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Rotational Kinematics Angular Position q Degrees and revolutions: Angular Position q Arc length s, measured in radians: s q r q (radians) q (degrees) /180 q (rev) 2 Angular Velocity w Sign of w A small sphere attached to a light rigid rod rotates about an axis perpendicular to and fixed to the other end of the rod. Relative to the positive direction of the axis of rotation, the angular positions of the sphere are negative, its angular velocity is negative, and its angular acceleration is negative. The sphere is 1. rotating clockwise and slowing down. 71% 2. rotating counterclockwise and slowing down. 3. rotating clockwise and speeding up. 4. rotating counterclockwise and speeding up. 5. first rotating counterclockwise and then 10% 10% 10% clockwise. 0% 1 2 3 4 5 A small sphere attached to a light rigid rod rotates about an axis perpendicular to and fixed to the other end of the rod. Relative to the positive direction of the axis of rotation, the angular positions of the sphere are negative, its angular velocity is negative, and its angular acceleration is negative. The sphere is 1. rotating clockwise and slowing down. 71% 2. rotating counterclockwise and slowing down. 3. rotating clockwise and speeding up. 4. rotating counterclockwise and speeding up. 5. first rotating counterclockwise and then 10% 10% 10% clockwise. 0% 1 2 3 4 5 The angular speed of the minute hand of a clock, in rad/s, is 35% 1 1. . 30% 1800 1 25% 2. 60 . 3. . 1 30 4. π. 5. 120π. 5% 5% 1 2 3 4 5 The angular speed of the minute hand of a clock, in rad/s, is 35% 1 1. . 30% 1800 1 25% 2. 60 . 3. . 1 30 4. π. 5. 120π. 5% 5% 1 2 3 4 5 Connections Between Linear & Rotational Quantities Angular Acceleration a Comparison to 1-D Kinematics Angular Linear a constant a constant w w0 a t v v0 at 1 1 2 q q 0 w0 t a t 2 x x0 v0t at 2 2 w 2 w0 2 2a (q q0 ) v2 v02 2a x x0 And for a point at a distance R from the rotation axis: x = Rq v = wR a = aR By convention, q, w, a are positive if they are in the counterclockwise direction. Decelerating Windmill As the wind dies, a windmill that had been rotating at w = 2.1 rad/s begins to slow down at a constant angular acceleration of a = 0.45 rad/s2. How long does it take for the windmill to come to a complete stop? w a av t w w f wi (0) (2.1 rad/s) t 4.7 s a av a (0.45 rad/s ) 2 The fan blade shown is slowing down. Which option describes a and w? 48% 24% 24% 1. w>0 and a>0 2. w>0 and a<0 5% 3. w<0 and a>0 4. w<0 and a<0 1 2 3 4 The fan blade shown is slowing down. Which option describes a and w? 48% 24% 24% 1. w>0 and a>0 2. w>0 and a<0 5% 3. w<0 and a>0 4. w<0 and a<0 1 2 3 4 Rotational Kinematics If the angular acceleration is constant: The graphs below show angular velocity as a function of time. In which one is the magnitude of the angular acceleration constantly decreasing? 1. a 53% 2. b 42% 3. c 4. d 5% 5. e 0% 0% 1 2 3 4 5 The graphs below show angular velocity as a function of time. In which one is the magnitude of the angular acceleration constantly decreasing? 