Chapter 10

Rotational Motion(of a 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– Basic knowledge makes it 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– Basic knowledge makes it 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– Basic knowledge makes it 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.

• 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.

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

• 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)

• first three pages due next class, all due one after

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 DynamicsProblem Solving

• Lessons from translational dynamics?– Using 2nd Law framework diagram (see)

• Use of extended free body diagrams– For what purpose do simple free body

diagrams still work very well?

• Dealing with both translation and rotation – Tutorial on Dynamics of Rigid Bodies– Pure rotational motion problem solving (see)– Mixed trans./rot. motion problem solving (see)

Questions

• How could the moment of inertia of a particular object be determined?

• What considerations are important to keep in mind?

• Moment of inertia practice assignment.

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

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

Rotational kinetic energy & the Energy Representation

• Lessons from energy and problem solving– conservation of energy framework diagram

• Including rotational kinetic energy (see)

• All the trans./rot. frameworks so far (see)

• Questions – Group: List, prioritize, raise for class

discussion

“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 (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 wB.

• 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

Using Newton’s LawsThe Physical situation

Choose/identify objects and forces

Create simple FBDs

Choose inertial coordinate systems

Implement Newton’s Laws

Mathematical representation

SolutionProblem

Using Newton’s LawsThe Physical situation

Choose/identify objects and forces

Create simple FBDs

Choose inertial coordinate systems

Implement Newton’s Laws

Mathematical representation

SolutionProblem

return

1

2

3 4

5

76

89

10

Using Newton’s Laws for RotationThe Physical situation

Choose/identify objects and forces

Create extended FBDs

Choose ref. pt. and rotational axis

Implement Newton’s Rotational Laws

Mathematical representation

SolutionProblem

back

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Using Conservation of EnergyThe Physical situation

Choose/identify objects and forces

Sketch Wnon-con,vi,vf,ri,rj for relevant objects

Mechanical Energy Ledger

Implement Cons. of Mech. Energy

Mathematical representation

SolutionProblem

back

Using Conservation of EnergyThe Physical situation

Choose/identify objects and forces

Sketch of vi,vf,ri,rj,ωi,ωf for relevant objects

Mechanical Energy Ledger

Implement Cons. of Mech. Energy

Mathematical representation

SolutionProblem

back

back