Ch 01



|CHAPTER 8 |ROTATIONAL KINEMATICS |

CONCEPTUAL QUESTIONS

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1. REASONING AND SOLUTION The figures below show two axes in the plane of the paper and located so that the points B and C move in circular paths having the same radii (radius = r).

[pic]

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2. REASONING AND SOLUTION When a pair of scissors is used to cut a string, each blade of the scissors does not have the same angular velocity at a given instant during the cut. The angular speed of each blade is the same; however, each blade rotates in the opposite direction. Therefore, it is correct to conclude that the blades have opposite angular velocities at any instant during the cut.

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3. REASONING AND SOLUTION Just before the clock is unplugged, the second hand is rotating so that its angular velocity is clockwise. The second hand moves with a constant angular velocity, so the angular acceleration is zero. When the plug is pulled, the second hand will continue to rotate with its clockwise angular velocity, but it will slow down. Therefore, the angular acceleration must be opposite to the angular velocity, or counterclockwise.

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4. REASONING AND SOLUTION The tangential speed, vT, of a point on the earth's surface is related to the earth's angular speed [pic] according to [pic], Equation 8.9, where r is the perpendicular distance from the point to the earth's rotation axis. At the equator, r is equal to the earth's radius. As one moves away from the equator toward the north or south geographic pole, the distance r becomes smaller. Since the earth's rotation axis passes through the geographic poles, r is effectively zero at those locations. Therefore, your tangential speed would be a minimum if you stood as close as possible to either the north or south geographic pole.

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5. REASONING AND SOLUTION

a. A thin rod rotates at a constant angular speed about an axis of rotation that is perpendicular to the rod at its center. As the rod rotates, each point at a distance r from the center on one half of the rod has the same tangential speed as the point at a distance r from the center on the other half of the rod. This is true for all values of r for [pic] where L is the length of the rod.

b. If the rod rotates about an axis that is perpendicular to the rod at one end, no two points are the same distance from the axis of rotation. Therefore, no two points on the rod have the same tangential speed.

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6. REASONING AND SOLUTION The wheels are rotating with a constant angular velocity.

a. Since the angular velocity is constant, each wheel has zero angular acceleration, α =

0 rad/s. Since the tangential acceleration aT is related to the angular acceleration through Equation 8.10, [pic], every point on the rim has zero tangential acceleration.

b. Since the particles on the rim of the wheels are moving along a circular path, they must have a centripetal acceleration. This can be supported by Equation 8.11, [pic], where [pic] is the magnitude of the centripetal acceleration. Since ω is nonzero, ac is nonzero.

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|7. REASONING AND SOLUTION The wheel and the two points under consideration |[pic] |

|are shown in the figure at the right. | |

| | |

|a. Each point must undergo the same angular displacement in the same time | |

|interval. | |

|Therefore, at any given instant, the angular velocity of both points is the | |

|same. | |

| | |

|b. Each point must increase its angular velocity at the same rate; therefore, | |

|each point has the same angular acceleration. | |

c. The tangential speed is given by [pic]. Since both points have the same angular speed ω, but point 1 is further from the center than point 2, point 1 has the larger tangential speed.

d. The tangential acceleration is given by [pic]. Since both points have the same angular acceleration α, but point 1 is further from the center than point 2, point 1 has the larger tangential acceleration.

e. The centripetal acceleration is given by [pic]. Since both points have the same angular speed ω, but point 1 is further from the center than point 2, point 1 has the larger centripetal acceleration.

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8. REASONING AND SOLUTION For a building that is located on the earth's equator, the points in the building rotate about the center of the earth. The tangential speed of any point is given by [pic], where, in this case, r is measured from the center of the earth. The top floor of the building has a larger value of r than the other floors; therefore, the top floor has the greater tangential speed.

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9. REASONING AND SOLUTION The centripetal acceleration of any point in the space station is given by [pic] (Equation 8.11),where r is the distance from the point to the axis of rotation of the space station. If the centripetal acceleration is adjusted to be g at a given value of r (such as the astronaut's feet), the centripetal acceleration will be different at other values of r. Therefore, if the adjustment is made so that the centripetal acceleration at the astronaut's feet equals g, then the centripetal acceleration will not equal g at the astronaut's head.

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10. REASONING AND SOLUTION The tangential speed for any point on the tip of a clock hand is given by [pic]. The angular speeds of the second hand, the minute hand, and the hour hand differ greatly, with the second hand having the largest angular speed and the hour hand having the smallest. If one desired to create a clock in which the tips of the second hand, the minute hand, and the hour hand moved with the same tangential speed, the lengths of the arms would also have to differ greatly, with the second hand having the smallest arm and the hour hand having the largest. Such a clock would not be very practical.

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11. REASONING AND SOLUTION Any point on a rotating object possesses a centripetal acceleration that is directed radially toward the axis of rotation. This also applies to a tire on a moving car and is true regardless of whether the car has a constant linear velocity or whether it is accelerating.

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12. REASONING AND SOLUTION The bicycle wheel has an angular acceleration. The arrows are perpendicular to the radius of the wheel. The magnitude of the arrows increases with increasing distance from the center in accordance with [pic], or [pic]. The arrows in the picture could represent either the tangential velocity or the tangential acceleration. The arrows are not directed radially inward; therefore, they cannot represent the centripetal acceleration.

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13. REASONING AND SOLUTION The speedometer of a truck uses a device that measures the angular speed of the tires. The angular speed is related to the linear speed of the truck by [pic].

Suppose two trucks are traveling side-by-side along a highway at the same linear speed v, and one truck has larger wheels than the other. The angular speed of the larger-diameter wheels is less than that of the smaller-diameter wheels. Since the speedometer uses a device that measures the angular speed of the tires, the speedometer reading on the truck with the larger wheels will indicate a smaller linear speed than that on the truck with the smaller wheels.

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14. REASONING AND SOLUTION When a fan is shut off, the fan blades gradually slow down until they eventually stop. The angular acceleration is never really constant because it gradually decreases, becoming zero at the instant the blades stop rotating. In solving problems, it is sometimes convenient to approximate such motion as having constant angular acceleration. This is a fairly good approximation for a short time interval just after the switch is turned off.

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15. REASONING AND SOLUTION Three examples that involve rotation about an axis that is not fixed:

(1) The motion of a Frisbee. The axis of the rotating Frisbee moves through the air

with the Frisbee.

(2) The motion of the earth in its orbit. The rotation axis of the earth changes

position as the earth revolves around the sun.

(3) The motion of a twirling baton that has been thrown into the air. The rotation axis constantly changes location as the baton rises and falls.

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