Rotational Motion

(1) Kinematics

Everything’s analogous to linear kinematics

Define angular properties properly

and derive the equations of motionby analogy

Motion on a Wheel

Computer Hard Drive

A computer hard drive typically rotates at 5400

rev/minute

Find the: •

Angular Velocity in rad/sec

•

Linear Velocity on the rim (R=3.0cm)•

Linear Acceleration

It takes 3.6 sec to go from rest to 5400 rev/min, with constant angular acceleration.

•

What is the angular acceleration?

Computer Hard Drive

Numbers worked out:

Examples

Consider two points on a rotating wheel. One on the inside (P) and the other at the end (b):

•

Which has greater angular velocity?

•

Which has greater linear velocity?

bR1R2

Rotation and

Translation

Rolling without Slipping

•

In reality, car tires both rotate and translate

•

They are a good example of something which rolls (translates, moves forward, rotates) without slipping

•

Is there friction? What kind?

Derivation

•

The trick is to pick your reference frame correctly!

•

Think of the wheel as sitting still and the ground moving past it with speed V.

Velocity of ground (in bike frame) = -ωR

=> Velocity of bike (in ground frame) = ωR

Bicycle comes to RestA bicycle with initial linear velocity V0

decelerates uniformly (without slipping) to rest over a distance d. For a wheel of radius R:

a)

What is the angular velocity at t0

=0?b)

Total revolutions before it stops?

c)

Total angular distance traversed by wheel?

(d) The angular acceleration?(e) The total time until it stops?

Torque vs. Force

•

Torque: rot. Force•

Remember: a ∝

F

•

α ∝ ?

•

α ∝ F•

α ∝ R⊥

•

α ∝ τ = R⊥

F

Torque –

More general

•

τ

= R⊥

F

•

τ

= R F⊥

•

τ

= R F sinθ

Torque –

More general

•

+: clockwise•

Two Torques, opposite

•

τ

= -R1

F1

+ R2

F2

sin 60°

A better way to define Torque

This gives us the magnitudeand the direction

FRrrr

×= τ

Vector Cross Product

A B vs.B A

A A

:CheckB" intoA Swing"

Rule-Hand-Right fromDirection SinB AC

B A C

rrrr

rr

rrr

××

×

Θ=

×=

Example of Cross ProductThe location of a body is length r from the origin and at an angle θ from the x-

axis. A force F

acts on the body purely in the y

direction.

What is the Torque on the body?

z

x

yθ

Rotational Dynamics

•

What plays the role of mass in rotation?

•

F = ma = mRα•

τ

= R F = mR2α

•

Rotational inertia: mR2

•

Στi

= (Σmi

Ri2) α

•

I = Σmi

Ri2

•

Στ

= I α•

(Στ)CM

= ICM

αCM

Calculating Moments of Inertia

( ) 233

2

121

24243

31 2

2

2

2

Mllll

Ml

MdRRl

MI Rl

l

l

l=⎟⎟

⎠

⎞⎜⎜⎝

⎛+===

−−∫

∫= dmRI 2 dRl

MdRdm l == ρ

A few helpful theorems

•

Parallel Axis TheoremI = ICM

+ M h2

•

Perpendicular Axis TheoremIz

= Ix

+ IyOnly valid for flat object!

Angular Momentum

Angular Momentum

MomentumL = Iω

p = mv

Στ

= Iα

= dL/dt

ΣF = ma = dp/dt

Στ=0 ⇒ L=const.

ΣF=0 ⇒ p=const.

Total Angular Momentum is conserved if Στ=0.

Note: L

= I ω, Angular Momentum is a vector

Rotating Kinetic Energy

•

K = Σ(1/2mi

vi2) = Σ(1/2 mi

Ri2 ω2)

= ½

Σ(mi

Ri2) ω2

= ½

I ω2

•

Rotational Kinetic Energy: ½

I ω2

•

W= F dl= F⊥Rdθ= τdθ

•

W=1/2 I ω22

-

1/2 I ω1

2

Rotation and Translation

•

Translation: K = ½

mv2

•

Rotation: K = ½

Iω2

•

Both (e.g. rolling): –

K = ½

mvCM

2

+ ½

Iω2

this is what we did before

Atwood’s Machine Revisited

A pulley with a fixed center (at point O), radius R0

and moment of inertia I,

has a

massless rope wrapped around it (no slipping). The rope has two masses, m1

and m2 attached to its ends. Assume m2

>m1

This is

what we

looked

at be

fore:

Now:

A pulley with a fixed center (at point O), radius R0

and moment of inertia I,

has a

massless rope wrapped around it (no slipping). The rope has two masses, m1

and m2 attached to its ends. Assume m2

>m1

Or :

Vector, Right Hand Rule

Why does the Bicycle Wheel Turn to the Right?

Angular Momentum

dtLdI

dtLd

dtLd

dtId

dtdII

rrr

rrrrrr

==

=====

∑

∑

ατ

ωωατ

)()(

ωrr

IL =Newton’s Law for rotational motion:

Remember,

This gives us the magnitudeand the direction of Torque

FRrrr

×= τ

Angular Momentum

p r rrr×=L

Angular Motion of a ParticleDetermine the angular momentum, L, of a particle,with mass m

and

speed v, moving in uniform circular motion with radius r.

Conservation of Angular Momentum

ωrr

IL =

Const 0 if

=→=

=∑L

dtLd

τ

τr

r

Man on a Disk

A person with mass m

stands on the

edge of a disk with radius R

and

moment ½MR2. Neither is moving. The person then starts moving on the disk with speed V.Find the angular velocity of the disk.

A bullet strikes a cylinderA bullet of speed V

and mass m

strikes a solid cylinder of mass M

and inertia ½MR2,

at radius R

and sticks. The cylinder is anchored at point 0 and is initially at rest.

What is ω of the system after the collision?

Is energy Conserved?

Kepler’s

2nd

Law2nd

Law:

Each

planet moves so that an imaginary line drawn from the Sun to the planet sweeps out area in equal periods of time.

Static Equilibrium

This is what we are familiar with:

Fnet

= 01D 2D

This is what we need to look at:

τnet

= 0

Crane example:

??

Another example:

Good example that requires consideration of both forces and torques

Center of Gravity –

remember CM !

Hook’s law –

same as before!

F

= -kΔL