ROTATIONAL MOTION I. Motion with constant angular velocity A wheel is spinning counterclockwise at a constant rate about a fixed axis. The diagram at right represents a snapshot of the wheel at one instant in time. A. Draw arrows on the diagram to represent the direction of the velocity for each of the points A, B. and C at the instant A shown. Explain your reasoning. Is the time taken by points 3 and C to move through one TOP view complete circle greater than. less than. or the some as the Wheel spins commrdka'.\" time taken by point A? On the basis of your answer above. determine how the speeds of points A, B. and C compare. Explain. B. Mark the position of each of the labeled points at a later time when the wheel has completed one half of a turn. Sketch a velocity vector at each point. For each labeled point. how does the velocity compare to the velocity at the earlier time in part A? Discuss both magnitude and direction. Top view Wheel spins counterclockwise Is there one single linear velocity vector that applies to every point on the wheel at all times? Explain. Rotational motion H C. Suppose the wheel makes one complete revolution in 2 seconds. 1. For each of the following points, nd the change in angle (A0) of the position vector during one second. (i.e.. Find the angle between the initial and nal position vectors.) - point/l - pointB - pointC 2. Find the rate of change in the angle for any point on the wheel. The rate you calculated above is called the angular speed of the wheel,or equivalently. the magnitude of the angular velocity of the wheel. The angular velocity is defined to be a vector that points along the axis of rotation and is conventionally denoted by the symbol EB (the Greek letter omega). To determine the direction of the angular velocity vector. we imagine an observer on the axis of rotation who is looking toward the object. If the observer sees the object rotating counterclockwise, the angular velocity vector is directed toward the observer; if the observer sees it rotating clockwise, the angular velocity vector is directed away from the observer. D. Would two observers on either side of a rotating object agree on the direction of the angular velocity vector? Explain. Would two observers who use different points on an object to determine the angular velocity agree on the magnitude of the angular velocity vector? Explain. E. The diagrams at right show top and side views of the spinning wheel in pan A. Axis of rotation i | I l I l I I On each diagram. draw a vector to represent the angular velocity of the wheel. (Use the convention that 6) Top view '\"Q'Cies a vecmr Wheel spins counterclockwise pomltng out of the page and indicates a vector pointing into the page.) I | I I I I Side view A\\u&u&|uuul I'll-'6'". F. In the space at right sketch the position vectors for . pOinl C at lhe beginning and at the end or a small \"me Sketch ofposilion vectors al interval Ar. r, and :0 + Ar I. Label the change in angle (A6) and the distance between the center of the wheel and point C (rt). Sketch the path taken by point C during this time interval. What is the distance that point C travels during At? Express your answer in terms of rt and A0. 2. Use your answer above and the definition of linear speed to derive an algebraic expression for the linear speed of point C in terms of the angular speed at of the wheel. What does your equation imply about the relative linear speeds for points farther and farther out on the wheel? Is this consistent with your answer to part A? ll. Motion with changing angular velocity A. Let 53., represent the initial angular velocity of a wheel. In each case described below. determine the magnitude of the change in angular velocity low] in terms of Iw,|. I. The wheel is made to spin faster. so that eventually. a fixed point on the wheel is going around twice as many times each second. (The axis of rotation is fixed.) 2. The wheel is made to spin at the same rate but in the opposite direction. B. Suppose the wheel slows down uniformly. so that Itiil decreases by 8:: radls every 4 s. (The wheel continues spinning in the same direction and has the same orientation.) Specify the angular acceleration E of the wheel by giving its magnitude and, relative to (13. its direction. In linear kinematics we nal_ the acceleration vector by rstconstructing a change in velocity vector AU and then dividing that by At. Describe the analogous steps that you used above to nd the angular acceleration a