Second Law of Rotation

• Newton’s First Law of Rotation• Newton’s Second Law of Rotation• Rotational Inertia / Moment of Inertia• Rotational Inertia of extended objects• Examples

– Potter’s wheel

– Pulley and rope

– Pulley and bucket

– Atwood’s machine

Newton’s First Law of Rotation

• Translational Equilibrium

• Rotational Equilibrium

• Both required for equilibrium (Ch 9)

• Seesaw example

Newton’s Second Law of Rotation

• Derive for one point:

• Newton’s Second Law for translation

• Newton’s Second Law for Rotation

r

F

Rotational 2nd Law for many points

• For collection of objects (or extended object)

– Internal torques redistribute to individual particles

r1

m1

m2

r2

Rotational Inertia

• “Moment of Inertia” or “Rotational Inertia”

• Keeps cropping up in all equations!

• Like “Rotational Mass”

• Product of mass and how its distributed around axis

• Units Kg m2

• Simple for point object, requires integral calculus for extended objects.

Rotational Inertia for Disk• For point or hollow ring

• For disk or cylinder

• Integrate from 0 to R

• Put in density

Rotational Inertia for various objects

• Don’t memorize – If you need these we’ll provide them

Summary - Rotational Inertia for objects

• For a point object or a ring

– all mass concentrated at one radius

• For any other object

– mass distributed at different radii

– Don’t worry, we’ll give you the fraction

Example – Potter’s wheel• Torque

• Angular acceleration

• Time

Example – pulley and rope

• Torque from tension

• Total Torque

• Angular acceleration If wheel was uniform disk

• Moment of inertia

Example – pulley and bucket

• Rotation 2nd Law for pulley

• Translation 2nd Law for bucket

(note sign change to sync direction)

• Connection between translational and angular acceleration

pulley and bucket (cont)

• Rotation and Translation 2nd Law

• Eliminating FT

• Rearranging

• Solving for α

pulley and bucket (cont)• Putting in numbers

• Acceleration of bucket

Review - Atwood’s Machine• Example 4-13

• Free body diagram for each block

• Tensions same by 3rd law

• Accelerations locked together (- sign)

• Equation of motion for each

• Solve for acceleration

• Solve for tension

• But what about Pulley?

Atwood’s Machine with Pulley• Tensions can now be different

• 2nd Law for m1

• 2nd Law for m2 (reverse direction)

• 2nd Law for pulley

• Linear acceleration of blocks and angular acceleration of pulley synchronized!

Atwood’s Machine with Pulley (2)• Substitute T2 and T1 in pulley equation

• Write a in terms of α

• Rearrange

Atwood’s Machine with Pulley (3)• Angular acceleration of pulley

• Linear acceleration of blocks

• Note if I = 0

(same as before!)

Atwood’s Machine with Pulley (4)

• Velocity after blocks have moved distance h

(we’ll need this when we

do it by energy)