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Chapter Review

Work and Energy

Work and Power · Energy Principles

Work as a Scalar Product

Work is the dot product of force and displacement vectors, measuring the energy transferred by a force acting over a distance. Only the force component along the displacement contributes.

Key Points

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    W = Fs cos θ — positive when force aids motion (θ < 90°), negative when opposing (θ > 90°), zero when perpendicular (θ = 90°)
  • •
    Work is a scalar quantity despite being derived from two vectors
  • •
    One Joule = work done by a 1 N force over 1 m displacement in the force's direction
  • •
    Centripetal force, normal force on flat surfaces, and tension in circular motion all do zero work (perpendicular to displacement)
  • •
    Pushing a wall does zero work — no displacement means no energy transfer regardless of effort
Formula

Work Done by Variable Forces

When force varies with position, total work equals the integral of force over displacement — geometrically, the area under the F vs. x graph.

Key Points

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    W = ∫F(x)dx from x₁ to x₂
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    Area above the x-axis is positive work; area below is negative work
  • •
    For a spring force F = kx, the work done is ½kx²
  • •
    Graphical method: approximate area using summation of F_i cos θ_i Δd_i strips
Formula

Kinetic Energy and the Work-Energy Theorem

Kinetic energy is energy due to motion, and the work-energy theorem states that the net work done on an object equals the change in its kinetic energy.

Key Points

  • •
    KE = ½mv² — depends on the square of velocity, so doubling speed quadruples KE
  • •
    W_net = ΔKE = ½mv_f² − ½mv_i²
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    Positive net work → object speeds up; negative net work → object slows down
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    Only net (resultant) work changes KE — individual forces may cancel
  • •
    Braking distance scales with v² — double the speed means 4× the stopping distance
  • •
    Friction work is always negative (removes energy from the system)
Formula

Gravitational Potential Energy

Near Earth's surface, gravitational PE depends on height above a chosen reference level. Only changes in PE (ΔU = mgΔh) have physical meaning.

Key Points

  • •
    U_g = mgh — valid only near the surface where g is approximately constant
  • •
    Reference level is arbitrary; PE can be negative if the object is below it
  • •
    On inclines, use vertical height h = L sin θ, not ramp length L
  • •
    Work done by gravity is path-independent — depends only on vertical height change
Formula

Elastic Potential Energy

Energy stored in a spring displaced from equilibrium follows a quadratic relationship with displacement, derived from Hooke's Law.

Key Points

  • •
    U_s = ½kx² — compression and extension store equal energy for the same |x|
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    Quadratic dependence: doubling displacement stores 4× the energy
  • •
    Restoring force follows Hooke's Law: F = −kx
  • •
    In simple harmonic motion, elastic PE and KE interchange continuously while total energy remains constant
Formula

Conservation of Mechanical Energy

In an isolated system with only conservative forces, total mechanical energy (KE + PE) is constant — energy transforms between forms but is never created or destroyed.

Key Points

  • •
    KE_i + PE_i = KE_f + PE_f (no non-conservative forces)
  • •
    For a falling object: v = √(2gh) — final speed is independent of mass
  • •
    With friction: mgh = ½mv² + f·d — friction converts mechanical energy to heat
  • •
    Energy lost to friction as percentage: (f·d / mgh) × 100%
  • •
    Conservative forces (gravity, spring, electric) → path-independent work; non-conservative forces (friction, air resistance) → path-dependent
Formula

Power

Power measures how quickly work is done or energy is transferred, with instantaneous power given by the dot product of force and velocity.

Key Points

  • •
    Average power: P = W/t; Instantaneous power: P = F·v
  • •
    1 Watt = 1 J/s; 1 horsepower ≈ 746 W
  • •
    1 kWh = 3.6 × 10⁶ J — a unit of energy, not power
  • •
    Same work in half the time requires double the power
  • •
    P = Fv assumes constant velocity; for acceleration use P = ΔW/Δt
Formula

Absolute Gravitational PE and Escape Velocity

Far from Earth's surface, gravitational PE follows a 1/r dependence with zero defined at infinity. Escape velocity is the minimum speed to overcome this gravitational binding.

Key Points

  • •
    U = −GMm/r — negative because the object is gravitationally bound; zero at infinity
  • •
    Near-surface approximation: −GMm/(R+h) + GMm/R ≈ mgh when h ≪ R
  • •
    Escape velocity: v_esc = √(2gR) = √(2GM/R) ≈ 11.2 km/s for Earth
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    Escape velocity is independent of the escaping object's mass
  • •
    Derived by setting KE = |U_surface|: ½mv² = GMm/R, mass cancels
Formula

Dimensional Analysis and Proportionality

Dimensional analysis verifies equations, while proportionality reasoning allows quick predictions of how changes in one variable affect another.

Key Points

  • •
    Dimensions of work: ML²T⁻²; dimensions of power: ML²T⁻³
  • •
    Work and energy share the same dimensions — 1 J = 10⁷ erg
  • •
    From W = Fs cos θ: doubling F doubles W (at constant s, θ)
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    From P = Fv: tripling v triples P (at constant F)
  • •
    From KE = ½mv²: doubling v quadruples KE (at constant m)

Formulas

Work Definition

Energy transferred by a force acting at angle θ to displacement.

Work-Energy Theorem

Net work equals the change in kinetic energy.

Kinetic Energy

Energy of motion; quadratic in velocity.

Gravitational PE (near surface)

Energy stored due to height in uniform gravity.

Elastic PE

Energy stored in a displaced spring.

Conservation of Energy

Total mechanical energy is constant without non-conservative forces.

Power

Rate of doing work or transferring energy.

Escape Velocity

Minimum speed to leave a planet's gravitational field.