AgentSkillsCN

06 Energy Work

运用库仑定律与场论分析电荷、电场力、电场强度与电势。 当学生需要解决电荷相互作用、电场与电势相关问题时，此方法尤为适用。

SKILL.md
--- frontmatter
id: energy-work
subject: physics
display_name: Work and Energy
description: |
  Analyzes energy transformations and applies work-energy theorem to solve mechanics problems.
  Use when students need to solve problems involving kinetic energy, potential energy, work, or power.

grade_band: 9-12
khan_tags: [physics, energy, work, conservation-of-energy, power]
standards:
  - NGSS.HS-PS3-1
  - NGSS.HS-PS3-2

objectives:
  - Calculate work done by constant and variable forces
  - Apply the work-energy theorem
  - Calculate kinetic and potential energy
  - Apply conservation of mechanical energy
  - Solve problems involving non-conservative forces
  - Calculate power and efficiency

prerequisites:
  - newtons-laws
  - forces-friction
  - kinematics-1d

estimated_time_minutes: 75

validator:
  type: numeric_solver
  config:
    unit_library: physics
    default_tolerance: 0.02
    require_units: true

sources:
  - name: OpenStax College Physics
    chapter: 7
    url: https://openstax.org/books/college-physics/pages/7-introduction-to-work-energy-and-energy-resources
    license: CC-BY-4.0

Work and Energy

Misconceptions

MisconceptionCorrection
"If I push hard but nothing moves, I did work"Work requires displacement; W = Fd cos(θ). No displacement = no work
"Energy is a force"Energy is a scalar quantity representing capacity to do work; force is a vector
"Heavier objects have more kinetic energy"KE depends on both mass AND velocity: KE = ½mv². A fast light object can have more KE
"Potential energy is energy that might be used"PE is stored energy due to position or configuration; it's real energy, not hypothetical
"Conservation of energy means energy never changes"Total energy is conserved, but it transforms between types (KE ↔ PE ↔ thermal)

Key Concepts

Work

Work is energy transferred by a force acting through a displacement.

For constant force:

  • •W = F · d = Fd cos(θ)
  • •θ is the angle between force and displacement
  • •Work can be positive (force in direction of motion) or negative (opposite)
  • •Unit: Joule (J) = N·m = kg·m²/s²

Special cases:

  • •Force parallel to motion: W = Fd
  • •Force perpendicular to motion: W = 0
  • •Force opposite to motion: W = -Fd

Kinetic Energy

Energy of motion:

  • •KE = ½mv²
  • •Always positive (mass and v² are both positive)
  • •Depends on reference frame

Work-Energy Theorem

The net work done on an object equals its change in kinetic energy:

  • •W_net = ΔKE = ½mv_f² - ½mv_i²

Potential Energy

Energy stored due to position or configuration:

Gravitational PE:

  • •PE_g = mgh (near Earth's surface)
  • •PE_g = -GMm/r (general formula)
  • •Reference point is arbitrary but must be consistent

Elastic PE (spring):

  • •PE_s = ½kx²
  • •k = spring constant (N/m)
  • •x = displacement from equilibrium

Conservation of Mechanical Energy

If only conservative forces do work:

  • •E_total = KE + PE = constant
  • •KE_i + PE_i = KE_f + PE_f

Conservative forces: gravity, spring force, electric force Non-conservative forces: friction, air resistance, applied forces

Work by Non-Conservative Forces

When friction or other non-conservative forces are present:

  • •W_nc = ΔKE + ΔPE = ΔE_mechanical
  • •Energy "lost" to friction becomes thermal energy

Power

Rate of doing work or energy transfer:

  • •P = W/t = ΔE/t
  • •P = Fv (instantaneous power)
  • •Unit: Watt (W) = J/s

Equations

code
[1] W = Fd cos(θ) (work by constant force)
[2] KE = ½mv² (kinetic energy)
[3] W_net = ΔKE = ½mv_f² - ½mv_i² (work-energy theorem)
[4] PE_g = mgh (gravitational potential energy)
[5] PE_s = ½kx² (elastic potential energy)
[6] KE_i + PE_i = KE_f + PE_f (conservation of energy)
[7] W_nc = ΔKE + ΔPE (work by non-conservative forces)
[8] P = W/t = Fv (power)

Worked Examples

Example 1: Work-Energy Theorem

Problem: A 5 kg box initially at rest is pushed with 40 N for 3 m on a frictionless surface. What is its final speed?

Solution:

  1. •Work done: W = Fd = 40 × 3 = 120 J
  2. •Work-energy theorem: W = ½mv_f² - ½mv_i²
  3. •120 = ½(5)v_f² - 0
  4. •v_f² = 240/5 = 48
  5. •v_f = 6.93 m/s

Example 2: Conservation of Energy

Problem: A 2 kg ball is dropped from 10 m. What is its speed just before hitting the ground?

Solution:

  1. •Initial: KE_i = 0, PE_i = mgh = 2 × 9.8 × 10 = 196 J
  2. •Final: KE_f = ½mv_f², PE_f = 0
  3. •Conservation: KE_i + PE_i = KE_f + PE_f
  4. •0 + 196 = ½(2)v_f² + 0
  5. •v_f = √(196) = 14 m/s

Example 3: Spring Energy

Problem: A spring (k = 500 N/m) compressed 0.2 m launches a 0.5 kg ball. How high does it rise?

Solution:

  1. •Initial PE_spring = ½kx² = ½(500)(0.2)² = 10 J
  2. •All spring PE converts to gravitational PE
  3. •mgh = 10 J
  4. •h = 10/(0.5 × 9.8) = 2.04 m

Explanation Patterns

  1. •Identify the system and determine if energy is conserved (are there non-conservative forces?)
  2. •Choose a reference point for potential energy (usually ground level or lowest point)
  3. •Identify initial and final states - what types of energy exist at each?
  4. •Apply the appropriate energy equation:
    • •No friction: KE_i + PE_i = KE_f + PE_f
    • •With friction: KE_i + PE_i + W_nc = KE_f + PE_f
  5. •Check your answer: Does the energy accounting make sense?

Common Problem Types

  1. •Free fall: PE converts to KE (or vice versa for upward motion)
  2. •Inclined plane: Component of gravity does work, PE changes with height
  3. •Spring problems: Elastic PE converts to KE or gravitational PE
  4. •Friction problems: Mechanical energy decreases by work done against friction
  5. •Power problems: Calculate rate of energy transfer or work done
  6. •Pendulum: PE and KE exchange at different positions