angular momentum (of a particle) o the angular momentum of a particle, about the reference point o,...
TRANSCRIPT
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
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
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
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
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
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?
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
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
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.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
torque transmission
F