Rucete ✏ AP Physics C In a Nutshell
4. Work, Energy, and Power
This chapter covers work, kinetic and potential energy, the work-energy theorem, conservative and nonconservative forces, Hooke’s law, conservation of mechanical energy, energy diagrams, and power. It explains how forces transfer energy and how to use energy methods for solving physical problems.
Work
• Work by a single force: W = F · Δr = FΔr cos(θ)
• Work by multiple forces: sum the work done by each force or calculate using net force.
• Work is a scalar and can be positive, negative, or zero depending on the angle between force and displacement.
• Units of work: joules (J) = N·m = kg·m²/s².
Cases of Work
• Positive work: force component parallel to displacement (speeds up object).
• Negative work: force component antiparallel to displacement (slows down object).
• Zero work: force perpendicular to displacement (no speed change).
Graphical Interpretation of Work
• Work done equals the area under a force vs. displacement graph (F vs. x).
Work–Kinetic Energy Theorem
• Net work done = change in kinetic energy:
Wnet = ΔKE = ½mv² – ½mv₀²
• Restates Newton’s second law in terms of energy.
Conservative and Nonconservative Forces
• Conservative forces (gravity, spring force): energy transformations are reversible; total mechanical energy is conserved.
• Nonconservative forces (friction, air resistance): energy dissipates into non-recoverable forms (heat, sound).
• For conservative forces: W = –ΔU
• Path independence: work done depends only on start and end points, not path taken.
Gravitational Potential Energy
• Near Earth's surface:
Ug = mgh
• Change in gravitational potential energy depends only on vertical displacement (Δy), not on the path taken.
Spring Force and Elastic Potential Energy
• Hooke’s Law: restoring force exerted by a spring:
F = –kx
• Elastic potential energy stored in a spring:
Us = ½kx²
• x is the displacement from the equilibrium position.
Conservation of Mechanical Energy
• If only conservative forces act, total mechanical energy (KE + PE) is constant:
KEi + PEi = KEf + PEf
• Mechanical energy transforms between kinetic and potential forms, but the total remains unchanged.
Nonconservative Forces and Modified Energy Conservation
• If nonconservative forces (like friction) act:
Wnc = ΔKE + ΔPE
• Wnc represents energy lost to heat, sound, deformation, etc.
• Energy is still conserved globally but not within mechanical energy alone.
Power
• Power measures the rate of doing work:
P = W/Δt
• Units: watts (W); 1 W = 1 J/s.
• For constant force and velocity:
P = Fv cos(θ)
• Instantaneous power can vary throughout motion if force or velocity changes.
In a Nutshell
Work transfers energy into or out of systems, changing kinetic or potential energy. When only conservative forces act, total mechanical energy remains constant. Nonconservative forces dissipate energy into other forms. Understanding energy transformations and power output is essential for solving a wide range of physics problems efficiently.