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PHYSICS 231 Lecture 15: Rotations

PHYSICS 231 Lecture 15: Rotations. Remco Zegers Walk-in hours Thursday 11:30-13:30 Helproom. Radians & Radius. Circumference=2 r. Part s= r. r. s. . =s/r.  in radians! 360 o =2 rad = 6.28… rad  (rad) = /180 o  (deg). Angular speed and acceleration. Average angular velocity.

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PHYSICS 231 Lecture 15: Rotations

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  1. PHYSICS 231Lecture 15: Rotations Remco Zegers Walk-in hours Thursday 11:30-13:30 Helproom PHY 231

  2. Radians & Radius Circumference=2r Part s=r r s  =s/r  in radians! 360o=2 rad = 6.28… rad  (rad) = /180o  (deg) PHY 231

  3. Angular speed and acceleration Average angular velocity Instantaneous Angular velocity Angular velocity : rad/s rev/s rpm (revolutions per minute) PHY 231

  4. Answer: in 1 year, the earth makes one full orbit around the sun. rad/s rpm rev/s = 2 rad/1 year 1 rev/1 year 1 rev/1 year = 2 rad/(3.2E+7 s) 1 rev/(3.2E+7 s) 1 rev/(5.3E+5 min) = 2.0E-07 rad/s 3.1E-8 rev/s 1.9E-6 rpm example What is the angular velocity of earth around the sun? Give in rad/s, rev/s and rpm PHY 231

  5. Angular acceleration Definition: The change in angular velocity per time unit Average angular acceleration Instantaneous angular acceleration Unit: rad/s2 PHY 231

  6. Equations of motion Linear motion Angular motion (t)= (0)+(0)+t½t2 (t)= (0)+t X(t)=x(0)+v(0)t+½at2 V(t)=V(0)+at PHY 231

  7. First 5s: (5)= (0)+(0)t+½t2 = /2+ 0 +½1(5)2 = /2+12.5=14.1 rad (=806o) (5)= (0)+t = 1t= 5 rad/s Next 5s: (5)= 14.1+5t = 39.1 rad (=22400=6.2 rev) t example A person is rotating a wheel. The handle is initially at =90o. For 5s the wheel gets an constant angular acceleration of 1 rad/s2. After that The angular velocity is constant. Through what angle will the wheel have rotated after 10s. =0  PHY 231

  8. Angular Linear velocities V V is called the tangential velocity PHY 231

  9. Angular linear acceleration velocity Change in velocity Change in velocity per time unit acceleration The linear acceleration equals the angular acceleration times the radius of the orbit PHY 231

  10. 12 m/s 10 m/s  a) v= r = 2*6=12 m/s b) Must be the same: 2 rad/s c) v=r = 2*5=10 m/s a example The angular velocity of a is 2 rad/s. a) What is its tangential velocity? If b is keeping pace with a, b) What is its angular velocity? c) What is its linear velocity? b 5m 6m PHY 231

  11. b) = ((0)+ (10))/2=18/2=9 rad/s c) v=r = 9*0.01=0.09 m/s d=vt=0.09*10=0.9 m A rolling coin 0=18 rad/s =-1.80 rad/s d d d=0.02 m • For how long does the coin roll? • What is the average angular velocity? • c) How far does the coin roll before coming to rest? a) (t)= (0)+t 0=18-1.8t t=10 s PHY 231

  12. gears If 1=3 rad/s, how fast Is the bike going? b) What if r1=0.1 m ? r3=0.7 m r1=0.3 r2=0.15 b) V1= 1r1=3*0.1=0.3 m/s V1=V2 (because of the chain) 2=V2/r2=0.3/0.15=2 rad/s 3=2 (connected) V3=3r3=2*0.7=1.4 m/s V1= 1r1=3*0.3=0.9 m/s V1=V2 (because of the chain) 2=V2/r2=0.9/0.15=6 rad/s 3=2 (connected) V3=3r3=6*0.7=4.2 m/s PHY 231

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