Rotational Energy and Momentum

• Rotational contribution to Energy• Examples

– Rolling down incline– Atwood’s Machine again

• Rotational Energy vs. Angular Momentum• Angular Momentum• Conservation of Angular Momentum• Examples

– Colliding disks– One object changing Rotational Inertia– Colliding point objects and disks

Rotational energy contribution

• For rotating collection of objects (or extended object)

• For translating and rotating object

• Both just get added together, as one big scalar!

r1

m1

m2

r2

Example - Rotational energy contribution

• Conservation of energy

•For rolling motion

Procession of Objects

• For objects rolling down incline plane

• (reminds me of)

Higher moments of inertia

Review - Atwood’s Machine with Pulley

• Velocity after blocks have moved distance h (2nd Law)

(we’ll need this when we

do it by energy)

Atwood’s Machine with Pulley (Energy)• Initial energy - m2 up, m1 down

• Final energy - m1 up, m2 down

• Equating (no external work done)

same as before

Rotational Energy and Angular Momentum

• Rotational Energy– Scalar contribution to TOTAL energy - thrown in with everything else!

–

– Usually no potential energy ½ Iω2 (torsion spring).

– Angular and linear velocity related for rolling motion.

• Angular Momentum– Vector quantity SEPARATE from linear momentum.

– Conserved in:

1. rotational collisions between extended objects.

2. rotational inertia changes (no linear analogy)

3. rotational collisions between extended objects and point masses.

Angular momentum• Rotational analog of Impulse-Momentum

• Collision of two rotating objects

• 3 Types of angular collisions1. rotational collisions between extended objects. (phonograph)

2. rotational inertia changes (figure skater)

3. rotational collisions between point mass and extended objects. (“Gravity”)

Type 1 - Two identical disks

• Problem 59 – A nonrotating cylindrical disk of moment of inertia I is dropped onto an identical

disk rotating at angular speed ω. What is final common angular speed?

Type 1 – Disk and rod• Problem 60

– A uniform disk turns at 2.4 rev/s around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped on to the disk with their centers superimposed. What is final common angular speed in rev/s?

Conversion to angular velocity cancels

Type 2 – Person on Platform

• Angular momentum of single object

• Since conversion to radians cancels

Type 2 – Figure Skater

• Initial, final angular velocities

• Angular momentum of single object

Pull your arms in close!

Type 2 – Person on Stool• Victims Volunteers anyone?• Estimate

– 65 kg person– .15 m radius– .9 m outstretched arms– 2 2-kg masses at arms length

• Tucked in

• Spread out

Type 2 – Example 8-15

• Angular momentum of object

Type 2 – Walking on Merry-Go Round

• Conservation of angular momentum

• Energy before and after

Work done on him as he walked to outside edge

Type 3 – Linear collision with Disk

• Conservation of angular momentum

• What happens when they step off?

No change. The change occurs when they combine with difference angular velocities, not when they separate with the same angular velocity

(same with railroad cars Decoupled)

Type 3 – Collision with Asteroid

• Conservation of angular momentum

• Neglect asteroid Iast after it becomes embedded

• Percent change

• For the asteroid

Collision with Asteroid (cont)• Percent change

• For asteroid

• Simplify

Asteroid =100 tonne

Angular Momentum - Vector

• Bicycle Wheel• Stability of Bicycle• Gyroscope• Crossing stream on narrow log• Polanowski’s story

Summary - Translation vs. Rotation

Translation Rotation

Translation vs. Rotation (cont)

Translation Rotation