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Lecture 6

Key Concepts:

  1. Conservation of Energy: Energy is never "lost," only converted from one form to another (e.g., mechanical to thermal and sound).
  2. Work as Energy Transfer: Work is the process of transferring energy to or from an object via a force.
  3. Work-Kinetic Energy Theorem: The crucial link between the net work done on an object and the change in its speed.
    • Positive Net Work -> Speeds Up
    • Negative Net Work -> Slows Down
    • Zero Net Work -> Constant Speed
  4. Work and Direction: Only the component of force parallel to the displacement does work. Perpendicular forces do zero work.
  5. Work as Area: The work done by a force is the area under the Force vs. Position (F-x) graph.

Key Formulas:

  • Kinetic Energy: K = ½mv² (Units: Joules, J)

  • Work (Constant Force): W = Fs cos(φ) W = F · s (Dot Product) (Units: Joules, J)

  • φ is the angle between the force and displacement vectors.

  • Work (Varying Force, 1-D): W = ∫ F(x) dx (from x_initial to x_final)

  • Work Done by a Spring (from x=0): W_stretch = ½kx²

  • k is the spring constant in N/m.

  • Work-Kinetic Energy Theorem: W_total = ΔK = K_final - K_initial W_total = ½mv_f² - ½mv_i²

  • Power: P_avg = W / Δt P_instantaneous = F · v = Fv cos(φ) (Units: Watts, W. 1 W = 1 J/s)

Useful Tips & Tricks for Exams:

  • Check the sign of Work: Always ask: "Is the force helping or hindering the motion?" If helping, W is positive. If hindering, W is negative. If perpendicular, W is zero.
  • Total Work vs. Individual Work: W_total relates to the change in kinetic energy. An individual force's work does not. Don't confuse them.
  • Centripetal Force Does No Work: In uniform circular motion, the force pointing to the center does zero work because it's always perpendicular to the velocity.
  • Graphical Problems: If you see a Force vs. Position graph, immediately think "Area = Work". Break complex shapes into simple rectangles and triangles.
  • Initial/Final States: When using the Work-Kinetic Energy theorem, clearly define your start and end points. "At rest" means K=0. "Constant speed" means ΔK=0.

This sheet covers all the core ideas from this lecture. Study it well, and you'll be in great shape for your exam. Do you have any final questions about Work and Energy?