Rotational Dynamics of Rigid Solids

Rotational Dynamics of Rigid Solids - Rotational Dynamics...

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Rotational Dynamics of Rigid Solids C George Kapp 2002 Rev 2008 Table of Contents Page 1. Introduction ......................................................................... …2 2. Coordinate System ................................................................... 4 3. Circular to Angular Transformations ........................................ 4 4. The Cause Variable .................................................................. 6 5. Mathematical Interlude. .......................................................... 8 6. Angular Relationships; The Strategy. ..................................... 10 7. Cause and Effect .................................................................... 10 8. Moment of Inertia, a Macro Quantity ...................................... 12 9. Newton’s Second Law for Rotation. ........................................ 12 10. Angular Impulse and Momentum. ......................................... 14 11. Energy Relationships; The Strategy. ...................................... 15 13. Work; An Alternative View. .................................................... 16 14. Parallel Axis Theorem ............................................................. 16 15. Rotational Equilibrium ........................................................... 17 16. Examples ............................................................................... 19 17. Appendix I, Moment of Inertia Computations ......................... 27 18. Appendix II. The Combination Method for determining the Total Kinetic Energy…………………….……………37 1
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Rotational Dynamics of Rigid Solids 1 Introduction. It is this author’s intent to start with the laws of Newton, applied to a collection of particles, and deduce all laws of rotation. To provide a crystal clear path from one to the next. In consideration of this journey, the student must be aware that the basis for true understanding and clarity is as much a mater of perspective and interpretation, as it is an exercise in mathematics and the application of fundamental laws of nature. front cm axis Y axis Consider the above situation: We have a rigid block. We give it a 180 degree rotation about each of the two axes shown, the center of mass axis, and the Y axis which is parallel to the cm axis. For the cm axis, the result is, back cm axis Y axis back and for the Y axis, the result is, Y axis We compare the result of each operation. Many observers would consider the results as dissimilar – truly visual inspection would agree. The cm 2
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rotation produces no translation of the center of mass of the box whereas the Y rotation produces considerable displacement of the center of mass of the box. And yet, BOTH were 180 degree rotations. A mater of perspective and interpretation. We require a distinction between position, and orientation. While the change in position of the box is different for each event, both events show the same change in orientation of the box. The orientation has changed by 180 degrees. We will designate the objects cm position with the usual position vector R cm , and define the objects orientation as its angular position θ . It can also be seen from the example that a change in angular position (a rotation) about a given axis is equivalent to a change in angular position (a rotation) about any other parallel axis . It is important to create a clear understanding of the above paragraph here and now. We often use words such as “rotate” and “spin”; and in most cases our mind associates an axis through the body of the object. When we say “rotate” or “spin”, we must THINK change in orientation – change in angular position. We must also be cautioned to guard against our mind’s selection of axis. Consider the following question: Is a ball rolling along on a flat table spinning about an axis which passes thru the point of contact?
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