Physics · Conservation Laws
Conservation of Energy
Energy cannot be created or destroyed — only transformed. This single law, disguised in countless forms, governs every process in the universe from nuclear reactions to pendulums to metabolism.
§ 1
Work
Work is the energy transferred to an object by a force acting through a displacement. Only the component of force parallel to the displacement does work.
WORK
$$W = \mathbf{F}\cdot\mathbf{d} = Fd\cos\theta$$
$\theta$ = angle between force and displacement. $W < 0$ if opposing motion.
$$W = \int_a^b F(x)\,dx$$
Variable force: work = area under the $F$–$x$ graph. Generalises to any path.
KEkinetic energyJ = kg m² s⁻²
PEpotential energyJ
Qthermal energyJ
Etotal energyJ
Wwork doneJ
PpowerW = J s⁻¹
§ 2
Kinetic energy
Kinetic energy is the energy an object possesses due to its motion. It depends on the square of speed — doubling speed quadruples KE.
KINETIC ENERGY — DERIVED FROM WORK
$$E_k = \frac{1}{2}mv^2$$
Apply constant net force $F$ over displacement $d$: $W = Fd$.
Definition of work
From kinematics: $v^2 = u^2 + 2ad$, so $d = (v^2-u^2)/2a$.
SUVAT identity
$W = Fd = ma\cdot\dfrac{v^2-u^2}{2a} = \dfrac{1}{2}m(v^2-u^2) = \Delta E_k$
Substituting $F=ma$ and simplifying — this is the work–energy theorem
$$\boxed{E_k = \tfrac{1}{2}mv^2}$$
Setting $u=0$ gives the KE from rest. All work done by net force goes into $E_k$.
§ 3
Potential energy
Potential energy is stored energy associated with position or configuration. It is only defined for conservative forces — forces where the work done is path-independent.
FORMS OF POTENTIAL ENERGY
$$E_p = mgh$$
Gravitational PE near Earth's surface. $h$ = height above chosen reference level.
$$E_s = \frac{1}{2}kx^2$$
Elastic PE in a spring. $k$ = spring constant (N m⁻¹), $x$ = extension from equilibrium.
Conservative vs non-conservative forces: gravity and spring force are conservative — the work done depends only on start and end positions, not the path. Friction is not conservative — it dissipates energy as heat, and the work done depends on path length.
§ 4
The work–energy theorem
The net work done on an object equals its change in kinetic energy. The area under the $F$–$x$ graph is the work done — and therefore the change in KE.
WORK–ENERGY THEOREM
$$W_\text{net} = \Delta E_k = E_{k,f} - E_{k,i} = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$$
§ 5
Conservation of mechanical energy
When only conservative forces act, the total mechanical energy $E = E_k + E_p$ is constant. Energy converts between kinetic and potential — but the sum never changes.
CONSERVATION OF MECHANICAL ENERGY
$$E_k + E_p = \text{constant} \quad\Leftrightarrow\quad \frac{1}{2}mv^2 + mgh = \frac{1}{2}mv_0^2 + mgh_0$$
Noether's theorem (1915): every continuous symmetry of a physical system corresponds to a conserved quantity. Conservation of energy arises from time-translation symmetry — the laws of physics are the same today as they were yesterday. This is arguably the deepest result in all of physics.
§ 6
The pendulum
A pendulum is a perfect demonstration of energy conservation — gravitational PE converts to KE at the bottom and back, with the total remaining constant (in the ideal case). It also provides a clean derivation of simple harmonic motion for small angles.
PENDULUM ENERGY + PERIOD
$$E = mgh = mgL(1-\cos\theta)$$
PE at angle $\theta$ from vertical. $L$ = length. At bottom, all PE → KE.
$$T = 2\pi\sqrt{\frac{L}{g}} \quad (\theta \ll 1)$$
Period — independent of mass and amplitude (for small angles). Huygens 1656.
Independence of mass: the period doesn't depend on mass because both the restoring force ($mg\sin\theta$) and the inertia ($m$) are proportional to $m$ — they cancel. Galileo reportedly discovered this by watching a chandelier swing in Pisa Cathedral.
§ 7
Spring energy and SHM
A mass on a spring is the archetype of simple harmonic motion. The spring stores elastic potential energy; the mass carries kinetic energy. They exchange perfectly at frequency $\omega = \sqrt{k/m}$.
SPRING ENERGY + SHM
$$E = \frac{1}{2}kx^2 + \frac{1}{2}mv^2 = \frac{1}{2}kA^2$$
Total energy = constant = $\frac{1}{2}kA^2$ where $A$ = amplitude. At $x=A$, all energy is elastic PE.
$$\omega = \sqrt{\frac{k}{m}},\quad T = 2\pi\sqrt{\frac{m}{k}}$$
Angular frequency and period. Unlike a pendulum, period depends on mass but not amplitude.
§ 8
Power
Power is the rate of energy transfer. The same work can be done at any power — what changes is how quickly.
POWER
$$P = \frac{W}{t} = \frac{\Delta E}{\Delta t}$$
Average power = energy transferred per unit time.
$$P = Fv$$
Instantaneous power = force × velocity. A car engine delivers constant power; as speed increases, traction force decreases.
Terminal velocity of a constant-power vehicle: $P = Fv$, and drag force $F_\text{drag} = kv^2$. At terminal velocity $P = kv^3$, so $v_\text{max} = (P/k)^{1/3}$. Doubling engine power increases top speed by only $2^{1/3} \approx 26\%$.
§ 9
Ramp with friction — energy accounting
When friction acts, mechanical energy is not conserved — some is converted to thermal energy. But total energy is still conserved: $\Delta E_k + \Delta E_p + \Delta E_\text{thermal} = 0$.
ENERGY WITH FRICTION
$$E_{k,i} + E_{p,i} = E_{k,f} + E_{p,f} + Q_\text{friction}$$
$$Q_\text{friction} = f_k\,d = \mu_k mg\cos\theta\cdot d$$