Understanding Work, Energy, and Conservation: A Comprehensive Review

1. Review from Last Lecture • Work • Kinetic Energy • Potential Energy • For conservative forces • Work-Energy Theorem • Total Energy • Constant for conservative forces • Stability • At a fixed point

2. Examples: Gravitational Field • Find maximum r for given initial v • Particle will escape if denominator is zero

3. Impulse • Imagine the force involved in hitting a golf ball • Force changes very quickly • How can we use Newton’s Laws if we don’t know the force at every time? • Integrate with respect to time • Call this the “impulse”

4. Impulse • What exactly is force the derivative of? • Call p = mv the momentum of a particle • The impulse, then, is the change in momentum

5. System of Particles 1 • Consider a system of several particles • Each particle obeys Newton’s Laws: • This force can be split into internal and external parts • Now sum up the forces on all the particles • For the internal forces we have • Thus 2 3

6. System of Particles 1 • Continuing with our system • Internal forces cancel, so • The sum of derivatives is the derivative of the sums • We can treat the motion of a system of particles as the change in the total system’s momentum! 2 3

7. Center of Mass 1 • If what does refer to? • Well so • Thus • This “average” position is called the Center of Mass 2 3

8. Center or Mass • What’s the Center of Mass for a continuous object? • Change sum into an integral (using density r) • Note that the total mass is also an integral • So CM is still really the “average” position • “Add” up the positions of each little bit of mass • Divide by the “number” of little bits of mass

9. Center of Mass • Can find center of mass of an object by hanging it from two places • Center of mass will always be below the hanging point

10. Center of Mass • Example: CM of a right triangle

11. Conservation of Momentum • Since the total momentum of a system of particles is only changed by external forces, the momentum of an isolated system is constant • For this we say, “momentum is conserved” • The total momentum is conserved even though parts of the system may be doing wild things

12. Elastic and Inelastic Collisions • In an elastic collision the total kinetic energy is conserved • Momentum is conserved in any collision • Example: • What are signs of final velocities?

13. Elastic and Inelastic Collisions • Example (cont.): • Consider reference frame where CM is at rest

14. 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

15. Elastic and Inelastic Collisions • Example: Ballistic Pendulum

16. Collisions in 2D • Use Conservation of Momentum in each direction • Consider case where one particle is at rest • In CM frame particles are back-to-back!

17. 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

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

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