Rotational kinetic energy

Moment of inertia, I

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

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.

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

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

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

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

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

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

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

Total kinetic energy

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?

Rotation around a point not at the center of mass

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

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

Crab nebula

The Crab pulsar, flashing 30 times a second.

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

The Crab pulsar spins with a period of 33 ms. Its radius is about 25 km. Its mass is roughly 4 x 1030 kg (around twice the mass of the Sun!). How much rotational kinetic energy does it have?

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?

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.

Want to know more?... MIT Open Course: Classical Mechanics Walter Lewin http://v.163.com/special/opencourse/classicalmechanics.html The second half of lecture 19 is all about the Crab pulsar.