apply-conservation-laws

Apply Conservation Laws

Solve physics problems systematically using conservation laws — identifying the conserved quantity, defining the system boundary, confirming conservation conditions are met, and applying the conservation equation to find unknowns.

Why This Is Best Practice

Adopted by: Noether's theorem (1915) established that every symmetry of a physical system corresponds to a conserved quantity — the mathematical foundation of all conservation laws. Conservation laws are the foundation of classical mechanics (Newton-Euler equations), quantum mechanics (commuting observables), special relativity (four-momentum conservation), and particle physics (Standard Model). Impact: Conservation laws reduce multi-variable dynamics problems to algebraic equations — solving what would otherwise require complex differential equations. Goldstein (2002) demonstrates that the Lagrangian/Hamiltonian formalism unifies conservation law application across all classical physics. In engineering, conservation of momentum is the basis for rocket propulsion analysis, jet engine thrust calculation, and hydraulic force calculations — domains where Newton's laws applied directly become intractable.

Steps

1. Choose the relevant conservation law

Match the conservation law to the problem type:

Conservation law	Applies when	Symmetry
Energy	No non-conservative forces (friction, drag) OR account for heat/work	Time translation invariance
Linear momentum	No net external force on system	Translational invariance
Angular momentum	No net external torque on system	Rotational invariance
Charge (Kirchhoff's current law)	Always (charge is always conserved)	Gauge invariance
Mass (continuity equation)	Non-relativistic fluid flow	—
Baryon number / lepton number	Nuclear/particle physics	—

2. Define the system boundary precisely

The conservation law applies to the entire system, not to any part of it:

• Conservation of momentum: the system must include all objects experiencing internal forces; external forces (gravity, table contact) are accounted for separately
• Conservation of energy: the system must include all forms of energy (kinetic, potential, thermal, chemical)

Mistakes almost always trace to an incorrectly defined system.

3. Apply conservation of energy

Mechanical energy conservation (no friction):

KE₁ + PE₁ = KE₂ + PE₂
½mv₁² + mgh₁ = ½mv₂² + mgh₂

Work-energy theorem (with non-conservative forces):

ΔKE = W_net = W_conservative + W_non-conservative
KE₂ − KE₁ = −ΔPE + W_friction

Energy conservation with heat (first law of thermodynamics):

ΔU = Q − W

Where ΔU = internal energy change, Q = heat added to system, W = work done BY system.

Relativistic energy:

E² = (pc)² + (mc²)²
E = γmc²  (total energy, including rest mass)
KE = (γ−1)mc²

4. Apply conservation of linear momentum

Isolated system (no external net force):

Σp_before = Σp_after
m₁v₁ᵢ + m₂v₂ᵢ = m₁v₁f + m₂v₂f

Collisions — combine with energy to classify:

• Elastic: both KE and momentum conserved (billiard balls, atomic collisions)
• Inelastic: momentum conserved, KE not (most real collisions)
• Perfectly inelastic: objects stick together; maximum KE loss consistent with momentum conservation

For elastic 1D collision:

v₁f = (m₁−m₂)v₁ᵢ / (m₁+m₂) + 2m₂v₂ᵢ / (m₁+m₂)
v₂f = 2m₁v₁ᵢ / (m₁+m₂) + (m₂−m₁)v₂ᵢ / (m₁+m₂)

Impulse-momentum theorem:

J = FΔt = Δp

For variable force: J = ∫F dt = Δp.

5. Apply conservation of angular momentum

Isolated system (no external net torque):

L_before = L_after
Iω_before = Iω_after  (for rigid body about fixed axis)

With changing moment of inertia (figure skater effect): When I decreases (arms pulled in), ω increases: Iᵢωᵢ = Ifωf.

Angular momentum of particle about point O:

L = r × p = mvr·sin(θ)

Where θ = angle between r and v.

Kepler's second law is conservation of angular momentum: as a planet sweeps equal areas in equal times, r×v = constant.

6. Verify with dimensional analysis and limiting cases

After applying conservation law:

1. Check units: both sides of the equation must have identical units
2. Test limiting cases: if m₁ >> m₂ in a collision, m₁ should be nearly unaffected — verify
3. Check sign conventions: momentum vectors have direction; energy is scalar (always positive)
4. Sanity check: final KE should never exceed initial KE for an inelastic collision (energy is not created)

Common Mistakes

• Including external forces in "isolated system" and still applying momentum conservation: If there is a net external force (gravity on a falling object, friction from a table), linear momentum is NOT conserved — apply the impulse-momentum theorem instead.
• Forgetting rotational KE in spinning systems: Total KE = ½mv² + ½Iω². Forgetting the rotational term in a rolling-without-slipping problem gives wrong energy budgets.
• Using conservation of mechanical energy when friction is present: Friction converts mechanical energy to heat — mechanical energy is not conserved. Either include the heat term or use work-energy theorem with friction force.

When NOT to Use

• Chaotic systems where small perturbations grow exponentially: conservation laws still hold but cannot predict long-term trajectories (e.g., three-body gravitational problem, turbulent flow).