Constants I Periodic Table O w(t) Learning Goal: To understand that contact between rolling objects and what they roll against imposes constraints on the change in position (velocity) and angle (angular velocity). O -w(t) The way in which a body makes contact with the world often imposes a constraint relationship between its O -ru(t) possible rotation and translational motion. A ball rolling on a road, a vo-vo unwinding as it talls, and a baseball leaving the pitcher's hand are all examples of constrained rotation and translation. In a similar manner, the rotation of one body and the translation of another may be constrained, as happens when a fireman unrols a hose from its storage drum. v(t) = O rult) Situations like these can be modeled by constraint equations, relating the coupled angular and linear motions. Although these equations fundamentally involve position (the angle of the wheel at a particular distance down the road), t is usually the relationship of velocities and accelerations that are relevant in solving a problem involving such constraints. The velocities are needed in the conservation equations for momentum and angular momentum, and the accelerations are needed for the dynamical equations. o 0) Submit It is important to use the standard sign conventions postive for counterclockwise rotation and positive for motion toward the right Otherwise, your dynamical equations will have to be modified. Unfortunately, a frequent result will be the appearance of negative signs in the constraint equations. (Eigure 1) Part C Complete previous part(s) Consider a measuring tape unwinding from a drum of radius r. The center of the drum is not moving: the tape unwinds as its free end is puled away from the drum. Neglect the thickness of the tape, so that the radius of the drum can be assumed not to change as the tape unwinds. In this case, the standard conventions for the angular velocity w and for the (translational) velocity v of the end of the tape result in a constraint equation with a positive sign (e.g. ifu>0, that is, the tape is unwinding, then w>0 also). • Part D Perhaps the trickiest aspect of working with constraint equations for rotational motion is determining the correct sign for the kinematic quantties. Consider a tire of radius r rolling to the right, without slipping, with constant x velocity t. (Ejgure 2)Find w, the (constant) angular velocity of the tire. Be careful of the signs in your answer, recal that positive angular velocity corresponds to rotation in the counterclockwise direction. o O -re, o - O ru, Submit RenuatAnswer • Part E Assume now that the angular velocity of the tire, which continues to roll without slipping, is not constant, but rather that the tire accelerates with constant angular acceleration a. Find a, the linear acceleration of the tire. Figure 1 of 2 > a, = 0 O -ar O ar Submit Regutst Answer Provide Feedback Next> P Pearson