1. Rotational kinetic energy

5. Moment of inertia, I

7. where
is the moment of inertia (惯性矩 ), which depends on the distribution of mass in the object.

8. Translational motion:
The higher an object’s mass, the more work you must do to increase its linear speed.
Rotational motion:
The higher an object’s moment of inertia, the more work you must do to increase its angular speed.

9. Example: Moment of inertia of a bicycle wheel
Assume that all the atoms in the wheel are the same distance R from the center.
Let m be the mass of each atom. Then

10. Example: Moment of inertia of a thin rod
Mass of each piece:

11. Moment of inertia of each piece:
Example: Moment of inertia of a thin rod
Moment of inertia of each piece:

12. Total moment of inertia:
Example: Moment of inertia of a thin rod
Total moment of inertia:

13. Total moment of inertia:
Example: Moment of inertia of a thin rod
Total moment of inertia:

14. Total moment of inertia:
Example: Moment of inertia of a thin rod
Total moment of inertia:

15. Moments of inertia have been calculated for many different shapes.
Don’t memorize them! You can always look them up somewhere.

16. Total kinetic energy

17. A disk and hoop have the same mass and the same radius.
Example: Downhill race between a disk (圆盘) and a hoop (圈)
A disk and hoop have the same mass and the same radius.
They start rolling from rest, at the same time.
Which goes faster?

18. Rotation around a point not at the center of mass

19. Rotation around a point which is not the center of mass
Parallel axis theorem

20. Which will hit the floor first?
Bare stick
Stick + brick
Both at the same time

21. Crab nebula

22. The Crab pulsar, flashing 30 times a second.

23. X-ray image of the Crab pulsar
In the center is a spinning neutron star (中子星)

24. The Crab pulsar spins with a period of 33 ms.
Its radius is about 25 km.
Its mass is roughly x 1030 kg (around twice the mass of the Sun!).
How much rotational kinetic energy does it have?

25. Answer: 2 x 1043 J.
The Sun generates about 1 x 1034 J of energy in one year.
For how many years could the rotational energy in the neutron star power the Sun?

26. Where does the energy for these powerful X-rays come from?
From the rotational kinetic energy of the neutron star.
We know this because the spinning rate is slowing down by 38 ns every day.

27. Want to know more?... MIT Open Course: Classical Mechanics
Walter Lewin
The second half of lecture 19 is all about the Crab pulsar.