Rotational kinetic energy
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Transcript of Rotational kinetic energy
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
Want to know more?...