Learning Objectives
- Define work done by a constant and variable force
- Understand kinetic and potential energy and the work-energy theorem
- Differentiate between conservative and non-conservative forces
- Understand the concept of power and its applications
- Apply the law of conservation of energy and analyze collisions
Key Concepts
Work Done
Work by a constant force: W = F · d = Fd cos θ, where θ is the angle between force and displacement.
If θ = 0°: W = Fd (maximum positive work). If θ = 90°: W = 0 (no work). If θ = 180°: W = -Fd (negative work).
Work by a variable force: W = ∫F · dx (area under F-x graph)
SI unit of work: joule (J) = N·m. 1 erg = 10⁻⁷ J, 1 eV = 1.6 × 10⁻¹⁹ J.
Kinetic Energy
Energy possessed by a body due to its motion: KE = ½mv²
In terms of momentum: KE = p²/2m
Work-Energy Theorem: The net work done on a body equals the change in its kinetic energy.
W_net = ΔKE = ½mv² - ½mu²
Potential Energy
Energy due to position or configuration of a body.
Gravitational PE: U = mgh (near Earth's surface)
Elastic PE (spring): U = ½kx² (where k is spring constant, x is extension/compression)
Relation: F = -dU/dx (force is the negative gradient of potential energy)
Conservative and Non-Conservative Forces
Conservative Force: Work done is independent of path, depends only on initial and final positions. Total work in a closed path = 0. Examples: gravity, spring force, electrostatic force.
Non-Conservative Force: Work done depends on path. Examples: friction, air resistance, viscous force.
Conservation of Mechanical Energy
For conservative forces: KE + PE = constant (Total mechanical energy is conserved).
½mv₁² + mgh₁ = ½mv₂² + mgh₂
When non-conservative forces act: W_nc = ΔKE + ΔPE
Power
Average Power: P_avg = W/t
Instantaneous Power: P = dW/dt = F · v = Fv cos θ
SI unit: watt (W) = J/s. 1 horsepower (hp) = 746 W. 1 kWh = 3.6 × 10⁶ J.
Collisions
Elastic Collision (1D): Both momentum and kinetic energy are conserved.
v₁ = [(m₁ - m₂)u₁ + 2m₂u₂] / (m₁ + m₂)
v₂ = [(m₂ - m₁)u₂ + 2m₁u₁] / (m₁ + m₂)
Special case: equal masses -- velocities are exchanged.
Perfectly Inelastic Collision: Bodies stick together. Maximum KE is lost.
(m₁ + m₂)v = m₁u₁ + m₂u₂
Coefficient of Restitution: e = (v₂ - v₁)/(u₁ - u₂). e = 1 (elastic), e = 0 (perfectly inelastic).
Summary
Work is the scalar product of force and displacement. The work-energy theorem connects net work to change in kinetic energy. Potential energy is associated with conservative forces. The total mechanical energy is conserved when only conservative forces act. Power is the rate of doing work. Collisions are classified as elastic (KE conserved) or inelastic (KE not conserved), but momentum is conserved in all collisions.
Important Terms
- Work: W = Fd cos θ, scalar quantity measured in joules
- Kinetic Energy: Energy of motion, KE = ½mv²
- Potential Energy: Energy of position/configuration
- Conservative Force: Force for which work is path-independent
- Power: Rate of doing work, P = W/t = Fv
- Elastic Collision: Collision where kinetic energy is conserved
- Coefficient of Restitution: Ratio of relative velocity after and before collision
Quick Revision
- W = Fd cos θ; W = ∫F dx for variable force
- KE = ½mv² = p²/2m; Work-Energy Theorem: W_net = ΔKE
- PE: gravitational = mgh, elastic = ½kx²
- Conservative forces: work independent of path; E = KE + PE = constant
- P = dW/dt = F · v; 1 hp = 746 W
- Elastic collision: both p and KE conserved; Inelastic: only p conserved