6.3 Conservative and Non-conservative Forces

Forces shape our physical world, determining how objects move and interact. This section explores conservative and non-conservative forces, highlighting their key differences and impacts on energy in systems.

We'll dive into potential energy, a crucial concept linked to conservative forces. Understanding these principles is vital for grasping energy conservation and work in various scenarios, from simple gravity to complex mechanical systems.

Types of Forces and Energy

Conservative vs non-conservative forces

• Conservative forces work done independent of path taken enables potential energy definition and reversible work performed (gravitational force, elastic force, electrostatic force)
• Non-conservative forces work done depends on path taken prevents potential energy definition and causes irreversible work (friction, air resistance, tension in moving rope)

Potential energy and conservative forces

• Potential energy stored in system due to position or configuration only defined for conservative forces represented by U or PE
• Work done by conservative force equals negative change in potential energy $W = -\Delta U$
• Force derived from potential energy $F = -\frac{dU}{dx}$
• Types include gravitational potential energy $U_g = mgh$ and elastic potential energy $U_e = \frac{1}{2}kx^2$

Energy Conservation and Work

Conservation of mechanical energy

• Total mechanical energy remains constant in closed system with only conservative forces $E_{initial} = E_{final}$
• $KE_i + PE_i = KE_f + PE_f$
• Problem-solving steps:
  1. Identify initial and final states
  2. Calculate initial and final kinetic and potential energies
  3. Apply conservation of energy equation
  4. Solve for unknown variables

Non-conservative forces and energy

• Mechanical energy not conserved when non-conservative forces present typically dissipating energy as heat or sound
• Work-energy theorem change in kinetic energy equals total work done on system $\Delta KE = W_{total} = W_{conservative} + W_{non-conservative}$
• Non-conservative forces generally decrease mechanical energy of system (friction converts kinetic energy into thermal energy)

Work of forces in scenarios

• Conservative forces work path-independent calculated using change in potential energy $W_c = -\Delta U$
  • Gravity work $W_g = -mg(h_f - h_i)$
  • Spring work $W_s = -\frac{1}{2}k(x_f^2 - x_i^2)$
• Non-conservative forces work path-dependent calculated directly using force and displacement $W_{nc} = \int F \cdot ds$
  • Constant friction force work $W_f = f_k d$
  • Air resistance work requires knowledge of force as function of velocity