Angular Momentum

Evidence of first pattern

• Person in spinning chair (demo with bricks)

• Rubber Stopper (demo from centripetal force lab)

• Playground low tech Merry Go Round (video side F Chapter 18)

• Ice Skater (your memory)

• Acrobat (transparency)

Informal Statement offirst pattern:

• As _ decreases, _ increases.

Informal Statement offirst pattern:

• As R decreases, _ increases.

Informal Statement offirst pattern:

• As R decreases, increases.

Evidence of Second Pattern

• A _ _ _ _ _ _ _ _ top is stable.

• A _ _ _ _ _ _ _ _ bike wheel is stable.

• A gyroscope is stable when it is _ _ _ _ _ _ _ _ .

Evidence of Second Pattern

• A spinning top is stable.

• A _ _ _ _ _ _ _ _ bike wheel is stable.

• A gyroscope is stable when it is _ _ _ _ _ _ _ _.

Evidence of Second Pattern

• A spinning top is stable.

• A spinning bike wheel is stable.

• A gyroscope is stable when it is _ _ _ _ _ _ _ _ .

Evidence of Second Pattern

• A spinning top is stable.

• A spinning bike wheel is stable.

• A gyroscope is stable when it is spinning.

Informal Statement ofsecond pattern:

• Spinning things are more _ _ _ _ _ _ than non-spinning things.

• It is tougher to change the direction of spinning things.

Informal Statement ofsecond pattern:

• Spinning things are more stable than non-spinning things.

• It is tougher to change the direction of spinning things.

Formal statement(includes both patterns)

• Angular momentum: L = I. [ Recall: P = mv ]

• If = 0 (closed system), then L is constant.

[ Recall: If F = 0, then P is constant. ]

“Conservation of Angular Momentum”

• What are the units for L?

Torque and Angular Momentum

(Recall: A force can change linear momentum.)

• A Torque can change angular momentum.

(Recall: F = P / T or F · T = P)

• = L / T

For the same , the change of the L is less noticeable if the L is large, so

• Xzylo (or paper airplane shaped like a pipe)• Throw a football with a spiral.• Bikes are most stable when moving fast.• A spinning basketball can be balanced on a finger.• Tops are stable when spinning.• Gyroscopes tend to stay lined up.

Example One:Centripetal Force Apparatus

• Draw the system from the side and from the top (show the radius in both drawings).

• L = I = Mr2

• LBEFORE = LAFTER

• MR2 = Mr2

Example Two: Playground Merry Go Round

• The person (40 kg) starts at the edge, and moves to 0.5 m from the center.

• The disk is 100 kg.

• The radius of the disk is 2.0 m.

• Initial speed is 1 rad/s

• Final speed = ?

Solve for final angular speed.Lo = L´

person + disk = person + disk

mR2 + (1/2)MR2 = mR’2´ + (1/2)MR2´

(40)221 + (1/2)(100)(2)21

=

(40)(0.5)2´ + (1/2)(100)(2)2´

´ = 1.7 rad/s (faster than before)

Closing Demonstration

• Hold spinning bicycle wheel while standing on a table that can spin.

• The total angular momentum of the system is a constant.

• If the person changes the L of the wheel, then the L of the person must change!!!

Rotational Energy

• Everyone would guess that a spinning object has energy, even if it’s not getting anywhere.

• Kinetic or Potential?

• How much? [It can’t be KE = (1/2)Mv2 , because it’s not getting anywhere.]

Build the Equation by Analogy

• Mass goes to ___ .

Build the Equation by Analogy

• Mass goes to I (rotational inertia).

Build the Equation by Analogy

• Mass goes to I.

• Speed (v) goes to ___ .

Build the Equation by Analogy

• Mass goes to I.

• Speed (v) goes to (rotational speed).

Build the Equation by Analogy

• Mass goes to I.

• Speed (v) goes to (rotational speed).

• KE = (1/2)Mv2 goes to KE = _____.

Build the Equation by Analogy

• Mass goes to I.

• Speed (v) goes to (rotational speed).

• KE = (1/2)Mv2 goes to KE = (1/2)I2

Example: A Compact Disc (CD)

• How much KE does it have when it’s spinning?

• KE = (1/2)I2

• So, what’s I and what’s ?

• Moment of Inertia for a disk = …

• I = (1/2)mr2

• Mass = 16 grams (= 0.016 kg)

• Radius = 6 cm (= 0.06 meter)

• I = (1/2)mr2 = (1/2)(.016)(.06)2 = 0.000029 kg•m2

What else do we need?

• Get this: The disc player needs information at a constant rate, so the angular speed needs to vary!

• = {144 rotations/min 240 RPM}

• = {2.4 rotations/sec 4.0 RPS}

• = {15 radians/s 25 rad/s}

• So, the average is about 20 rad/s

Finish Up: KE = (1/2)I2

• KE = (1/2)(.000029)(20)2

• KE = 0.0058 Joules

What is the ‘take-away’?Just like mgh, (1/2)Mv2 , and (1/2)kx2 … rotational energy needs to be _ _ _ _ _ _ _ _ when we use an

equation using energy:

W = E E2 = E1

What is the ‘take-away’?Just like mgh, (1/2)Mv2 , and (1/2)kx2 …

rotational energy needs to be included when we use an equation using energy:

W = E E2 = E1

‘Menu’ for the Review Game

Fx = 0 Fy = 0 = 0

Fx = Max Fy = May = I

I = MR2 Lo = L´ E2 = E1