1. Initially, a ball has an angular velocity of 5.0 rad/s counterclockwise. Some time later, after rotating through a total angle of 5.5 radians, the ball has an angular velocity of 1.5 rad/s clockwise.
   (a) What is the angular acceleration? What is the average angular velocity? How much time did this take?
   (b) At some point the angular velocity of the ball had to have been zero. At what angle from it's initial orientation did this occur and how long did it take?
   

2. From the angle versus time graph given to the side, determine the average angular velocity in each segment A to E. Is the angular acceleration in each segment zero, negative, or positive? What is the average angular velocity over the entire time interval?
   
3. From the angular velocity versus time graph below, determine the angular acceleration in each segment. What is the average angular velocity in each segment? What is the angular displacement in each segment? What is the average angular velocity over the entire time interval?
   
4. In the diagram below, a rope is threaded around three pulleys of radii, R₁ = 0.30 m, R₂ = 0.20 m, and R₃ = 0.10 m. There is no slippage.
   (a) If the rope moves with constant speed v = 5.0 m/s, what is the angular velocity of each pulley? What is the frequency and period of each pulley?
   (b) The rope accelerates from 5.0 m/s to 10.0 m/s in 10.0 s, what is the average angular velocity of each pulley for this time? What is the average angular acceleration of each pulley? How many revolutions does each pulley turn in that 10.0 s?
   
   

5. Two rotating disks, each with a visible dot, are show at the same instant in time in the figure below. The disk on the left has zero initial angular velocity and an angular acceleration of a = 12.0 rad/s². The disk on the right has a constant angular velocity of w = 18.0 rad/s. When will the two disks have turned through the same total angle? How many revolutions will this have taken for each disk? If the disks had different radii, would there be any change to your answers?
   
   

6. A ball is thrown into the air at an angle of 25° to the horizontal with a velocity of 11.0 m/s. As the ball leave the pitcher's hand it has an angular velocity of 35.0 rad/s. How many revolutions does the ball make before it returns to the same level? Note that the x, y, and rotational motions are all completely independent. Also note that the linear velocity given is not the velocity in the equation v = wR.

7. A motorcycle has tires with a diameter of 44.0 cm. Cruising down the highway, they are rotating at 1150 rpm (revolutions per minute). What is the angular velocity in radians per second? What is the frequency and period of the tires? What is the tangential velocity of the edges of the tires (and hence the linear velocity of the motorcycle) in m/s and km/h?
   

8. In the diagram below, three forces are applied to a 3-4-5 triangle. The forces are F₁ = 91.7 N, F₂ = 150 N, and F₃ = 67.7 N. F₃ is applied at the middle of side AB. (a) Find the net torque about point A. (b) Find the net torque about point B. (c) Find the net torque about point C.
   
9. Metal nuts on machinery bolts often have a recommended torque to which they should be tigtened by a wrench. If the nuts are not tightened enough, they may work loose. If tightened too much they may strip the bolt and be impossible to remove. In the diagram below, what maximum force F can you apply to tighten the bolt to 300 m-N?
   

Questions? mikec@kwantlen.bc.ca