Chapter 11 Angular Momentum

Introduction When studying angular motion, angular momentum plays a key role. Newton’s 2 nd for rotation can be expressed in terms of angular momentum. When the net torque is zero, angular momentum is conserved. (similar to net force and linear momentum).

11.1 The Vector Product and Torque We have seen the product of two vectors result in a scalar value. The product of two vectors can also be a vector (as with Torque, τ = r F ) Vector (Cross) Product- The product of two vectors A and B, defined as a third vector C. and magnitude

11.1 The direction of vector C is found by the right hand rule (curl fingers from A to B) Incidentally, the magnitude of the cross product is equal to the area of a parallelogram created by the parent vectors.

11.1 Properties of the Cross Product – NOT commutative (order matters, changes the direction of vector C) – If A and B are parallel/antiparallel then – If A and B are perpendicular then

11.1 Properties cont’d – Cross Products are distributive – The derivative of a cross product with respect to a variable like time, follows the derivative product rule. (maintaining the multiplicative order)

11.1 Cross products with unit vectors

11.1 Vector A x B is given by (See Board Work for Proof)

11.1 Quick Quizzes p. 339 Examples

11.2 Angular Momentum Developing Angular Momentum – We know Newton’s 2 nd Law in terms of changing momentum of a particle (mass m, position r, momentum p) – Lets cross product both sides with position vector r to find the net torque on the particle

11.2 Now lets add to the right side a term equal to zero Product Rule

11.2 Angular Momentum – Dimensions of ML 2 T -1, units kg. m 2 /s – Magnitude of an object’s angular momentum (Following cross product magnitude eqn) Net Torque- time rate of change of angular momentum

11.2 Quick Quizzes p 341 Ex 11.3

11.3 Angular Momentum of a Rotating Rigid Object For a rotating object, every particle moves about the axis of rotation with angular velocity. ( ω ) That particle’s angular momentum is But remember so

11.3 We can now define angular momentum of a rotating object as And remember

11.3 Quick Quiz p. 344 Examples 11.5, 11.6

11.4 Conservation of Angular Momentum Just with linear systems where the net force is zero and linear momentum is conserved, Angular momentum is conserved with zero net torque. Therefore L is a constant and L i = L f (both magnitude and direction)

11.4 Since angular momentum is conserved with zero net torque, a spinning object is considered to be very stable. Applications- – Gyroscopes – Motorcycle/Bicycle Wheels – Rifling/Arrow Fletching – Football Spiral

11.4 More on the football, with zero net torque the axis of rotation should remain fixed in space. Sometimes the axis of rotation remains tangent to the trajectory.

11.4 While gravity provides no net torque, air resistance can (depends on v 2 and shape do) The faster its thrown the more likely the ball is to orient itself to reduce air resistance. (Rotation Axis follows the trajectory)

11.4 Now angular momentum is conserved, what will happen to a rotating object if the M.o.I changes. I and ω are inversely proportional to each other. Figure skating is a prime example.

11.4

Quick Quizzes p 346 Examples End of CH 11