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.

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• 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.

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• 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

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• 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.

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(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.

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(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.

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(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.

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• 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

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• 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.

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(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.

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