Concepts & Calculations

Note: Each of these problems consists of Concept Questions followed by a related quantitative Problem. The Concept Questions involve little or no mathematics. They focus on the concepts with which the problems deal. Recognizing the concepts is the essential initial step in any problem-solving technique.

| 69 | Concept Questions (a) In general, is the direction of an object’s average angular velocity the same as its initial angle , its final angle , or the difference between its final and initial angles? (b) The table that follows lists four pairs of initial and final angles of a wheel on a moving car. Decide which pairs give a positive average angular velocity and which give a negative average angular velocity. Provide reasons for your answers.

| Initial angle | Final angle
(a) | 0.45 rad | 0.75 rad
(b) | 0.94 rad | 0.54 rad
(c) | 5.4 rad | 4.2 rad
(d) | 3.0 rad | 3.8 rad

Problem The elapsed time for each pair of angles is 2.0 s. Review the concept of average angular velocity in Section 8.2 and then determine the average angular velocity (magnitude and direction) for each of the four pairs of angles in the table. Check to see that the directions (positive or negative) of the angular velocities agree with the directions found in the Concept Question.

| 70 | Concept Questions (a) In general, does the average angular acceleration of a rotating object have the same direction as its initial angular velocity , its final angular velocity , or the difference between its final and initial angular velocities? (b) The table that follows lists four pairs of initial and final angular velocities for a rotating fan blade. Determine the direction (positive or negative) of the average angular acceleration for each pair. Provide reasons for your answers.

| Initial angular velocity | Final angular velocity
(a) | +2.0 rad/s | +5.0 rad/s
(b) | +5.0 rad/s | +2.0 rad/s
(c) | −7.0 rad/s | −3.0 rad/s
(d) | +4.0 rad/s | −4.0 rad/s

Problem The elapsed time for each of the four pairs of angular velocities is 4.0 s. Find the average angular acceleration (magnitude and direction) for each of the four pairs. Be sure that your directions agree with those found in the Concept Question.

| 71 | Concept Question In the table are listed the initial angular velocity and the angular acceleration of four rotating objects at a given instant in time.

| Initial angular velocity | Angular acceleration
(a) | +12 rad/s | +3.0 rad/s2
(b) | +12 rad/s | −3.0 rad/s2
(c) | −12 rad/s | +3.0 rad/s2
(d) | −12 rad/s | −3.0 rad/s2

In each case, state whether the angular speed of the object is increasing or decreasing in time. Account for your answers.
Problem For each of the four pairs in the table, determine the final angular speed of the object if the elapsed time is 2.0 s. Compare your final angular speeds with the initial angular speeds and make sure that your answers are consistent with your answers to the Concept Question.

| 72 | Concept Questions Does the tip of a rotating fan blade have a tangential acceleration when the blade is rotating (a) at a constant angular velocity and (b) at a constant angular acceleration? Provide reasons for your answers. Problem A fan blade is rotating with a constant angular acceleration of +12.0 rad/s2. At what point on the blade, measured from the axis of rotation, does the magnitude of the tangential acceleration equal that of the acceleration due to gravity?

| 73 | Concept Questions During a time t, a wheel has a constant angular acceleration, so its angular velocity increases from an initial value of to a final value of . During this time, is the angular displacement less than, greater than, or equal to (a) or (b) ? In each case, justify your answer. (c) If you conclude that the angular displacement does not equal or , then how does it depend on , , and t?
Problem A car is traveling along a road, and the engine is turning over with an angular velocity of +220 rad/s. The driver steps on the accelerator, and in a time of 10.0 s the angular velocity increases to +280 rad/s. (a) What would have been the angular displacement of the engine if its angular velocity had remained constant at the initial value of +220 rad/s during the 10.0 s? (b) What would have been the angular displacement if the angular velocity had been equal to its final value of +280 rad/s during the 10.0 s? (c) Determine the actual value of the angular displacement during this period. Check your answers to see that they are consistent with those to the Concept Questions.

| 74 | Concept Questions Two identical dragsters, starting from rest, accelerate side-by-side along a straight track. The wheels on one of the cars roll without slipping, while the wheels on the other slip during part of the time. (a) For which car, he winner or the loser, do the wheels roll without slipping? Why? For the dragster whose wheels roll without slipping, is there (b) a relationship between its linear speed and the angular speed of its wheels, and (c) a relationship between the magnitude of its linear acceleration and the magnitude of the angular acceleration of its wheels? If a relationship exists in either case, what is it?
Problem A dragster starts from rest and accelerates down the track. Each tire has a radius of 0.320 m and rolls without slipping. At a distance of 384 m, the angular speed of the wheels is 288 rad/s. Determine, (a) the linear speed of the dragster and (b) the magnitude of the angular acceleration of its wheels.

| 75 | Concept Questions A propeller is rotating about an axis perpendicular to its center, as the drawing shows. The axis is parallel to the ground. An arrow is fired at the propeller, travels parallel to the axis, and passes through one of the open spaces between the propeller blades. The vertical drop of the arrow may be ignored. There is a maximum value for the angular speed of the propeller beyond which the arrow cannot pass through an open space without being struck by one of the blades. (a) If the arrow is to pass through an open space, does it matter if the arrow is aimed closer to or farther away from the axis (see points A and B in the drawing, for example)? Explain. (b) Does the maximum value of increase, decrease, or remain the same with increasing arrow speed v? Why? (c) Does the maximum value of increase, decrease, or remain the same with increasing arrow length L? Justify your answer.
Problem The angular open spaces between the three propeller blades are each (60.0°). Find the maximum value of the angular speed when the arrow has the lengths L and speeds v shown in the following table. Check to see that your answers agree with your answers to the Concept Questions.

| L | v
(a) | 0.71 m | 75.0 m/s
(b) | 0.71 m | 91.0 m/s
(c) | 0.81 m | 91.0 m/s

| 76 | Concept Questions At a county fair there is a betting game that involves a spinning wheel. As the drawing shows, the wheel is set into rotational motion with the beginning of the angular section labeled “1” at the marker at the top of the wheel. The wheel then decelerates and eventually comes to a halt on one of the numbered angular sections. (a) Given the initial angular velocity and the magnitude of the angular deceleration, which one of the equations of rotational kinematics would you use to calculate the angular displacement of the wheel? (b) When using the equations of rotational kinematics, can the initial angular velocity and the angular deceleration be expressed in rev/s and rev/s2, respectively, or is it necessary to use radian measure? Explain. (c) Does a greater initial angular velocity necessarily mean that the wheel comes to a halt on an angular section labeled with a greater number (8 versus 6, for example)? Provide a reason for your answer.
Problem The wheel in the drawing is divided into twelve angular sections, each of which is 30.0°. Determine the numbered section on which the wheel comes to a halt when the deceleration of the wheel has a magnitude of 0.200 rev/s2 and the initial angular velocity is (a) +1.20 rev/s and (b) +1.47 rev/s. Check to see that your answers are consistent with your answers to the Concept Questions.

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