Dynamics of Rotational MotionThe main problem of dynamics: How a net force affects(i) translational (linear) motion (Newtons’ 2nd law)

(ii) rotational motion ??? (iii) combination of translational and rotational motions ???

amF

m F

αz

Axis ofrotation

a

Lever arm l is the distance between theline of action and the axis of rotation,measured on a line that is to both.

Definition of torque: Fr

A torque applied to a door

Units: [ τ ] = newton·meter = N·m

τz > 0 if the force acts counterclockwiseτz < 0 if the force acts clockwise

Newton’s Second Law for Rotation about a Fixed Axis

(i) One particle moving on a circle: Ftan=matan and atan= rαz

rFtan= mr2 αz τz = I αz

Only Ftan contributes to the torque τz .

τz I(ii) Rigid body (composed of many particles m1, m2, …)

zzzi i

iiiz Irm

2

Only externaltorques(forces)count !

Example 10.3: a1x,T1,T2? a1x

a2yy

Pulley: T2R-T1R=Iαz

a1x=a2y=Rαz

X

T2-T1=(I/R2)a1x

Glider: T1=m1a1x

Object: m2g-T2=m2a1x

221

212

221

211

221

21

/

)(

/

/

RImm

gmMmT

RImm

gmmT

RImm

gma x

Work-Energy Theorem and Power in Rotational MotionRotational work:

2

1

tantan

dW

dRdFdsFdW

zR

zR

zzRz WconstFor )( 12

Work-Energy Theorem for Rigid-Body Rotation:

122

122

2

12

1

2

1RRzzR KKIIdIW

Power for rotational work or energy change:

)( vFPofganaloPdt

d

dt

dWP zzRz

RR

zzz

z dIdt

ddId Proof:

Rigid-Body Rotation about a Moving Axis

22

2

1

2

1 cmcm IMvK Proof:

General Theorem: Motion of a rigid body is alwaysa combination of translation of the center of mass and rotation about the center of mass.

iiicm

ii

icmi i

icmiii

rmvm

vvvvmvmK

222

222

'2

1

2

1

)'2'(2

1

2

1

Rolling withoutslipping: vcm= Rω, ax = R αz

Energy:

General Work-Energy Theorem:

E – E0 = Wnc , E = K + U

i

cmii

ii

dt

rdrm

dt

d

vmncesi

0'

'

'

Rolling Motion Rolling Friction

Sliding and deformation of a tire also cause rolling friction.

Combined Translation and Rotation: Dynamics

i

zcmizi

cmi IandaMF

Note: The last equation is valid only if the axis through the center of mass is an axis of symmetry and does not change direction.

Exam Example 24: Yo-Yo has Icm=MR2/2 and

rolls down with ay=Rαz (examples 10.4, 10.6; problems 10.20, 10.75)

Find: (a) ay, (b) vcm, (c) T Mg-T=May τz=TR=Icmαz

ay=2g/3 , T=Mg/3

ay

3

42

gyayvcm

y

Exam Example 25: Race of Rolling Bodies (examples 10.5, 10.7; problem 10.22, problem 10.29)

β v a

Data: Icm=cMR2, h, βFind: v, a, t, and min μs

preventing from slipping

y

xSolution 1: Conservation of Energy Solution 2: Dynamics(Newton’s 2nd law) androlling kinematics a=Rαz

RvandcMRIfor

MvcIMvMgh

UKUKUK

/

)1(2

1

2

1

2

1

0,0,

2

222

212211

c

ghv

1

2

x = h / sinβ

v2=2axc

ga

1

sin

g

ch

v

x

v

xt

)1(2

sin

12

fs

c

c

Mg

Mg

c

c

F

fMg

c

cMaMgf

N

sss

1

tan

cos

sin

1minsin

1sin

FN

cMafcMRaIRf

c

gaMafMgF

szcmsz

sx

1

sinsin

c

ghh

c

gaxv

1

2

sin1

sin22

Angular Momentum(i) One particle: sinmvrLvmrprL

Fr

dt

Ldamr

dt

vdmrvm

dt

rd

dt

Ld

0)( vvm

(ii) Any System of Particles: i

i dt

LdLL

It is Newton’s 2nd law for arbitrary rotation.

Note: Only external torques count since .0 terna lin

(iii) Rigid body rotatingaround a symmetry axis:

(nonrigid or rigid bodies)

ILIrmLL zziiiz

2

Unbalanced wheel: torque of friction in bearings.

zzzzz II

dt

dI

dt

dLandconstI

Impulse-Momentum Theorem for Rotation

Principle of Conservation of Angular MomentumTotal angular momentum of a system is constant (conserved),if the net external torque acting on the system is zero:

00

ifconstLdt

Ld

dt

Ld

Example: Angular acceleration due to sudden decrease of the moment of inertia

ff

f IIncesiI

I 000

0

For a body rotating around a symmetry axis:

I1ω1z = I2ω2z

ω0 < ωfOrigin of Principles of Conservation

There are only three general principles of conservation (of energy, momentum, and angular momentum) and they are consequences of the symmetry of space-time(homogeneity of time and space and isotropy of space).

In[28]:= fs_ : NIntegrate1 Sqrt1 x x s x 6 s, x, 0, s;Plot fs, 1, s, 0, 1

Out[29]=

0 .2 0 .4 0 .6 0 .8 1 .0

1 .0

1 .1

1 .2

Hinged board (faster than free fall)

h=L sinα

Mg

m Ball: 02 sin)/2(/22/ gLvhtmvmgh ball

Board: I=(1/3)ML2

0

0

sin

0 02

0

0 0

0

2220

))(sin1(sin6

1

sinsin3

)sin(sin3

2

)/(

322

xx

dx

t

t

d

g

Lt

L

g

dt

d

dtdMLIhhMg

ball

cup

cup

0sin

ball

cup

t

t

00 50Critical3/2

Gyroscopes and Precession

cF

ca

)0(00 L

)(00 L

w

n

Dynamics of precession: dtLd

Precession is a circular motion of the axisdue to spin motion of the flywheel about axis.

Precession angular speed:

I

mgr

LdtL

Ld

dt

d

z

z

Circular motion of the center of mass requiresa centripetal force

Fc = M Ω2 r supplied by the pivot.

Nutation is an up-and-downwobble of flywheel axis that’s superimposed on the

precession motion if Ω ≥ ω.

Period of earth’s precession is 26,000 years.

Analogy between Rotational and Translational Motions

Physical Concept Rotational TranslationalDisplacement θ s

Velocity ω v

Acceleration α a

Cause of acceleration Torque τ Force F

Inertia Moment of inertia I = Σmr2

Mass m

Newton’s second law Στ = I α ΣF = ma

Work τ θ Fs

Kinetic Energy (1/2) Iω2 (1/2) mv2

Momentum L = I ω p = mv