Linear Momentum • Momentum is a vector describing how difficult it is to stop a moving object • Total momentum is the sum of individual momenta • p ⃗=mv ⃗ • Units are kg·m/s or N·s Angular Momentum • Angular momentum (L ⃗) is a vector describing how difficult it is to stop a rotating object • Total angular momentum is the sum of individual angular momenta • A mass with velocity v ⃗ moving at some position r ⃗ about point Q has angular momentum L ⃗_Q • Units are kg·m2/s Calculating Angular Momentum • L ⃗_Q=r ⃗×p ⃗=r ⃗×mv ⃗=(r ⃗×v ⃗ )m Spin Angular Momentum • For an object rotating about its center of mass • L ⃗=Iω ⃗ • This is known as an object’s spin angular momentum • Spin angular momentum is constant regardless of your reference point Example 1: Object in Circular Orbit • Find the angular momentum of a planet orbiting the sun. Assume a perfectly circular orbit Example 2: Angular Momentum of a Point Particle • Find the angular momentum for a 5-kg point particle located at (2,2) with a velocity of 2 m/s east • About point O at (0,0) • About point P at (2,0) • About point Q at (0,2) Angular Momentum and Net Torque Conservation of Angular Momentum • Spin angular momentum, the product of an object’s moment of inertia and its angular velocity about the center of mass, is conserved in a closed system with no external net toques applied • L ⃗=Iω ⃗ Example 3: Ice Skater Problem • An ice skater spins with a specific angular velocity. She brings her arms and legs closer to her body, reducing her moment of inertia to half its original value. What happens to her angular velocity? What happens to her rotational kinetic energy? Example 4: Combining Spinning Discs • A disc with moment of inertia 1 km·m2 spins about an axle through its center of mass with angular velocity 10 rad/s. An identical disc which is not rotating is slide along the axle until it makes contact with the first disc. • If the two discs stick together, what is their combined angular velocity? Example 5: Catching While Rotating • Angelina spins on a rotating pedestal with an angular velocity of 8 radians per second. • Bob throws her an exercise ball, which increases her moment of inertia from 2 kg·m2 to 2.5 kg·m2 • What is Angelina’s angular velocity after catching the exercise ball? (Neglect any external torque from the ball) 2005 Free Response Question 3 2014 Free Response Question 3
