RDY - Introduction to Rotational Dynamics

Rotational Dynamics Overview

Introduction

Have you had the following experiences: Trying to balance on a stationary bicycle, trying to balance a stationary coin on its side, or trying to stand a toy top on its point without spinning it? Clearly rotation enters the picture to turn these situations from virtually impossible to extremely easy. There is nothing in physics that we have discussed so far that would account for this difference. Now that we have learned the difference between Uniform Circular Motion and Rotational Kinematics, and have discovered how to deal with the kinetic energy of rotating distributed masses, we are prepared to develop our last conservation principle in Newtonian mechanics, that of conservation of angular momentum. Does a wheel always roll down a hill? Not always. Conservation of angular momentum brings these seemingly impossible events into the realm of reality. In fact, the concepts you will use in this unit find application in the quantum mechanical world, a deeper topic in physics that you may study beyond this course. So it is now time to recast Newton's second law and linear momentum into the topic of rotational dynamics.

Essential Questions

1. What is torque?
2. What rotational quantities are conserved and under what conditions?
3. What is the rotational analog of force?
4. What is the rotational analog of mass?
5. What is Newton's second law for rotation?
6. What units are used to discuss angular dynamic quantities?
7. How is an object's center of mass calculated?
8. Why do you tip over (or not tip over) when on a bicycle?
9. How is angular momentum calculated?
10. Are there rotational analogs to work and power?

Key Terms

1. angular momentum - a vector quantity that represents the product of an objects moment of inertia and its angular velocity. Also the cross product of displacement vector (from an axis of rotation) to the linear moment of an object.
2. cross product - also known as the vector product, is a product of two vectors, A{bold} and B{bold}, denoted as A{bold} x B{bold}, which results in a vector normal to the plane defined by A{bold} and B{bold}, with a magnitude |A{bold}|•|B{bold}| sin{theta}, where {theta} is the angle between A{bold} and B{bold}.
3. precession - a change in the orientation of the rotational axis of a rotating object.
4. right hand rule - a mnemonic used in physics and mathematics that denotes the commonly accepted three dimensional Cartesian coordinate system in which the thumb represents x, the index finger represents y and the middle finger represents z axes. Likewise it also represents the commonly agreed upon order of operations when performing a cross product of two vectors, in which the resultant vector C{bold} is A{bold} x B{bold}, with the thumb pointing in the direction of A{bold}, the index finger point in the direction of B{bold} and the middle finger pointing in the direction of C{bold}.
5. torque - represented by , is the tendency of a force to tangentially accelerate an object about an axis, formally defined as {tau} = r{bold} x F{bold} where r{bold} is the displacement vector from the axis of rotation to the point at which the force, F{bold}, is applied to the object.

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