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CHAPTER 8 ROTATIONAL KINEMATICS CONCEPTUAL QUESTIONS ____________________________________________________________________________________________ 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 ). A B C r r Axis A B C r r Axis ____________________________________________________________________________________________ 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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128 ROTATIONAL KINEMATICS 3. REASONING AND SOLUTION Just before the battery is removed, 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 battery is removed, 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. ____________________________________________________________________________________________ 4. REASONING AND SOLUTION The tangential speed, v T , of a point on the earth's surface is related to the earth's angular speed ϖ according to T v r ϖ = , 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. ____________________________________________________________________________________________ 5. SSM 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 0 < r ( L /2) 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. ____________________________________________________________________________________________ 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 a T is related to the angular acceleration through Equation 8.10, T a r α = , every point on the rim has zero tangential acceleration.
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