Angular Momentum (of a particle) O The angular momentum of a particle, about the reference point O,...
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![Page 1: Angular Momentum (of a particle) O The angular momentum of a particle, about the reference point O, is defined as the vector product of the position, relative.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d215503460f949f5dfa/html5/thumbnails/1.jpg)
Angular Momentum(of a particle)
r
l
p
O
The angular momentum of a particle, about the reference point O, is defined as the vector product of the position, relative to the reference point, and momentum of the particle
prl
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Torque , about the reference point O, due to a force F exerted on a particle, is defined as the vector product of the position relative to the reference point and force
r F
Torque
r
F
O
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Newton's second law V(angular momentum of a particle)
dt
d l
pr
dt
d dt
d
dt
d prp
r
dt
dpr
netFr
net
(In an inertial reference frame) the net torque, exerted on a particle, is equal to the rate of change of its angular momentum
netdt
d
l
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L
Example. Kepler’s second law
rd
L
rdA
The gravitational torque (about the sun) exerted by the sun on the planets is a zero vector.
.constvr
mm2
1
dt
dA
dt
d
2
1 rr
m2
L
![Page 5: Angular Momentum (of a particle) O The angular momentum of a particle, about the reference point O, is defined as the vector product of the position, relative.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d215503460f949f5dfa/html5/thumbnails/5.jpg)
Newton's second law VI(angular momentum of a system)
dt
dL
i
idt
dl
i
i
dt
d l
i
i,net i
i,exti
iint,
ii,ext ext
(In an inertial reference frame) the net external torque, exerted on a system of particles, is equal to the rate of change of its (total) angular momentum
ddt ext
L
![Page 6: Angular Momentum (of a particle) O The angular momentum of a particle, about the reference point O, is defined as the vector product of the position, relative.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d215503460f949f5dfa/html5/thumbnails/6.jpg)
Example. What is the final angular velocity?
a
initial
b
final
i
?
Initial total angular momentum (magnitude)
aL =a amv2 aa m2 2
Final total angular momentum
L mv mb b2
bb = b 2 2
From conservation of angular momentum (zero external torque): ab
b
a
2
2
Puzzle: Total kinetic energy
01m2
m2
2
m2K 2
2
22
2222
tot
a
ab
b
aa
ab
Who performed the work?
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Rigid Body
A system in which the relative position of all particles is time independent is called a rigid body.
A
iv
i
irA
The motion can be considered as a superposition of the translational motion of a point and the rotational motion around the point.
AiAi rvv
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Angular Momentum and Angular Velocity l
L
In general, each component of the total angular momentum depends on all the components of the angular velocity.
i
vrL iii m
iirr
ii m i
i2ii rm irr
r’
i
2ii
i
2i
2ii
iii
2iiz 'rmzrmzzrmL
;
iiiix xzmL ;
iiiiy yzmL
![Page 9: Angular Momentum (of a particle) O The angular momentum of a particle, about the reference point O, is defined as the vector product of the position, relative.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d215503460f949f5dfa/html5/thumbnails/9.jpg)
effect of symmetry
i
2iiz 'rmL
0xzmLi
iiix
0yzmLi
iiiy
Only for object with appropriate symmetry the direction of angular momentum is consistent with the direction of angular velocity of the object
i
2ii 'rmI
is called the moment of inertia (rotational inertia) of the body about the axis of rotation.
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Newton’s Law VII(for rotational motion of a rigid body)
dt
dI
dt
dL
extI
For symmetrical rigid bodies, the angular acceleration is proportional to the net external torque.
extI
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Fixed and Instantaneous Axis of Rotation(Newton’s second law VIII)
F
F
The angular acceleration, of an object rotating about a fixed axis or instantaneous, is proportional to the component, along the axis of rotation, of the net external torque.
Idt
dI
dt
dL ,ext ,extI
torque
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Moment of Inertia(rotational inertia)
A
A
system of particles:
I m rA i ii
'2
continuous body
body
2A dm'rI
r’
dm
ri’
mi
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Example. Moment of inertia of a uniform thin rod
dxL
MxI 2
y0
L
L
0
3
3
x
L
M 2ML3
1y
dx
x
L
cmI
2/L
2/L
2 dxL
Mx
2/L
2/L
3
3
x
L
M 2ML12
1
about an end
about the center
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Example. Moment of inertia of a uniform circle
d
dr
r
circle
2A dmrI
drrd
R
Mr 2
2
0
R
0
2
R
0
2
0
32 drdr
R
M
4
R
R
M2
4
22MR
2
1
![Page 15: Angular Momentum (of a particle) O The angular momentum of a particle, about the reference point O, is defined as the vector product of the position, relative.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d215503460f949f5dfa/html5/thumbnails/15.jpg)
Parallel - axis theorem
AC
dmr
'r
D
AI body
2dmr body
2dm'rD
bodybody
2
body
2 dm'2dm'rdmD rD
0D
2IMD C2
If the moment of inertia of a rigid body about an axis through the center of mass is IC, then the moment of inertia, about a parallel axis separated by distance D from the axis that passes through the center of mass, is given by
IA = MD2 + Ic
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Center of a force
If a certain body exerts a force on several particles of a given system, the center of the force is defined by position such that for any point of reference
r f r fcf i
ii i
i
lift
weight
buoyancy
lift
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Example. Center of gravity
iW
ir
i
ii m gr
gr
i
iim
gr
cmM gr
Mcm
The center of gravity in a uniform gravitational field is at the center of mass.
Note: Not applicable to a nonuniform gravitational field
gravitationaltorque
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Equilibrium of a rigid body
A rigid object is in equilibrium, if and only if the following conditions are satisfied:
(a) the net external force is a zero vector,
(b) the net external torque is a zero vector.
AF1
F2
F3O
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Rotational kinetic energy
The total kinetic energy of a system rotating about the point of reference is called the rotational (kinetic) energy
K,o = Ki,o
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rotational energy and angular velocity
i
i,oo, KK i
2ii 'rm
2
1
i
2iivm
2
1
2
i
2ii 'rm
2
1 2I2
1
The rotational kinetic energy is related to the magnitude of angular velocity and the moment of inertia of the body
2o,o, I
2
1K
![Page 21: Angular Momentum (of a particle) O The angular momentum of a particle, about the reference point O, is defined as the vector product of the position, relative.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d215503460f949f5dfa/html5/thumbnails/21.jpg)
Total Kinetic Energy of a Rigid Body
i
2iitot vm
2
1K
i
2iAim
2
1rv
i
iAii
2ii
i
2Ai 2m
2
1m
2
1vm
2
1rvr
i
iiA
22i
ii
2A
ii m'rm
2
1vm
2
1rv
If the center of mass is at point A
2cmMv
2
1 2cm,I
2
1 0totK TK cm,K
![Page 22: Angular Momentum (of a particle) O The angular momentum of a particle, about the reference point O, is defined as the vector product of the position, relative.](https://reader030.fdocuments.net/reader030/viewer/2022032703/56649d215503460f949f5dfa/html5/thumbnails/22.jpg)
work and power in rotational motion
rd
F
d
d
dW rdF rF
d
Fr
d
d
ddW
The differential work in a rotational motion depends on the torque about the point of rotation
The power delivered to a rigid body depends on the applied torque and the angular velocity of the body
P
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Transformation of torque
AB
iF
i,A
i,B
i,A
Ar
Br
ii,A Fr
ii,BAB Fr
ii,BiAB FrF
iAB F
i,B
BtotA AB
F
conclusion (total force)
If the total force applied to a body is zero, the torque of this force about any point has the same value.
F
-F
d
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torque transmission
F