Review from Last Lecture

1. Review from Last Lecture • Impulse • Momentum • “What force changes” • Center of Mass • “The average position” • Conservation of Momentum • Constant momentum in isolated systems • Collisions

2. Elastic and Inelastic Collisions • In an inelastic collision the total kinetic energy is not conserved • Momentum is conserved in any collision • Example: case where particles stick together (“perfectly inelastic”)

3. Elastic and Inelastic Collisions • Example: Ballistic Pendulum

4. Elastic and Inelastic Collisions • In an elastic collision the total kinetic energy is conserved • Now have two constraints (momentum and energy) • First consider 1D (sign of v meaningful) • Separate by masses

5. Elastic and Inelastic Collisions • Separate by masses • Divide equations • Now solve

6. Collisions in 2D • Momentum conserved in each direction • Consider one particle at rest in elastic collision • Need one more constraint!

7. Rocket Propulsion • “Rockets can’t fly in vacuum. What do they have to push against?” • Nonsense. Rockets don’t push; they conserve momentum, and send parts (fuel) away from the body as fast as possible

8. Rocket Propulsion • How fast do rockets accelerate? • Start with v, with mass M+Dm • Some time Dt later, have expelled Dm with velocity ve. To conserve momentum, rest of rocket (M) must have additional velocity (in the other direction) of Dv = -ve Dm/M

9. Rocket Propulsion • How fast do rockets accelerate? • Thrust: (instantaneous) force on rocket

10. Rotation of Rigid Object • A rigid object is one in which the relative positions of all the parts is fixed • What happens when we rotate this object? • All points move in a circle about the axis • How far do they move? • Note that q is unitless (ratio of distances) but “measured” in radians

11. Rotation of Rigid Object • What if the object is “spinning” • It turns thru a given angle every second • Define the “angular velocity” • Angular velocity measured in radians per second (rad/s or s-1) • What if it’s “spinning up” • Angular acceleration

12. Rotation of Rigid Object • While q is a scalar, w and a are really vectors • Use “right hand rule” to determine w • For a, it’s in the same direction if the magnitude of w is increasing

13. Linear to Angular and Back • For each linear quantity (x,v,a) there is a corresponding angular quantity connected by r • Note that centripetal acceleration is now:

14. Linear to Angular and Back • This angular-linear correlation extends to the kinematic equations as well

15. Rotational Kinetic Energy • What is the kinetic energy of a spinning object? • Just the sum of it’s parts! • Call I the Moment of Inertia

16. Rotational Kinetic Energy • What is the kinetic energy of a rolling object? • Note that for a given energy, a larger I gives a smaller v. • Hoops on a ramp go slower than disks