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PHYSICS 231 Lecture 21: Angular momentum

PHYSICS 231 Lecture 21: Angular momentum. Remco Zegers Walk-in hour: Monday 9:15-10:15 am Helproom. Neutron star. In the previous episode.   =I (compare to  F=ma ) Moment of inertia I: I=(  m i r i 2 ) : angular acceleration I depends on the choice of rotation axis!!.

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PHYSICS 231 Lecture 21: Angular momentum

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  1. PHYSICS 231Lecture 21: Angular momentum Remco Zegers Walk-in hour: Monday 9:15-10:15 am Helproom Neutron star PHY 231

  2. In the previous episode.. =I (compare to F=ma) Moment of inertia I: I=(miri2) : angular acceleration I depends on the choice of rotation axis!! PHY 231

  3. Extended objects (like the stick) I=(miri2) =(m1+m2+…+mn)R2 =MR2 M PHY 231

  4. Some common cases PHY 231

  5. Falling bars (demo) mass: m L Ibar=mL2/3 Fg Compare the angular acceleration for 2 bars of different mass, but same length. =I=mL2/3 also =Fd=mgL/2 so =3g/(2L) independent of mass! Compare the angular acceleration for 2 bars of same mass, but different length =3g/(2L) so if L goes up,  goes down! PHY 231

  6. Example A monocycle (bicycle with one wheel) has a wheel that has a diameter of 1 meter. The mass of the wheel is 5 kg (assume all mass is sitting at the outside of the wheel). The friction force from the road is 25 N. If the cycle is accelerating with 0.3 m/s2, what is the force applied on each of the paddles if the paddles are 30 cm from the center of the wheel? 0.5m 0.3m F 25N PHY 231

  7. Rotational kinetic energy Consider a object rotating with constant velocity. Each point moves with velocity vi. The total kinetic energy is: KEr=½I2 Conservation of energy for rotating object: [PE+KEt+KEr]initial= [PE+KEt+KEr]final PHY 231

  8. Example. 1m Consider a ball and a block going down the same 1m-high slope. The ball rolls and both objects do not feel friction. If both have mass 1kg, what are their velocities at the bottom (I.e. which one arrives first?). The diameter of the ball is 0.4 m. Block: [½mv2+mgh]initial= [½mv2+mgh]final 1*9.8*1 = 0.5*1*v2 so v=4.4 m/s Ball: [½mv2+mgh+½I2]initial= [½mv2+mgh+½I2]final I=0.4*MR2=0.064 kgm2 and =v/R=2.5v 1*9.8*1 = 0.5*1*v2+0.5*0.064*(2.5v)2 so v=3.7 m/s Part of the energy goes to the rotation: slower! PHY 231

  9. Rotational kinetic energy KEr=½I2 Conservation of energy for rotating object: [PE+KEt+KEr]initial= [PE+KEt+KEr]final Example. 1m PHY 231

  10. Angular momentum Conservation of angular momentum If the net torque equals zero, the angular momentum L does not change Iii=Iff PHY 231

  11. Conservation laws: • In a closed system: • Conservation of energy E • Conservation of linear momentum p • Conservation of angular momentum L PHY 231

  12. Neutron star Sun: radius: 7*105 km Supernova explosion Neutron star: radius: 10 km Isphere=2/5MR2 (assume no mass is lost) sun=2/(25 days)=2.9*10-6 rad/s Conservation of angular momentum: Isunsun=Insns (7E+05)2*2.9E-06=(10)2*ns ns=1.4E+04 rad/s so period Tns=5.4E-04 s !!!!!! PHY 231

  13. The spinning lecturer… A lecturer (60 kg) is rotating on a platform with =2 rad/s (1 rev/s). He is holding two 1 kg masses 0.8 m away from his body. He then puts the masses close to his body (R=0.0 m). Estimate how fast he will rotate (marm=2.5 kg). PHY 231

  14. ‘2001 a space odyssey’ revisited A spaceship has a radius of 100m and I=5.00E+8 kgm2. 150 people (65 kg pp) live on the rim and the ship rotates such that they feel a ‘gravitational’ force of g. If the crew moves to the center of the ship and only the captain would stay behind, what ‘gravity’ would he feel? PHY 231

  15. L The direction of the rotation axis L The rotation in the horizontal plane is reduced to zero: There must have been a large net torque to accomplish this! (this is why you can ride a bike safely; a wheel wants to keep turning in the same direction.) The conservation of angular momentum not only holds for the magnitude of the angular momentum, but also for its direction. PHY 231

  16. L L Initial: angular momentum: Iwheelwheel Closed system, so L must be conserved. Final: -Iwheel wheel+Ipersonperson person= 2Iwheel wheel Iperson Rotating a bike wheel! A person on a platform that can freely rotate is holding a spinning wheel and then turns the wheel around. What will happen? PHY 231

  17. Demo: defying gravity! PHY 231

  18. Global warming The polar ice caps contain 2.3x1019 kg of ice. If it were all to melt, by how much would the length of a day change? Mearth=6x1024 kg Rearth=6.4x106 m PHY 231

  19. Does not move! 0.2L 0.5L L Two more examples What is the tension in the tendon? 30N 12.5N PHY 231

  20. A top A top has I=4.00x10-4 kgm2. By pulling a rope with constant tension F=5.57 N, it starts to rotate along the axis AA’. What is the angular velocity of the top after the string has been pulled 0.8 m? PHY 231

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