If we start over and perform
the same rotations in the reverse order,
the orientation of the object:
1. will be the same as before.
2. will be different than before.
3. depends on the choice of the point.
Some implications: Math, Quantum
Mechanics … interesting!!!
Translational - Rotational Motion
Analogy
• What do we mean here by “analogy”?
– Diagram of the analogy (on board)
– Pair learning exercise on translational
quantities and laws
– Summation discussion on translational
quantities and laws
• Introduction of angular/rotational quantities
• Formulation of the specific analogy
– Validation of analogy
Translational - Rotational Motion
Analogy (precisely)
If qti corresponds to qri for each translational
and rotation quantity,
then L(qt1,qt2,…) is a translational dynamics
formula or law, if and only if L(qr1,qr2,…) is
a rotational dynamics formula or law.
(To the extent this is not true, the analogy is
said to be limited. Most analogies are
limited.)
Angular quantities
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•
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Angle units: radians
Average and instantaneous quantities
Translational-angular connections
Example
Example
Vector nature of angular quantities
– Care needed (book rotation, other examples)
– Tutorial on rotational motion (handout)
Constant angular acceleration
• What is expected in analogy with the
translational case?
• And what is the mathematical and
graphical representation for the case of
constant angular acceleration?
• Example (Physlet E10.2)
Torque
• Pushing over a block?
• Dynamic analogy with translational motion
– When angular velocity is constant, what?...
– What keeps a wheel turning?
• Definition of torque magnitude
– 5-step procedure: 1.axis, 2.force and location,
3.line of force, 4.perpendicalar distance to
axis, 5. torque = r┴ F
– Question
– Ranking tasks 101,93
– Question
Torque and Rotational Inertia
• Moment of inertia
– Derivation involving torque and Newton’s 2nd
Law
– Intuition from experience (demo: PVC rods)
– Definition
• Ranking tasks 99,100,98
• …More later…
Rotational Dynamics
Problem Solving
• What are the lessons from translational
dynamics?
• Use of extended free body diagrams
– For what purpose do simple free body
diagrams still work very well?
• Dealing with both translation and rotation
• Examples
– inc. Tutorial on Dynamics of Rigid Bodies
Questions
• How could the moment of inertia of a
particular object be determined?
• What considerations are important to keep
in mind?
Determining moment of inertia
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•
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By experiment
From mass density
Use of parallel-axis theorem
Use of perpendicular-axis theorem
Question
– Ranking tasks 90,91,92
Rotational kinetic energy & the
Energy Representation
• Rotational work, kinetic energy, power
• Conservation of Energy
– Rotational kinetic energy as part of energy
– question
• Rolling motion
– question
• Rolling races
– question
• Jeopardy problems 1 2 3 4
• Examples
“Rolling friction”
• Optional topic
• Worth a look, comments only
The end
• Pay attention to the
Summary of
Rotational Motion.
•
A disk is rotating at a constant rate about
a vertical axis through its center. Point Q
is twice as far from the center of the disk
as point P is. Draw a picture. The
angular velocity of Q at a given time is:
1.
2.
3.
4.
twice as big as P’s.
the same as P’s.
half as big as P’s.
None of the above.
back
•
When a disk rotates counterclockwise at
a constant rate about the vertical axis
through its center (Draw a picture.), the
tangential acceleration of a point on the
rim is:
1.
2.
3.
4.
positive.
zero.
negative.
not enough information to say.
back
•
1.
2.
3.
4.
A wheel rolls without slipping along a
horizontal surface. The center of the
wheel has a translational speed v. Draw
a picture. The lowermost point on the
wheel has a net forward velocity:
2v
v
zero
not enough information to say
back
•
1.
2.
3.
4.
The moment of inertia of a rigid body
about a fixed axis through its center of
mass is I. Draw a picture. The moment
of inertia of this same body about a
parallel axis through some other point is
always:
smaller than I.
the same as I.
larger than I.
could be either way depending on the
choice of axis or the shape of the object.
back
•
A ball rolls (without slipping) down a long
ramp which heads vertically up in a short
distance like an extreme (and
dysfunctional) ski jump. The ball leaves
the ramp straight up. Refer to picture.
Assume no air drag and no mechanical
energy is lost, the ball will:
1. reach the original height.
2. exceed the original height.
3. not make the original height.
back
(5kg)(9.8m/s2)(10m) = (1/2)(5kg)(v)2 +
(1/2)(2/5)(5kg)(.1m)2(v/(.1m))2
Draw a picture and label relevant quantities.
back
(5kg)(9.8m/s2)(10m) = (1/2)(5kg)(v)2 +
(1/2)(2/5)(5kg)(.1m)2(v/(.1m))2
(5kg)(9.8m/s2)(h) = (1/2)(5kg)(v)2
Draw a picture and label relevant quantities.
back
(1/2)(5kg)(.1m/s)2 +
(1/2)(1/2)(5kg)(.2m)2(.1m/s/(.1m))2
= (1/2)(5kg)(v)2 +
(1/2)(1/2)(5kg)(.2m)2(v/(.2m))2
Draw a picture and label relevant quantities.
back
•
•
Suppose you pull up on the end of a
board initially flat and hinged to a
horizontal surface.
How does the amount of force needed
change as the board rotates up making
an angle Θ with the horizontal?
a. Decreases with Θ
b. Increases with Θ
c. Remains constant
back
• Several solid spheres of different radii,
densities and masses roll down an incline
starting at rest at the same height.
• In general, how do their motions compare
as they go down the incline, assuming no
air resistance or “rolling friction”?
Make mathematical arguments on the white
boards.
back
(1kg)(9.8m/s2)(1m)
= (1/2)(1/2)(.25kg)(.05m)2(v/.05m)2
+ (1/2)(1kg)v2
Draw a picture and label relevant quantities.
back
• Consider a board set up between on two
scales that measure the force on them.
And suppose the distance between the
scales is L and the weight of the board is
w B.
• What weight does each scale read?
• If an object of weight w is put on the board
a distance d from scale on the right, what
will the right and left scales read?
back