Additional Problems

| 56 | Our sun rotates in a circular orbit about the center of the Milky Way galaxy. The radius of the orbit is 2.2 × 1020 m, and the angular speed of the sun is 1.2 × 10−15 rad/s. (a) What is the tangential speed of the sun? (b) How long (in years) does it take for the sun to make one revolution around the center?

| 57 | ssm A 220-kg speedboat is negotiating a circular turn (radius = 32 m) around a buoy. During the turn, the engine causes a net tangential force of magnitude 550 N to be applied to the boat. The initial tangential speed of the boat going into the turn is 5.0 m/s. (a) Find the tangential acceleration. (b) After the boat is 2.0 s into the turn, find the centripetal acceleration.

| 58 | A gymnast is performing a floor routine. In a tumbling run she spins through the air, increasing her angular velocity from 3.00 to 5.00 rev/s while rotating through one-half of a revolution. How much time does this maneuver take?

| 59 | ssm An electric fan is running on HIGH. After the LOW button is pressed, the angular speed of the fan decreases to 83.8 rad/s in 1.75 s. The deceleration is 42.0 rad/s2. Determine the initial angular speed of the fan.

| 60 | Review Multiple-Concept Example 7 in this chapter and Interactive LearningWare 8.2 at as aids in solving this problem. A train is rounding a circular curve whose radius is 2.00 × 102 m. At one instant, the train has an angular acceleration of 1.50 × 10−3 rad/s2 and an angular speed of 0.0500 rad/s. (a) Find the magnitude of the total acceleration (centripetal plus tangential) of the train. (b) Determine the angle of the total acceleration relative to the radial direction.

| 61 | Interactive Solution 8.61 at offers one approach to solving this problem. The drawing shows the blade of a chain saw. The rotating sprocket tip at the end of the guide bar has a radius of 4.0 × 10−2 m. The linear speed of a chain link at point A is 5.6 m/s. Find the angular speed of the sprocket tip in rev/s.

* | 62 | Multiple-Concept Example 7 explores the approach taken in problems such as this one. The blades of a ceiling fan have a radius of 0.380 m and are rotating about a fixed axis with an angular velocity of +1.50 rad/s. When the switch on the fan is turned to a higher speed, the blades acquire an angular acceleration of +2.00 rad/s2. After 0.500 s have elapsed since the switch was reset, what is (a) the total acceleration (in m/s2) of a point on the tip of a blade and (b) the angle between the total acceleration and the centripetal acceleration ? (See Figure 8-13b.)

* | 63 | ssm A compact disc (CD) contains music on a spiral track. Music is put onto a CD with the assumption that, during playback, the music will be detected at a constant tangential speed at any point. Since , a CD rotates at a smaller angular speed for music near the outer edge and a larger angular speed for music near the inner part of the disc. For music at the outer edge (r = 0.0568 m), the angular speed is 3.50 rev/s. Find (a) the constant tangential speed at which music is detected and (b) the angular speed (in rev/s) for music at a distance of 0.0249 m from the center of a CD.

* | 64 | The drawing shows a graph of the angular velocity of a rotating wheel as a function of time. Although not shown in the graph, the angular velocity continues to increase at the same rate until t = 8.0 s. What is the angular displacement of the wheel from 0 to 8.0 s?

* | 65 | A dentist causes the bit of a high-speed drill to accelerate from an angular speed of 1.05 × 104 rad/s to an angular speed of 3.14 × 104 rad/s. In the process, the bit turns through 1.88 × 104 rad. Assuming a constant angular acceleration, how long would it take the bit to reach its maximum speed of 7.85 × 104 rad/s, starting from rest?

* | 66 | Interactive LearningWare 8.2 at provides a review of the concepts that are important in this problem. A race car, starting from rest, travels around a circular turn of radius 23.5 m. At a certain instant, the car is still accelerating, and its angular speed is 0.571 rad/s. At this time, the total acceleration (centripetal plus tangential) makes an angle of 35.0° with respect to the radius. (The situation is similar to that in Figure 8-13b.) What is the magnitude of the total acceleration?

** | 67 | A bicycle is rolling down a circular hill that has a radius of 9.00 m. As the drawing illustrates, the angular displacement of the bicycle is 0.960 rad. The radius of each wheel is 0.400 m. What is the angle (in radians) through which each tire rotates?

Problem 67

** | 68 | Review Conceptual Example 2 before attempting to work this problem. The moon has a diameter of 3.48 × 106 m and is a distance of 3.85 × 108 m from the earth. The sun has a diameter of 1.39 × 109 m and is 1.50 × 1011 m from the earth. (a) Determine (in radians) the angles subtended by the moon and the sun, as measured by a person standing on the earth. (b) Based on your answers to part (a), decide whether a total eclipse of the sun is really “total.” Give your reasoning. (c) Determine the ratio, expressed as a percentage, of the apparent circular area of the moon to the apparent circular area of the sun.

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