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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 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 • • • • • 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