🎯 Key Points
- Work-Energy theorem: W_net = ΔKE; Power P = W/t = F·v·cos θ
- Conservative forces (gravity, spring): path-independent work, zero work in closed loop, F=−dU/dx
- Elastic collision: momentum AND KE conserved; Inelastic: only momentum conserved, KE is lost
- Coefficient of restitution e = (velocity of separation)/(velocity of approach); e=1 perfectly elastic, e=0 perfectly inelastic
- Equal-mass elastic collision (1D): velocities are completely EXCHANGED
As a pendulum swings, total mechanical energy (KE+PE) stays constant: at the highest points A and C, all energy is potential (zero speed); at the lowest point B, all of that energy has converted into kinetic energy (maximum speed).
Work
- W = F·d·cos(θ) (scalar product of force and displacement)
- Work done by conservative forces is path-independent
- Work done against friction = heat generated
Kinetic and Potential Energy
- KE = ½mv²
- Gravitational PE = mgh
- Elastic PE (spring) = ½kx²
- Work-Energy Theorem: W_net = ΔKE
Conservation of Energy
- Total mechanical energy (KE + PE) = constant for conservative systems
- Power: P = W/t = F·v
Collisions
- Elastic collision: both momentum and KE conserved
- Inelastic collision: only momentum conserved; KE lost
- Perfectly inelastic: objects stick together; maximum KE loss
- Coefficient of restitution e = relative velocity of separation / relative velocity of approach
Conservative and Non-Conservative Forces
- Conservative force: work done depends only on initial and final position, not on path (gravity, spring force, electrostatic force); work done in a closed path = 0
- Non-conservative force: work done depends on the path taken (friction, air resistance); work done in a closed path is not zero, mechanical energy is dissipated as heat
- For a conservative force, F = -dU/dx (force is negative gradient of potential energy)
Power
- Average power: P_avg = W/t (total work done divided by total time)
- Instantaneous power: P = dW/dt = F·v·cos(θ) (theta is angle between force and velocity)
- SI unit: watt (1 W = 1 J/s); commercial unit: 1 horsepower = 746 W; 1 kWh = 3.6 × 10⁶ J
Elastic Collisions in One Dimension
- For masses m₁ (velocity u₁) and m₂ (velocity u₂) colliding elastically, final velocities:
- v₁ = [(m₁ - m₂)u₁ + 2m₂u₂] / (m₁ + m₂)
- v₂ = [(m₂ - m₁)u₂ + 2m₁u₁] / (m₁ + m₂)
- Special case: if m₁ = m₂, velocities are exactly exchanged
- Special case: if m₂ is initially at rest and m₁ >> m₂, then v₁ ≈ u₁ and v₂ ≈ 2u₁ (light target shoots off at twice the incoming speed)
- Special case: if m₁ >> m₂ and target at rest, the heavy mass continues almost undisturbed
Potential Energy Curve
- Slope of U-x graph gives force: F = -dU/dx; positive slope means force acts in -x direction
- Points where dU/dx = 0 are equilibrium points (stable if U is minimum, unstable if U is maximum)
- A particle oscillates between turning points where total energy E equals U(x) (KE becomes zero there)
Work Done by a Variable Force
- When force changes with position, work = ∫F·dx = area under the Force–displacement (F–x) graph
- Spring force F = −kx is variable; work done in stretching from 0 to x = ½kx² (area of the triangle under the F–x line)
- For a force varying in direction too, W = ∫F·ds (line integral of the dot product along the path)
Motion in a Vertical Circle
- For a body on a string looping a vertical circle, minimum speed at the TOP: v_top = √(gr) (tension just zero, gravity supplies centripetal force)
- Minimum speed at the BOTTOM to complete the loop: v_bottom = √(5gr) (from energy conservation between top and bottom)
- Tension at any point: T = mv²/r − mg·cos(θ) contribution; tension is maximum at the lowest point, minimum at the highest point
- Difference in tension between lowest and highest points = 6mg (a standard result)
Two-Dimensional (Oblique) Collisions
- In 2D collisions, momentum is conserved SEPARATELY along two perpendicular directions (x and y)
- m₁u₁ = m₁v₁cos(θ₁) + m₂v₂cos(θ₂) along x; 0 = m₁v₁sin(θ₁) − m₂v₂sin(θ₂) along y
- For elastic collision of equal masses (one at rest), the two bodies move off at 90° to each other
Types of Equilibrium
- Stable equilibrium: potential energy is minimum; a small displacement produces a restoring force pushing the body back (d²U/dx² > 0)
- Unstable equilibrium: potential energy is maximum; a small displacement pushes the body further away (d²U/dx² < 0)
- Neutral equilibrium: potential energy is constant; body stays in the new position (d²U/dx² = 0)
🚀 JEE Advanced Edge
Oblique elastic collisions: When two equal masses collide elastically and one is initially at rest, after the collision the two velocity vectors are always perpendicular to each other (90° apart) — a powerful shortcut for billiard-ball-style 2D collision problems, derivable from simultaneously conserving momentum (vector) and KE (scalar).
Energy loss in perfectly inelastic collisions: KE lost = ½ × (m₁m₂)/(m₁+m₂) × (relative velocity)² — this "reduced mass" formula directly gives the energy dissipated as heat/deformation without needing to separately calculate initial and final KE.
Worked problem: A 4 kg block moving at 6 m/s collides perfectly inelastically with a stationary 2 kg block. Find the velocity after collision and the KE lost. Approach: Momentum conservation: 4×6 = (4+2)v → v = 4 m/s. KE before = ½(4)(36) = 72 J. KE after = ½(6)(16) = 48 J. KE lost = 72−48 = 24 J (matches the reduced-mass formula: ½ × (4×2)/6 × 36 = 24 J).