Rotation

1. Physics 7C lecture 10 • Rotation Thursday October 31, 8:00 AM – 9:20 AM Engineering Hall 1200

2. 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. 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. 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. 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 vex, burning rate λ=dm/dt and initial mass m0?

6. Rocket propulsion • What is the speed of rocket if we know the exhaust speed vex, burning rate λ=dm/dt and initial mass m0? between time t and t + dt, according to momentum conservation: (m+dm) v = m (v+dv) + dm(v-vex)

7. Rocket propulsion • What is the speed of rocket if we know the exhaust speed vex, burning rate λ=dm/dt and initial mass m0? between time t and t + dt, according to momentum conservation: (m+dm) v = m (v+dv) + dm(v-vex) m dv – v dm+ (v-vex) dm= 0 (m0- λ t) dv –vexλdt = 0 dv - λvex dt /(m0- λ t)= 0 v + vex ln(m0- λ t) = constant v = v0 + vexln (m0/(m0- λ t)) = v0 + vexln (m0/m)

8. 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. Introduction • How do we quantify the spinning of wind turbine?

10. Angular displacement • angular displacement: θ • unit: radian • direction: (right hand rule!)

11. Radian 2 π radian = 360 degree

12. Angular displacement • Motion of a spinning wheel

13. Angular displacement and velocity • How do we quantify the spinning of wind turbine? ω = dθ /dt

14. Angular velocity is a vector! • signs of angular displacement

15. Right hand rule • Angular displacement is a vector, use right hand rule to determine the direction.

16. Angular acceleration • α = dω/dt

17. Angular acceleration • calculate ω from α

18. These are very similar to linear motion • linear and angular motion:

19. Linear vs. angular motion • v =?

20. Linear vs. angular motion • what is the acceleration?

21. Linear vs. angular motion • radian vs. degree

22. Example • calculate the acceleration of the black point in the disk.

23. Example • calculate the acceleration of the black point in the disk.

24. Example • calculate the acceleration of the black point in the disk.

25. Speed of propeller • calculate the speed of the tip of the propeller.

26. Speed of propeller • calculate the speed of the tip of the propeller.