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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).
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Chapter 11 Angular Momentum
Introduction • When studying angular motion, angular momentum plays a key role. • Newton’s 2nd 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 = rF ) • 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 11.1-11.2
11.2 Angular Momentum • Developing Angular Momentum • We know Newton’s 2nd 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 ML2T-1, units kg.m2/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 Li = Lf (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 v2 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 ωare inversely proportional to each other. • Figure skating is a prime example.
11.4 • Quick Quizzes p 346 • Examples 11.7-11.9 • End of CH 11
