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