Rotational Kinematics Circular Motion Position: r constant Speed: v constant A Particle in Uniform Circular Motion Tperiod the time required for one complete rotation. 1 circumference 2 r
v 1 period T For a particle in uniform circular motion, the velocity vector v remains constant in magnitude, but it continuously changes its direction. Angular Position Degrees and revolutions: Angular Position Arc length s,
measured in radians: s r (radians) (degrees) /180 (rev) 2 Angular Velocity Sign of Connections Between Linear & Rotational Quantities Angular Acceleration
Comparison to 1-D Kinematics Angular Linear constant a constant 0 t v v0 at
1 0 0t t 2 2 2 0 2 2 ( 0 ) x x0 v0t 2 1 2 at 2 2
v v0 2a x x0 And for a point at a distance R from the rotation axis: x = Rv = Ra = R By convention, are positive if they are in the counterclockwise direction. Decelerating Windmill As the wind dies, a windmill that had been rotating at = 2.1 rad/s begins to slow down at a constant angular acceleration of = 0.45 rad/s2. How long does it take for the windmill to come to a complete stop?
av t f i (0) (2.1 rad/s) t 4.7 s 2 av ( 0.45 rad/s ) Angular Velocity & Acceleration
ACT t1 The fan blade shown is slowing down. Which option describes and ? (a) >0 and >0; (b) >0 and <0; (c) <0 and >0; (d) <0 and <0. Rotational Kinematics If the angular acceleration is
constant: 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 balls angular acceleration, assuming it to be constant? (b) How many revolutions does the ball make before being caught? 0 t
0 (157.0 rad/s) (154.7 rad/s) 3.03 rad/s 2 t (0.795 s) 0t 12 t 2 (157.0 rad/s)(0.795 s) 12 ( 3.03 rad/s 2 )(0.795 s) 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 wheels angular acceleration, assuming it to be constant. (b) How long does it take for the wheel to come to rest? 2 02 2 2 02 0 (3.40 rad/s) 2 0.736 rad/s 2
2 2(2 rad/rev)(1.25 rev) 0 t 0 0 (3.40 rad/s) t 4.62 s 2 ( 0.736 rad/s ) A Rotating Crankshaft A cars tachometer indicates the angular velocity 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 crankshafts 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? i 500 rpm (2 rad/rev)/(60 s/min)=52.4 rad/s f 2500 rpm 5i 262.0 rad/s f i (262.0 rad/s 52.4 rad/s) 69.9 rad/s 2 t (3.0 s) f i i t 12 t 2
1rev 2 0 (52.4 rad/s)(3.0 s) 12 (69.9 rad/s 2 ) 3.0 s 472 rad 75 rev 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 pulleys angular velocity to decrease with a constant angular acceleration = 2.10 rad/s2. (a) If the pulleys initial angular velocity is 0 = 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? 0 t t ( 0 ) / 0 (5.40 rad/s) / ( 2.10 rad/s 2 ) 2.57 s 0t 12 t 2 (5.40 rad/s)(2.57 s) 12 ( 2.10 rad/s 2 )(2.57 s) 2 6.94 rad s r 6.94rad 5.0 cm 34.7cm CD Speed CDs and DVDs turn with a variable that keeps the tangential speed vt constant.
Find the angular speed 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 r (1.25 m/s) 50.0rad 1rev r 2.50 cm:
7.96 rps (0.0250 m) s 2 rad (1.25 m/s) 20.8 rad 1rev r 6.00 cm: 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. acp r 2 vt2 / r at r
a at2 acp2 tan 1 a cp Speeding up / 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
cm ac 2 r 8.00 9.07cm 580.5 s 2 r s rad aT r 95 2 9.07cm 861.7 cm 2 s s 2 2
a ac aT a tan c aT 1 2 2 2 2 580.5 cm s 861.7 cm s 580.5 cm
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