Unit 3: Work, Energy, and Power
1. Work and Path Independence
Work is a measure of energy transfer that occurs when a force acts on an object over a displacement.
Work (scalar product):
W = F·Δr = |F||Δr|cosθ
Work (calculus form):
W = ∫F·dr = ∫F(x)dx
Key Concepts:
• Path independence for conservative forces
• Path dependence for non-conservative forces
• Work done by varying forces
• Positive and negative work
2. Conservative Forces and Potential Energy
Conservative forces are characterized by path-independent work and are associated with potential energy functions.
Gravitational Potential Energy:
ΔUg = mgΔh
Elastic Potential Energy:
Ue = ½kx²
General Potential Energy:
F = -∇U
F(x) = -dU/dx
Example:
For a spring stretched from x₁ to x₂:
ΔUe = ½k(x₂² - x₁²)
Work done by spring = -ΔUe
3. Kinetic Energy and the Work-Energy Theorem
The Work-Energy Theorem connects the concepts of work and kinetic energy, providing a powerful tool for problem-solving.
Kinetic Energy:
K = ½mv²
Work-Energy Theorem:
Wnet = ΔK
In calculus form:
∫F·dr = ½m(v₂² - v₁²)
Applications:
• Variable forces
• Complex motion
• Multi-object systems
• Non-uniform acceleration
4. Conservation of Energy
When only conservative forces do work, mechanical energy is conserved.
Conservation of Mechanical Energy:
E = K + U = constant
ΔK + ΔU = 0
For systems with non-conservative forces:
ΔE = Wnc
Example:
A 2kg mass on a spring (k = 100 N/m):
Initial: compressed 0.2m, v = 0
At equilibrium: x = 0
Find v using conservation of energy:
½kx₁² = ½mv₂²
v = √(k/m)x₁ = 2 m/s
5. Power and Energy Transfer Rates
Power represents the rate of energy transfer or the rate at which work is done.
Average Power:
P_avg = W/Δt
Instantaneous Power:
P = dW/dt = F·v
In terms of energy:
P = dE/dt
Important Concepts:
• Constant vs. variable power
• Efficiency
• Power in simple machines
• Relationship to force and velocity
6. Work Done by Variable Forces
AP Physics C emphasizes analysis of work done by forces that vary with position.
General form:
W = ∫F(x)dx
For a spring:
W = ∫(-kx)dx = -½kx²
For gravity near Earth's surface:
W = ∫(-mg)dy = -mgΔh
Example:
Force varying as F(x) = (2x + 3)N
Work from x = 0 to x = 2m:
W = ∫(2x + 3)dx from 0 to 2
W = (x² + 3x)|₀² = 10 Joules
7. Systems and Energy Transfer
Analysis of energy transfer between systems is crucial in AP Physics C.
System Considerations:
• Defining system boundaries
• Internal vs. external forces
• Energy transfer mechanisms
• Work by external forces
For a system:
ΔEsystem = Wexternal + Q
where Q represents heat transfer
8. Non-Conservative Forces and Energy Loss
Non-conservative forces like friction affect the conservation of mechanical energy.
Work done by friction:
Wf = -μkNd
Energy lost to friction:
ΔE = -μkNd
Example:
Block sliding down rough incline:
Initial energy: mgh
Final energy: ½mv² + μkmgdcosθ
Find v using: mgh = ½mv² + μkmgdcosθ
9. Energy in Simple Harmonic Motion
Simple harmonic motion provides an excellent example of energy conversion.
Total Energy:
E = ½kA² = ½mv_max²
Energy at any position:
E = ½kx² + ½mv²
Velocity from energy:
v = ±√((k/m)(A² - x²))
Key Points:
• Energy oscillates between K and U
• Total energy remains constant
• Maximum velocity at x = 0
• Maximum potential energy at x = ±A
Practice and Advanced Applications
To master Work, Energy, and Power in AP Physics C:
- Practice integrating variable forces
- Analyze complex energy transformations
- Study power in various situations
- Work with conservative and non-conservative forces
- Review past AP exam questions
Common Challenges:
• Identifying conservative vs. non-conservative forces
• Setting up proper integration limits
• Choosing appropriate system boundaries
• Analyzing complex energy transformations
• Dealing with variable forces