1. a 53% 2. b 42% 3. c 4. d 5% 5. e 0% 0% 1 2 3 4 5 Thrown for a Curve To throw a curve ball, a pitcher gives the ball an initial angular speed of 157.0 rad/s. When the catcher gloves the ball 0.795 s later, its angular speed has decreased (due to air resistance) to 154.7 rad/s. (a) What is the ball’s angular acceleration, assuming it to be constant? (b) How many revolutions does the ball make before being caught? w w0 a t w w0 (157.0 rad/s) (154.7 rad/s) a 3.03 rad/s 2 t (0.795 s) q w0t 1 a t 2 2 (157.0 rad/s)(0.795 s) 1 ( 3.03 rad/s 2 )(0.795 s) 2 2 123.9 rad 19.7 rev Wheel of Misfortune On a certain game show, contestants spin the wheel when it is their turn. One contestant gives the wheel an initial angular speed of 3.40 rad/s. It then rotates through 1.25 revolutions and comes to rest on BANKRUPT. (a) Find the wheel’s angular acceleration, assuming it to be constant. (b) How long does it take for the wheel to come to rest? w 2 w02 2a q w 2 w02 0 (3.40 rad/s) 2 a 0.736 rad/s 2 2q 2(2 rad/rev)(1.25 rev) w w0 a t w w0 0 (3.40 rad/s) t 4.62 s a (0.736 rad/s ) 2 A Rotating Crankshaft A car’s tachometer indicates the angular velocity w of the crank shaft in rpm. A car stopped at a traffic light has its engine idling at 500 rpm. When the light turns green, the crankshaft’s angular velocity speeds up at a constant rate to 2500 rpm in a time interval of 3.0 s. How many revolutions does the crankshaft make in this time interval? wi 500 rpm (2 rad/rev)/(60 s/min)=52.4 rad/s w f 2500 rpm 5wi 262.0 rad/s w f wi (262.0 rad/s 52.4 rad/s) a 69.9 rad/s 2 t (3.0 s) q f qi wi t 1 a t 2 2 1rev 0 (52.4 rad/s)(3.0 s) 1 (69.9 rad/s 2 ) 3.0 s 472 rad 75 rev 2 2 2 rad Time to Rest A pulley rotating in the counterclockwise direction is attached to a mass suspended from a string. The mass causes the pulley’s angular velocity to decrease with a constant angular acceleration a = 2.10 rad/s2. (a) If the pulley’s initial angular velocity is w0 = 5.40 rad/s, how long does it take for the pulley to come to rest? (b) Through what angle does the pulley turn during this time? (c) If the radius of the pulley is 5.0 cm, through what distance is the mass lifted? w w0 a t t (w w0 ) / a 0 (5.40 rad/s) / (2.10 rad/s2 ) 2.57 s q w0t 1 a t 2 2 (5.40 rad/s)(2.57 s) 1 ( 2.10 rad/s 2 )(2.57 s) 2 6.94 rad 2 s q r 6.94rad 5.0 cm 34.7cm CD Speed CDs and DVDs turn with a variable w that keeps the tangential speed vt constant. Find the angular speed w and the frequency that a CD must have in order to give it a linear speed vt = 1.25 m/s when the laser beam shines on the disk (a) at 2.50 cm from its center, and (b) at 6.00 cm from its center. vt w r (1.25 m/s) 50.0rad 1rev r 2.50 cm: w 7.96 rps (0.0250 m) s 2 rad (1.25 m/s) 20.8 rad 1rev r 6.00 cm: w 3.31 rps (0.0600 m) s 2 rad Rotational vs. Linear Kinematics Analogies between linear and rotational kinematics: Connections Between Linear & Rotational Quantities More Connections Between Linear & Rotational Quantities This merry-go-round has both tangential and centripetal acceleration. at ra acp rw 2 vt2 / r a at2 acp 2 Speeding up tan 1 acp / at The Microhematocrit Suppose the centrifuge is just starting up, and that it has an angular speed of 8.00 rad/s and an angular acceleration of 95.0 rad/s2. (a) What is the magnitude of the centripetal, tangential, and total acceleration of the bottom of a tube? (b) What angle does the total acceleration make with the direction of motion? 2 v2 rad ac w 2 r 8.00 9.07cm 580.5 s 2 cm r s rad aT a r 95 2 9.07cm 861.7 cm 2 s s a ac aT 2 2 580.5 cm s 861.7 cm s 2 2 2 2 1039 cm s 2 a 580.5 cm tan c 1 s2 34 0.593rad o aT 861.7 s 2 cm

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