u I. Angular velocity and linear velocity- ' ' I A wheel spins counterclockwise about a xed'ictioniers axle. The diagram represents a snapshot of the wheel at an instant in time: ' ' A. If the axle is perfectly frictionless. would you expect the rate at A which the wheel spins to increase, decrease, or remain constant as time goes on? - ' Top vi cw ' B. At each point (A, B, and C) draw a straight arrow to represent . _ _ ' Wheel spins rarrmer-rtodarrse, . the direction in which that point is moving at the instant shown. - Is the time taken by points 3 and C to move through one complete circle greater than, less than, or the same as the time taken by pointA? ' On the basis of your answer above. which point is traveling the greatest distance per unit time? "Which point is traveling the shortest distance per unit time? Explain; ' . C. Mark the position of each of the labeled points at a later time when the wheel has completed one half of a turn. - ' How does the motion of each point compare to the motion of that same point at the earlier time {in part B)? In particular. has the direction of motion changed? Has the linear speed cha11ged'?- ' Top view W'heel spins rormrer-rfarlarfse , . . D. Suppose the wheel makes one complete revolution in 2 seconds. Find the rate of change in the angle for point A. Express your answer in radians per second (rad.-"s).- ' "Would your answer also apply to the rate of change in the angle for point 3? Point C? Explain: ' The rate of change in the angle for the wheel is called the angular Wicca}: of the wheel. Angular velocity is conventionally denoted by the symbol a: (the Greek letter omega). - ' "E. Suppose that the wheel makes one complete revolution in 2 seconds (as before) and that the distance between the center of wheel and point A is 42 cm. ~ 1. Calculate: . the distance traveled by point A during one complete revolution of the wheel, and . the magnitude of the (linear) velocity of point At 2. Use your answers above to write a general algebraic expression (i.e., a formula) for the magnitude of the linear velocity v of any point on the wheel in terms of the angular velocity of the wheel and the distance / between that point and the axis of rotation. What does your equation imply about the relative linear velocities for points farther and farther out on the wheel? Is this consistent with your answer to part B (on the preceding page)?~ - II. Changing angular velocity A. Let wi represent the initial angular velocity of a wheel that you determined in part I.D (on the preceding page). In each case below, determine the change in angular velocity Aw. 1. 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 that a different wheel were to slow down uniformly as it spins, so that @ decreases by 8x rad/s every 4 s. (The wheel continues spinning in the same direction and has the same orientation.) Calculate the angular acceleration a of the wheel, giving both its magnitude and its direction relative to @. (Hint: How does linear acceleration relate to linear velocity?)~