Conservation laws in mechanics
Some quantities do not change even when everything else does. Those constants make it possible to solve in two lines problems that would take a full page of differential equations using Newton.
01Why it pays to conserve something
Solving a problem with Newton’s laws requires knowing the forces at every instant. Often they are not known —what exact force do two cars exert on each other during a crash?— and yet the problem can be answered, because certain quantities have the same value before and after no matter what happened in between.
- Mechanical energy, if the forces acting are conservative.
- Momentum, if the net external force is zero.
- Angular momentum, if the net external torque is zero.
Each has its condition: it is not always conserved, it is conserved when the condition holds.
02Work and kinetic energy
The work done by a force along a path is the line integral of that force; for a constant force and a straight-line displacement it reduces to W = F·d·cosα.
The net work of all the forces equals the change in kinetic energy. It always holds, with or without friction.
A force perpendicular to the displacement does no work: that is why the normal force and the centripetal force never change the speed. And power is work per unit time, P = F·v, the same quantity that in a circuit is P = V·I.
03Potential energy and conservative forces
A force is conservative when its work does not depend on the path, only on the endpoints. For such forces —and only for them— a potential energy can be defined:
| Force | Potential energy | Conservative |
|---|---|---|
| Weight | Ep = m·g·h | Yes |
| Elastic | Ep = ½·k·x² | Yes |
| General gravitational | Ep = −G·M·m/r | Yes |
| Electric | Ep = q·V | Yes |
| Friction | Does not exist | No: it dissipates |
If only conservative forces do work, Ec + Ep = constant (kinetic plus potential energy). With friction, the difference is exactly the heat dissipated:
The relation between a conservative force and its potential is F⃗ = −∇Ep: the force points in the direction in which the potential energy decreases fastest. It is the same relation as between the electric field and the potential, E⃗ = −∇V.
04Impulse and momentum
This is why an airbag works: the change in momentum in a crash is fixed, but if the same Δp is spread over ten times as much time, the peak force is divided by ten.
If the sum of external forces is zero, the total p⃗ does not change. During a collision, the internal forces are enormous compared with the weight or friction, so momentum is conserved to an excellent approximation even if the system is not truly isolated.
05Collisions
| Type | Coefficient e | Momentum | Kinetic energy |
|---|---|---|---|
| Elastic | \( e = 1 \) | Conserved | Conserved |
| Partially inelastic | 0 < e < 1 | Conserved | Partly lost |
| Perfectly inelastic | \( e = 0 \) | Conserved | Maximum loss: the bodies stick together |
The coefficient of restitution compares the speed of separation with the speed of approach, e = |v₂′ − v₁′| / |v₂ − v₁|. With it, the final velocities of a head-on collision are determined:
Move with the time slider: before the collision on the left, after it on the right. Watch what happens to the momentum and to the energy.
06The same thing, in electronics
- Energy in a capacitor and in an inductor. ½·C·V² and ½·L·I² are the exact twins of ½·k·x² and ½·m·v². An LC circuit oscillates by exchanging energy just like a pendulum.
- Conservation of charge. Kirchhoff’s current law is a conservation law, with the same logic as that of momentum.
- Impedance matching. The maximum power transfer from a source to a load has its exact mechanical analog in the elastic collision between equal masses, where one transfers all its energy to the other.
- Energy harvesting. A piezoelectric generator converts the work of a deformation into electrical energy; the balance is set up with the work–energy theorem.
07In the lab
Drop a ball from h₀ and measure the height of the first bounce h₁. Then e = √(h₁/h₀). Repeat with balls of different materials and tabulate the results.
With two carts and optical gates, measure the four velocities of a collision. Verify that m₁v₁ + m₂v₂ is conserved within the error, and calculate what fraction of the energy was lost.
Use a microcontroller to record force against time during a brief impact. The area under the curve is the impulse; compare it with m·Δv measured from video.
08Common mistakes
- Assuming kinetic energy is conserved in every collision. Only in the elastic one.
- Adding momenta as scalars. It is a vector: in a head-on collision, one of the two velocities enters with a negative sign.
- Using Ep = m·g·h with friction present without subtracting the dissipated heat.
- Confusing work with force. Holding a suitcase still does no work, however tiring it is.
- Defining a potential energy for a non-conservative force. It does not exist.
- Forgetting to choose a reference level for the height. What makes physical sense is the difference in potential energy, not its absolute value.
09Self-assessment
A 2 kg body falls 5 m. How fast is it moving when it reaches the ground?
m·g·h = ½·m·v² gives v = √(2·9.8·5) ≈ 9.9 m/s, regardless of the mass.
Two identical bodies, one at 5 m/s and the other at rest, collide elastically. What happens?
They exchange velocities: the first comes to rest and the second leaves at 5 m/s. It is the case of maximum energy transfer.
How much energy is lost in a perfectly inelastic collision between 2 kg at 4 m/s and 3 kg at rest?
v′ = 8/5 = 1.6 m/s. E before = 16 J, E after = ½·5·1.6² = 6.4 J: 9.6 J is lost, 60%.
Why does an airbag reduce injuries if Δp is the same?
Because J = F·Δt: by lengthening the time of impact, the peak force drops in the same proportion.
Does the normal force do work on a body sliding down an inclined plane?
No: it is perpendicular to the displacement, so its work is zero.
What is the electrical analog of ½·m·v²?
The energy of an inductor, ½·L·I², with inductance playing the role of mass and current the role of velocity.
10Further reading
- Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers with Modern Physics, Volume 1. 9th ed. (Spanish edition, “Física para ciencias e ingeniería”), Cengage Learning, 2014. Chapters 7 to 9: work, energy, momentum and collisions.
- Paul A. Tipler and Gene Mosca. Physics for Scientists and Engineers, Volume 1. 6th ed., W. H. Freeman, 2008 (Spanish edition, “Física para la ciencia y la tecnología”, Reverté, 2010). Good treatment of the coefficient of restitution and of collisions in two dimensions.
- Richard P. Feynman. The Feynman Lectures on Physics, Volume I. Addison-Wesley, 1963 (Spanish edition, “Física, volumen I”, Addison-Wesley / Pearson). Chapter 4, “Conservation of Energy”, explains what it really means for something to be conserved.
- Marcelo Alonso and Edward J. Finn. Physics. Addison-Wesley, 1992 (Spanish edition, “Física”, Addison-Wesley Iberoamericana, 1995). Develops mechanics starting from the conservation laws instead of Newton’s laws.