11. Rotational Vectors & Angular Momentum Angular Velocity & Acceleration Vectors Torque & the Vector Cross Product Angular Momentum Conservation of Angular Momentum Gyroscopes & Precession

Earth isn’t quite round. How does this affect its rotation axis, and what’s this got to do with ice ages? (The deviation from roundness is exaggerated.) Axis precesses with period ~26,000 yr.

Importance of rotation: Earth’s rotational axis  seasons. Angular momentum of protons in living tissues  MRI Rotating air  tornadoes. Rotating wheel  stabilizes bicycle.

11.1. Angular Velocity & Acceleration Vectors Right-hand rule Angular acceleration vector:  change direction  //     //   

11.2. Torque & the Vector Cross Product Right hand rule cross product

Cross Product Cross product C of vectors A & B: right-hand rule Dot product C of vectors A & B: Properties of cross product: Distributive Anti-commutative

GOT IT? 11.1. Which numbered torque vector goes with each pair of force-radius vectors? Neglect magnitudes. 3 2 1 4 Note: Wolfson gave 6 as the answer to (b).

11.3. Angular Momentum Linear momentum: Angular momentum: particle rigid body with axis of rotation along principal axis general case, I a tensor.  L &  can have different directions.

Example 11.1. Single Particle A particle of mass m moves CCW at speed v around a circle of radius r in the x-y plane. Find its angular momentum about the center of the circle, express the answer in terms of its angular velocity.

Torque & Angular Momentum System of particles:   rotational analog of 2nd law.

11.4. Conservation of Angular Momentum

Example 11.2. Pulsars  A star rotates once every 45 days. It then undergoes supernova explosion, hurling most of its mass into space. The inner core of the star, whose radius is initially 20 Mm, collapses into a neutron star only 6 km in radius. The rotating neutron star emits regular pulses of radio waves, making it a pulsar. Calculate the pulse rate ( = rotation rate ). Assume core to be a uniform sphere & no external torque. Before collapse: After collapse: 

Example 11.3. Playground A merry-go-round of radius R = 1.3 m has rotational inertia I = 240 kg m2 & is rotating freely at 1 = 11 rpm. A boy of mass mb = 28 kg runs straight toward the center at vb = 2.5 m/s & leaps on. At the same time, a girl of mass mg = 32 kg, running tangentially at speed vg = 3.7 m/s in the same direction as the merry-go-round also leaps on. Find the new angular speed 2 once both children are seated on the rim. Before : After : 

Demonstration of Conservation of Angular Momentum

GOT IT? 11.2. If you step on a non-rotating table holding a non-rotating wheel. if you spin the wheel CCW as viewed from above, which way do you rotate? If you then turn the wheel upside down, will your rotation rate increase, decrease, or remain the same? What about your direction of rotation? CW to keep L = 0. Same, CCW.

11.5. Gyroscopes & Precession Gyroscope: spinning object whose rotational axis is fixed in space. External torque required to change axis of rotation  Higher spin rate  larger L  harder to change orientation Usage: Navigation Missile & submarine guidance. Cruise ships stabilization. Space-based telescope like Hubble.

Precession Precession: Continuous change of direction of rotation axis, which traces out a circle. 

Erath’s Precession Earth’s precession (period ~ 26,000 y ) The equatorial bulge is highly exaggerated.

Perfect sphere  = 0  = 0 Oblate spheroid  <  The equatorial bulge is highly exaggerated.

GOT IT? 11.3. You push horizontally at right angles to the shaft of a spinning gyroscope. Does the shaft move upward, downward, in the direction you push, opposite the direction you push?

Bicycling Looking down at bike. Direction of bike’s motion L+  t wheel Wheel turns L Einstein leans  points into paper L //   wheel turns to Einstein’s left