Rotational Motion (rigid object about a fixed axis) Chapter 10 Rotational Motion (rigid object about a fixed axis)

What is meant by a “rigid object” 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 Help along the way The Bonus 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 Help along the way A fairy tale The Bonus 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 Help along the way The Bonus A puzzle 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 Angle units: radians Average and instantaneous quantities Translational-angular connections 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

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 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 Rolling races 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: 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: positive. zero. negative. not enough information to say. back

A wheel rolls without slipping along a horizontal surface 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

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: reach the original height. exceed the original height. 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