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## Angular Momentum

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**Evidence of first pattern**• Person in spinning chair (demo with bricks) • Rubber Stopper (demofrom centripetal force lab) • Playground low tech Merry Go Round (video side F Chapter 18) • Ice Skater (your memory) • Acrobat (transparency)**Informal Statement offirstpattern:**• As _ decreases, _ increases.**Informal Statement offirstpattern:**• As R decreases, _ increases.**Informal Statement offirstpattern:**• As R decreases, w 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 ofsecondpattern:**• Spinning things are more _ _ _ _ _ _ than non-spinning things. • It is tougher to change the direction of spinning things.**Informal Statement ofsecondpattern:**• 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 = Iw. [ Recall: P = mv ] • If St = 0 (closed system), then L is constant. [ Recall: If SF = 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: SF = DP / T or SF · T = DP) • St = DL / T**For the same t, 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 = Iw = Mr2w • LBEFORE = LAFTER • MR2w = Mr2w**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 mR2w + (1/2)MR2w = mR’2w´ + (1/2)MR2w´ (40)221 + (1/2)(100)(2)21 = (40)(0.5)2w´ + (1/2)(100)(2)2w´ w´ = 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 w (rotational speed).**Build the Equation by Analogy**• Mass goes to I. • Speed (v) goes to w (rotational speed). • KE = (1/2)Mv2 goes to KE = _____.**Build the Equation by Analogy**• Mass goes to I. • Speed (v) goes to w (rotational speed). • KE = (1/2)Mv2 goes to KE = (1/2)Iw2**Example: A Compact Disc (CD)**• How much KE does it have when it’s spinning? • KE = (1/2)Iw2 • So, what’s I and what’s w ? • 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! • w = {144 rotations/min 240 RPM} • w = {2.4 rotations/sec 4.0 RPS} • w = {15 radians/s 25 rad/s} • So, the average is about 20 rad/s**Finish Up: KE = (1/2)Iw2**• 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 = DEE2 = 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 = DEE2 = E1