1. Copyright © 2012 Pearson Education Inc. Rotation Physics 7C lecture 10 Thursday October 31, 8:00 AM – 9:20 AM Engineering Hall 1200

2. Copyright © 2012 Pearson Education Inc. External forces and center-of-mass motion When a body or collection of particles is acted upon by external forces, the center of mass moves as though all the mass were concentrated there.

3. Copyright © 2012 Pearson Education Inc. External forces and center-of-mass motion Fragments of a firework shell would fly at 100 m/s for 5 seconds before they burn out. If a shell reaches its max height of 1000 meter and explodes, are the audiences on the ground safe from burning fragments? Ignore air resistance.

4. Copyright © 2012 Pearson Education Inc. External forces and center-of-mass motion Fragments of a firework shell would fly at 100 m/s for 5 seconds before they burn out. If a shell reaches its max height of 1000 meter and explodes, are the audiences on the ground safe from burning fragments? Ignore air resistance. motion of center of mass: motion of fragments relative to center of mass:

5. Copyright © 2012 Pearson Education Inc. Rocket propulsion As a rocket burns fuel, its mass decreases, as shown in Figure below. What is the speed of rocket if we know the exhaust speed v ex, burning rate λ=dm/dt and initial mass m 0 ?

6. Copyright © 2012 Pearson Education Inc. Rocket propulsion between time t and t + dt, according to momentum conservation: (m+dm) v = m (v+dv) + dm (v-v ex ) What is the speed of rocket if we know the exhaust speed v ex, burning rate λ=dm/dt and initial mass m 0 ?

7. Copyright © 2012 Pearson Education Inc. Rocket propulsion between time t and t + dt, according to momentum conservation: (m+dm) v = m (v+dv) + dm (v-v ex ) m dv – v dm + (v-v ex ) dm = 0 (m 0 - λ t) dv –v ex λ dt = 0 dv - λ v ex dt /(m 0 - λ t)= 0 v + v ex ln(m 0 - λ t) = constant v = v 0 + v ex ln (m 0 /(m 0 - λ t)) = v 0 + v ex ln (m 0 /m) What is the speed of rocket if we know the exhaust speed v ex, burning rate λ=dm/dt and initial mass m 0 ?

8. Copyright © 2012 Pearson Education Inc. Introduction The north star is Polaris today, but 5000 years ago it was Thuban. What caused the change? What causes bodies to start or stop spinning? We’ll introduce some new concepts, such as torque and angular momentum, to deepen our understanding of rotational motion.

9. Copyright © 2012 Pearson Education Inc. Introduction How do we quantify the spinning of wind turbine?

10. Copyright © 2012 Pearson Education Inc. Angular displacement angular displacement: θ unit: radian direction: (right hand rule!)

11. Copyright © 2012 Pearson Education Inc. Radian 2 π radian = 360 degree

12. Copyright © 2012 Pearson Education Inc. Angular displacement Motion of a spinning wheel

13. Copyright © 2012 Pearson Education Inc. Angular displacement and velocity How do we quantify the spinning of wind turbine? ω = dθ /dt

14. Copyright © 2012 Pearson Education Inc. Angular velocity is a vector! signs of angular displacement

15. Copyright © 2012 Pearson Education Inc. Right hand rule Angular displacement is a vector, use right hand rule to determine the direction.

16. Copyright © 2012 Pearson Education Inc. Angular acceleration α = dω/dt

17. Copyright © 2012 Pearson Education Inc. Angular acceleration calculate ω from α

18. Copyright © 2012 Pearson Education Inc. These are very similar to linear motion linear and angular motion:

19. Copyright © 2012 Pearson Education Inc. Linear vs. angular motion v =?

20. Copyright © 2012 Pearson Education Inc. Linear vs. angular motion what is the acceleration?

21. Copyright © 2012 Pearson Education Inc. Linear vs. angular motion radian vs. degree

22. Copyright © 2012 Pearson Education Inc. Example calculate the acceleration of the black point in the disk.

23. Copyright © 2012 Pearson Education Inc. Example calculate the acceleration of the black point in the disk.

24. Copyright © 2012 Pearson Education Inc. Example calculate the acceleration of the black point in the disk.

25. Copyright © 2012 Pearson Education Inc. Speed of propeller calculate the speed of the tip of the propeller.

26. Copyright © 2012 Pearson Education Inc. Speed of propeller calculate the speed of the tip of the propeller.