Short Version 11 Rotational Vectors Angular Momentum 11

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Short Version : 11. Rotational Vectors & Angular Momentum

Short Version : 11. Rotational Vectors & Angular Momentum

11. 1. Angular Velocity & Acceleration Vectors Right-hand rule Angular acceleration vector: // change

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

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

Cross Product Cross product C of vectors A & B: = area of A-B

Cross Product Cross product C of vectors A & B: = area of A-B parallelogram. Given by right-hand rule, is a vector A-B plane. Dot product C of vectors A & B: Properties of cross product : 1. Distributive 2. Anti-commutative 3. NOT associative B is a vector in the A-B plane and A. A A×(A×B)

11. 3. Angular Momentum Linear momentum: Angular momentum: particle rigid body with axis of

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. In terms of I defined in chap 10 :

Example 11. 1. Single Particle A particle of mass m moves CCW at speed

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 2 nd law.

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

11. 4. Conservation of Angular Momentum Rotating Stool with Weights

11. 4. Conservation of Angular Momentum Rotating Stool with Weights

Conceptual Example 11. 1. Playground A merry-go-round is rotating freely when a boy runs

Conceptual Example 11. 1. Playground A merry-go-round is rotating freely when a boy runs straight toward the center & leaps on. Later, a girl runs tangentially in the same direction as the merry-go-round also leaps on. Does the merry-go-round’s speed increase, decrease, or stays the same in each case? Boy Girl Lb = 0 L = 0 I = Im + Ib L = Lg I = Im + Ig ?

Making the Connection A merry-go-round of radius R = 1. 3 m has rotational

Making the Connection A merry-go-round of radius R = 1. 3 m has rotational inertia I = 240 kg m 2 & 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 Rotating Stool & Bicycle Wheel

Demonstration of Conservation of Angular Momentum Rotating Stool & Bicycle Wheel

11. 5. Gyroscopes & Precession Gyroscope: spinning object whose rotational axis is fixed in

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. Gyroscopic Stability

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

Precession: Continuous change of direction of rotation axis, which traces out a circle. Gyroscope with Adjustable Weights

Rate of Precession occurs if L. z L precesses CCW around z. For L

Rate of Precession occurs if L. z L precesses CCW around z. For L constant: x const Rate of precession : y

“Torqueless” Precession L r L L// v v r L is conserved. Only L//

“Torqueless” Precession L r L L// v v r L is conserved. Only L// is conserved. Precession is due to torque caused by centripetal force. Torqueless Precession

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

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

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

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

GOT IT? 11. 3. You push horizontally at right angles to the shaft of

GOT IT? 11. 3. You push horizontally at right angles to the shaft of a spinning gyroscope. Does the shaft move (a) upward, (b) downward, (c) in the direction you push, (d) opposite the direction you push?

Looking down at bike. Bicycling Direction of bike’s motion L+ t wheel L Wheel

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