13 Dynamics (12/15) -- University Physics

13 Dynamics Dynamics Concepts and Principles To study the dynamics of an arbitrary rigid body we will break the motion down into a pure translation of the CM and a pure rotation about the CM. We will use particle dynamics, i.e., Newton’s second law applied to the CM of the object, to study the translational portion of the motion. The study of the rotational portion of the motion requires a pair of new concepts. We will “invent” these concepts through the use of an analogy with linear dynamics. In linear dynamics, Newton’s second law states that the linear acceleration of an object is proportional to the total force acting on the object and inversely proportional to the mass, or inertia, of the object. It would seem plausible that the angular acceleration of an object would depend on analogous concepts in the same manner. We will replace the concept of force, often thought of as the push or pull applied to an object, with a quantity measuring the twist applied to an object. We will call this new quantity torque, symbolized t. We will replace the concept of mass, the measure of the resistance of the object to changes in its linear velocity, with a quantity measuring the resistance of the object to changes in its angular velocity. We will call this new quantity rotational inertia, symbolized I. In summary, Before we go any further, however, let’s define these new concepts more clearly. Torque In simple English, torque measures the twist applied to an object. The question remains, however, how do we quantify twist? | Let’s examine a common device used to generate twist, a wrench. The magnitude, location, and orientation of the force applied to the wrench by the person’s hand are indicated. Each of these three parameters effects the amount of twist the person delivers to the wrench (and therefore to the bolt). | If you’ve turned many bolts in your life, two things about this person’s bolt-turning technique should grab you. First, why is this person applying the force at such a silly angle? She would generate much more twist if she applied the same magnitude force perpendicular to the wrench, rather than at an angle far from 900. Second, why is she not applying the force at the far edge of the wrench? She would generate far more twist if she applied the same magnitude force at the far edge of the wrench. If the preceding paragraph makes sense to you, you understand how to quantify torque. To maximize torque, you should: - Apply the force far from the axis of rotation (the bolt). - Apply the force perpendicular to the position vector between the axis of rotation and the force. - Apply a large magnitude force. Mathematically, this is summarized by: Note that this function has a maximum when r is large, F is large, and f = 900. Torque will also be assigned a direction, either clockwise or counterclockwise, depending upon the direction of the twist applied to the object. Rotational Inertia We have constructed a rotational analogy to Newton’s second law, Ou