Chapter Review

Work and Energy

Work, Power, Kinetic Energy and Potential Energy · Energy Conservation, Resistive Losses and Efficiency

Work and the Scalar Product

Work quantifies energy transfer by a force acting through a displacement. It is the dot product of force and displacement vectors, making it a scalar despite being derived from vectors.

Key Points

  • •
    — only the force component along displacement matters
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    θ = 0° → maximum positive work (force along motion); θ = 90° → zero work (perpendicular); θ = 180° → maximum negative work (opposing motion)
  • •
    Area under a vs graph equals total work done
  • •
    No displacement means no work, regardless of how hard a force is applied
  • •
    Work is measured in Joules (J) = kg·m²/s²
Formula

Conservative vs Non-Conservative Forces

Conservative forces (gravity, elastic spring, electrostatic) do work that depends only on endpoints, not the path taken. Non-conservative forces (friction, air resistance, tension) have path-dependent work.

Key Points

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    Work by gravity depends only on vertical height change , not on path length
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    Work done by a conservative force over any closed loop is zero
  • •
    Only conservative forces allow the definition of a potential energy function
  • •
    Friction always removes mechanical energy from a system (converts to heat)

Kinetic Energy and the Work-Energy Theorem

Kinetic energy is the energy of motion. The work-energy theorem states that net work done on an object equals its change in kinetic energy.

Key Points

  • •
    — doubling speed quadruples kinetic energy, tripling speed gives 9×
  • •
    Braking distance scales with : double the speed, need 4× the stopping distance
  • •
    Only the net (resultant) force matters for — individual forces may do positive or negative work
  • •
    → speeds up; → slows down; → constant speed
Formula

Gravitational Potential Energy

Gravitational PE is energy stored due to an object's height above a reference level in a uniform gravitational field near Earth's surface.

Key Points

  • •
    — valid only near Earth's surface where is constant
  • •
    PE is relative to a chosen reference level; only has physical meaning
  • •
    On an incline, use vertical height , not ramp length
  • •
    PE can be negative if the object is below the chosen reference level
Formula

Elastic Potential Energy

Elastic PE is energy stored in a deformed spring (or elastic object) due to displacement from its equilibrium position.

Key Points

  • •
    — double the displacement gives 4× the stored energy, not 2×
  • •
    Compression and extension of the same magnitude store equal energy ( removes sign)
  • •
    In SHM, elastic PE and KE continuously interchange while their sum remains constant
  • •
    Stiffer spring (larger ) stores more energy at the same displacement
Formula

Conservation of Mechanical Energy

In a system with only conservative forces, total mechanical energy () remains constant. When non-conservative forces act, energy is dissipated as heat or sound.

Key Points

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    when only conservative forces are present
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    Mass cancels for free-fall problems: regardless of mass
  • •
    With friction:
  • •
    Energy lost to friction as a percentage:
  • •
    PE and KE interconvert in equal amounts — decrease in one equals increase in the other
Formula

Resistive Losses and Efficiency

When friction or air resistance acts, mechanical energy is irreversibly converted to thermal energy. Efficiency measures how much input energy is converted to useful output.

Key Points

  • •
    Friction work: where is friction force and is path length
  • •
    Energy lost depends on the actual path traveled (non-conservative), not just endpoints
  • •
    Efficiency
  • •
    In many problems, mass cancels when calculating percentage energy lost
  • •
    Real systems always have due to resistive forces

Power

Power measures the rate at which work is done or energy is transferred. It distinguishes how quickly a task is completed, not just how much total work is involved.

Key Points

  • •
    — same work in less time requires more power
  • •
    for motion at constant velocity under a constant force
  • •
    1 horsepower ≈ 746 W; 1 kWh = J (a unit of energy, not power)
  • •
    kW is a rate (power); kWh is energy (power × time) — do not confuse them
  • •
    For accelerating objects, use instead of
Formula

Absolute Gravitational PE and Escape Velocity

Far from Earth's surface, varies with distance. Absolute gravitational PE uses dependence with zero defined at infinity. Escape velocity is the minimum speed to break free from a planet's gravity.

Key Points

  • •
    — negative because object is gravitationally bound; zero at infinity
  • •
    is the near-surface approximation of the difference when
  • •
    is independent of the escaping object's mass
  • •
    Escape velocity applies to unpowered projectiles, not continuously thrusting rockets
  • •
    Earth: km/s; Moon: km/s
Formula

Formulas

Work Done

Force component along displacement times distance.

Kinetic Energy

Energy of motion; proportional to square of speed.

Gravitational PE

Energy stored due to height in uniform gravity.

Elastic PE

Energy stored in a spring; quadratic in displacement.

Conservation (with losses)

Energy balance when friction removes mechanical energy.

Power

Rate of doing work; also equals force times velocity.

Absolute Gravitational PE

PE at distance r from center; zero at infinity.

Escape Velocity

Minimum speed to leave a planet's gravity; mass-independent.