Transcript Chapter 10 - galileo.harvard.edu

Chapter 10
Rotational Motion
(rigid object about
a fixed axis)
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 about a specific vertical
axis by 900 and then about a specific
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 axis.
• A book is rotated about a specific vertical
axis by 900 and then about a specific
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 axis.
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 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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Radians
Average and instantaneous quantities
Translational-angular connections
Example
Example
Vector nature (almost) of angular
quantities
– Tutorial on rotational motion
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
– Example (You create one)
Torque and Rotational Inertia
• Moment of inertia
– Derivation involving torque and Newton’s 2nd
Law
– Intuition from experience
– 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
Determining moment of inertia
How?
(Count the ways…)
Determining moment of inertia
•
•
•
•
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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
– Proposed experiment
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 ski jump. The
ball leaves the ramp straight up. Draw a
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