Conservation Laws

A conservation law says that some quantity does not change in an isolated system. Conservation laws are among the deepest organizing principles in physics — they constrain what can happen even when the detailed dynamics are unknown.

Concept

A quantity $Q$ is conserved if, for an isolated system, $\frac{dQ}{dt} = 0.$

The most important conservation laws in school physics are:

• Energy — Total energy (kinetic + potential + internal + radiation + mass-energy) is constant if no external work is done.
• Linear momentum — $\vec{p} = m\vec{v}$ conserved if net external force is zero.
• Angular momentum — $\vec{L} = \vec{r} \times \vec{p}$ conserved if net external torque is zero.
• Electric charge — Total charge of an isolated system never changes.
• Mass-energy — In modern physics, mass is a form of energy ($E = mc^2$); the combined quantity is conserved.

These laws come from deep symmetries (Noether's theorem):

Where Conservation "Fails" (and Doesn't)

• In radioactive decay, kinetic energy of products is less than rest energy of reactants, BUT the missing mass appears as energy ($E = mc^2$). Mass-energy together is conserved.
• In an inelastic collision, kinetic energy is not conserved, but total energy (incl. heat, sound) is.
• "Apparent" violations always trace to ignoring some form of energy or to an external force/torque acting on the system.

Worked Example

Q: A $2\,\text{kg}$ block moving at $5\,\text{m/s}$ collides head-on and sticks to a stationary $3\,\text{kg}$ block. Find the final velocity and the energy lost.

Solution:

By conservation of linear momentum: $m_1 u_1 + m_2 u_2 = (m_1 + m_2) v$ $2(5) + 3(0) = 5 v \Rightarrow v = 2\,\text{m/s}.$

Initial KE: $\tfrac{1}{2}(2)(5)^2 = 25\,\text{J}$.

Final KE: $\tfrac{1}{2}(5)(2)^2 = 10\,\text{J}$.

Energy lost (to heat, sound, deformation): $25 - 10 = 15\,\text{J}$. Total energy is still conserved — the 15 J is just no longer kinetic.

Common Confusions

• "Momentum is conserved in every collision." — Only when net external force is zero.
• "Kinetic energy is conserved in every collision." — Only in elastic collisions.
• "Mass is conserved." — Approximately, but in nuclear/particle physics mass and energy interconvert; the combined mass-energy is what's conserved.
• "Conservation of charge means an electron can't be destroyed." — An electron and a positron can annihilate, but the net charge is still zero before and after.

Key Takeaways

• Conservation laws follow from symmetries via Noether's theorem.
• Total energy, momentum, angular momentum, and charge are conserved in isolated systems.
• Energy can transform into different forms; track ALL forms before concluding it is "lost."
• Mass and energy together are conserved, not separately.
• Use conservation laws when forces are complicated or unknown — they bypass detailed dynamics.