Rotational kinetic energy

244

233

222

211 2

1

2

1

2

1

2

1rot

vmvmvmvmK

2244

2233

2222

2211 2

1

2

1

2

1

2

1rot

rmrmrmrmK

2244

233

222

2112

1rot

rmrmrmrmK

Moment of inertia, I

2

2

1rot

IK

2

2

1rot

IK

where

N

iiirmI

1

2 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.

2

2

1rot

IK 2COM2

1trans

MvK

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

2

2321

23

22

21

MRI

RmmmI

RmRmRmI

Example: Moment of inertia of a thin rod

Mass of each piece: L

xMM

Example: Moment of inertia of a thin rod

Moment of inertia of each piece:

xxL

Mx

L

xMxMI nnn

222

Example: Moment of inertia of a thin rod

Total moment of inertia:

N

nn

N

n

xxL

MII

1

2

1

Example: Moment of inertia of a thin rod

Total moment of inertia:

2

2

2L

Ldxx

L

MI

Example: Moment of inertia of a thin rod

Total moment of inertia:

2

12

1MLI

Moments of inertia have been calculated for many different shapes.

Don’t memorize them! You can always look them up somewhere.

22COMtot

rottranstot

2

1

2

1 IMvK

KKK

Total kinetic energy

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

rottranstot KKK

Rotation around a point which is not the center of mass

2COM

2COMtot

2COM

2COMtot

2COM

2COMtot

2

12

1

2

12

1

2

1

IMrK

IrMK

IMvK

COM2COM IMrI Parallel axis theorem

Which will hit the floor first?

(1) Bare stick

(2) Stick + brick

(3) 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?

2

5

2MRI

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.

http://v.163.com/special/opencourse/classicalmechanics.html

The second half of lecture 19 is all about the Crab pulsar.

MIT Open Course:Classical Mechanics

Walter Lewin

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