11. Rotational Vectors & Angular Momentum

1. Angular Velocity & Acceleration Vectors

2. Torque & the Vector Cross Product

3. Angular Momentum

4. Conservation of Angular Momentum

5. Gyroscopes & Precession

Earth isn’t quite round.

How does this affect its rotation axis,

and what’s this got to do with ice ages?

(The deviation from roundness is exaggerated.)

Axis precesses with period ~26,000 yr.

Importance of rotation:

• Earth’s rotational axis seasons.

• Angular momentum of protons in living tissues MRI

• Rotating air tornadoes.

• Rotating wheel stabilizes bicycle.

11.1. Angular Velocity & Acceleration Vectors

Right-hand ruleAngular acceleration vector:

0limt t

ω

αd

d t

ω

change direction

// //

11.2. Torque & the Vector Cross Product

τ r

τ F

τ r F

Right hand rule

cross product

sinr F ˆ

ˆ

τ r

τ F

Cross Product

Cross product C of vectors A & B: C A B

sinC A B

Given by right-hand rule,C

Dot product C of vectors A & B: C A B cosA B

Properties of cross product :

1. Distributive

2. Anti-commutative

3. NOT associative

A B C A B A C

A B B A

A A B

A A 0

A A B 0

is a vector in the A-B plane and A.

is a vector A-B plane.

= area of A-B parallelogram .

ˆ ˆ ˆ

x y z

x y z

A A A

B B B

x y z

A B

ˆ ˆ ˆy z z y z x x z x y y xA B A B A B A B A B A B x y z

ˆ ˆ ˆ

0

0x y

x y

A A

B B

x y z

A B ˆx y y xA B A B z

GOT IT? 11.1.

Which numbered torque vector goes with each pair of force-radius vectors?

Neglect magnitudes.

123 4

Note: Wolfson gave 6 as the answer to (b).

11.3. Angular Momentum

Linear momentum: mp v

Angular momentum: L r p m r v

I ω rigid body with axis of rotation along principal axis

particle

I ω general case, I a tensor. L & can have different directions.

Example 11.1. Single Particle

A particle of mass m moves CCW at speed v around a circle of radius r in the x-y

plane.

Find its angular momentum about the center of the circle,

express the answer in terms of its angular velocity.

m L r v

ˆm r v k

2 ˆm r k

2m r ω

I ω2I m r

Torque & Angular Momentum

ii

L LSystem of particles: i ii

r p

i

idd

dt dt LL

i ii i

i

d d

dt dt

r p

p r

ii

i

d

dt p

r 0ii i i

dm

dt

rp v v

i ii

r F ii

τ

d

dt

Lτ rotational analog of 2nd law.

11.4. Conservation of Angular Momentum

Example 11.2. Pulsars

A star rotates once every 45 days.

It then undergoes supernova explosion, hurling most of its mass into space.

The inner core of the star, whose radius is initially 20 Mm,

collapses into a neutron star only 6 km in radius.

The rotating neutron star emits regular pulses of radio waves, making it a pulsar.

Calculate the pulse rate ( = rotation rate ).

Assume core to be a uniform sphere & no external torque.

20 0 0

2

5L m r

20

02

r

r

Before collapse:

22

5L m r After

collapse:

0L L

23

2

20 10 1/

456

kmrev day

km

52.5 10 /rev day 3 /rev s

Conceptual Example 11.1. Playground

A merry-go-round is rotating freely when a boy runs straight toward the center & leaps on.

Later, a girl runs tangentially in the same direction as the merry-go-round also leaps on.

Does the merry-go-round’s speed increase, decrease, or stays the same in each case?

Lb = 0 L = 0

I = Im + Ib

BoyGirl

L = Lg

I = Im + Ig

?

Making the Connection

A merry-go-round of radius R = 1.3 m has rotational inertia I = 240 kg m2

& is rotating freely at 1 = 11 rpm.

A boy of mass mb = 28 kg runs straight toward the center at vb = 2.5 m/s & leaps on.

At the same time, a girl of mass mg = 32 kg, running tangentially at speed vg = 3.7 m/s

in the same direction as the merry-go-round also leaps on.

Find the new angular speed 2 once both children are seated on the rim.

0 1 g gL I m R v

1

2 2

g g

b g

I m R v

I m m R

Before :

2 22 2 2b gL I m R m R After :

0L L

2

2 22

1240 11 32 1.3 3.7 / / 60 / min

2

240 28 32 1.3

kg m rpm kg m m s rev rad s

kg m kg kg m

12 rpm

Demonstration of Conservation of Angular Momentum

GOT IT? 11.2.

If you step on a non-rotating table holding a non-rotating wheel.

(a)if you spin the wheel CCW as viewed from above, which way do you rotate?

(b)If you then turn the wheel upside down, will your rotation rate increase, decrease, or

remain the same?

What about your direction of rotation?

(a) CW to keep L = 0.

(b) Same, CCW.

11.5. Gyroscopes & Precession

Gyroscope: spinning object whose rotational axis is fixed in space.

External torque required to change axis of rotation

Higher spin rate larger L harder to change orientation

Usage:

• Navigation

• Missile & submarine guidance.

• Cruise ships stabilization.

• Space-based telescope like Hubble.

Precession

Precession: Continuous change of direction of rotation axis,

which traces out a circle.

d

dt

Lτ g r F

r L L L

Rate of Precession

Precession occurs if L.

ˆˆsin

z L

L precesses CCW around z.

ˆˆˆ

sin

d

d t L

L

z LFor L constant:

ˆ sin cos , sin sin , cos L ˆˆ sin sin , sin cos , 0 z L

const

sin sin , sin cos , 0 sin , cos , 0L

sinL

Rate of precession :

x

y

z

ˆd

mgd t

L

τ r z sinrmg ˆr L

Earth’s precession (period ~ 26,000 y )

The equatorial bulge is highly exaggerated.

Erath’s Precession

The equatorial bulge is highly exaggerated.

= 0

Perfect sphere

Oblate spheroid

<

= 0

GOT IT? 11.3.

You push horizontally at right angles to the shaft of a spinning gyroscope.

Does the shaft move

(a)upward,

(b)downward,

(c)in the direction you push,

(d)opposite the direction you push?

Bicycling

g τ r F points into paper

L // wheel turns to biker’s left

L

Direction of bike’s motion

wheel

L+ t

Looking down at bike.

Biker leans

Wheel turns