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
• 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 DynamicsProblem 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
• 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. twice as big as P’s.
2. the same as P’s.
3. half as big as P’s.
4. 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. positive.
2. zero.
3. negative.
4. not enough information to say.
back
• 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:
1. 2v2. v3. zero4. not enough information to say
back
• 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:
1. smaller than I.2. the same as I.3. larger than I.4. 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