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# Chp. 9 - Chapter 9 Rotation Conceptual Problems 1 Two...

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837 Chapter 9 Rotation Conceptual Problems 1 Two points are on a disk that is turning about a fixed-axis through its center, perpendicular to the disk and through its center, at increasing angular velocity. One point is on the rim and the other point is halfway between the rim and the center. ( a ) Which point moves the greater distance in a given time? ( b ) Which point turns through the greater angle? ( c ) Which point has the greater speed? ( d ) Which point has the greater angular speed? ( e ) Which point has the greater tangential acceleration? ( f ) Which point has the greater angular acceleration? ( g ) Which point has the greater centripetal acceleration? Determine the Concept ( a ) Because r is greater for the point on the rim, it moves the greater distance. ( b ) Both points turn through the same angle. ( c ) Because r is greater for the point on the rim, it has the greater speed. ( d ) Both points have the same angular speed. ( e ) Both points have zero tangential acceleration. ( f ) Both have zero angular acceleration. ( g ) Because r is greater for the point on the rim, it has the greater centripetal acceleration. 2 True or false: ( a ) Angular speed and linear speed have the same dimensions. ( b ) All parts of a wheel rotating about a fixed axis must have the same angular speed. ( c ) All parts of a wheel rotating about a fixed axis must have the same angular acceleration. ( d ) All parts of a wheel rotating about a fixed axis must have the same centripetal acceleration. ( a ) False. Angular speed has the dimensions [ ] T 1 whereas linear speed has dimensions [] T L . ( b ) True. The angular speed of all points on a wheel is d θ / dt. ( c ) True. The angular acceleration of all points on the wheel is d ω / dt. ( d ) False. The centripetal acceleration at a point on a rotating wheel is directly proportional to its distance from the center of the wheel 3 Starting from rest and rotating at constant angular acceleration, a disk takes 10 revolutions to reach an angular speed . How many additional revolutions at the same angular acceleration are required for it to reach an angular speed of 2 ? ( a ) 10 rev, ( b ) 20 rev, ( c ) 30 rev, ( d ) 40 rev, ( e ) 50 rev?

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Chapter 9 838 Picture the Problem The constant-acceleration equation that relates the given variables is θ α ω ω Δ + = 2 2 0 2 . We can set up a proportion to determine the number of revolutions required to double ω and then subtract to find the number of additional revolutions to accelerate the disk to an angular speed of 2 . Using a constant-acceleration equation, relate the initial and final angular velocities to the angular acceleration: θ α ω ω Δ + = 2 2 0 2 or, because 2 0 ω = 0, θ α ω Δ = 2 2 Let Δ θ 10 represent the number of revolutions required to reach an angular speed : 10 2 2 θ α ω Δ = (1) Let Δ 2 represent the number of revolutions required to reach an angular speed : ( ) ω θ α ω 2 2 2 2 Δ = (2) Divide equation (2) by equation (1) and solve for Δ 2 : ( ) 10 10 2 2 2 4 2 θ θ ω ω θ ω Δ = Δ = Δ The number of additional revolutions is: () rev 30 rev 10 3 Δ 3 Δ Δ 4 10 10 10 = = = θ θ θ and ) ( c is correct.
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