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)
