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Key Concepts of Work, Energy, Power

Dec 3, 2024

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Lecture Notes on Work, Energy, and Power

Work

  • Definition: Work is the component of force in the direction of displacement times the displacement.
    • Formula: ( W = F \times d \times \cos(\theta) )
    • Example Calculation:
      • ( F = 20 ) N, ( d = 3 ) m, ( \theta = 30^\circ )
      • ( W = 20 \times 3 \times \cos(30^\circ) \approx 52 \text{ J} )
  • Types of Work:
    • Positive: Force and displacement in the same direction.
    • Zero: No displacement or force perpendicular to displacement.
    • Negative: Force opposite to displacement (e.g., friction).

Graphical Representation

  • Work can be represented by the area under a force vs. position graph.

    • Example: Constant force of 3 N from position 2 to 4.
      • Area = 3 \times (4-2) = 6 J
    • Non-constant force example decomposed into triangle and rectangle.
      • Area total = 8 J

  • Net Work: Sum of work done by all forces in a system.

Work-Energy Theorem

  • Definition: Net work done on an object is equal to the change in its kinetic energy.
    • Formula: ( W_{net} = \Delta KE = KE_{final} - KE_{initial} )
    • Derivation involves Newton's laws and kinematic equations.
    • Example: Calculate final velocity for a box with ( W_{net} = 30 ) J.
      • Given ( m = 10 ) kg, ( v_i = 0 ), solve for ( v_f ).

Gravitational Potential Energy

  • Work Against Gravity: Lifting an object at constant speed.
    • Formula: ( PE = mgh )
    • Property: Path independent, only depends on start and end points.

Conservative vs. Non-Conservative Forces

  • Conservative Forces: Work done depends only on initial and final positions.

    • Examples: Gravitational force, spring force.
    • Conservation of Mechanical Energy: Total mechanical energy remains constant.
      • Mechanical Energy = Potential Energy + Kinetic Energy

  • Non-Conservative Forces: Work depends on path taken.

    • Examples: Friction, air resistance.
    • Add or remove mechanical energy from a system.

Power

  • Definition: Rate at which work is done.
    • Formula: ( P = \frac{W}{t} ) or ( P = F \cdot v )
    • Unit: Watts (W)
    • Example Calculations:
      • Elevator: ( P = 1.3 \times 10^4 ) W
      • Moving object: ( F = 12 ) N, ( v = 3 ) m/s, ( P = 36 ) W

These notes summarize the key concepts of work, energy, and power discussed in the lecture, including their definitions, formulas, and example calculations.

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