Conservation of Energy
Source lecture(s): SC133 Lec 9
Intuition
Energy is the universe's strictest accountant. It changes form constantly — motion to height, height to spring compression, everything eventually to heat — but the total never budges. In mechanics this gives a superpower: compare any two moments of a process without solving for anything in between. Roller coaster, pendulum, planet — if you know the energy budget, you know the speeds.
The statement
For a system where only conservative forces do work:
With non-conservative forces (friction, drag) the mechanical energy leaks into thermal energy, but the total still balances:
Energy conservation isn't broken by friction — mechanical energy just stops being the whole story.
Where it comes from
Mechanical energy conservation is the work–energy theorem plus the definition of potential energy: \(W_\text{cons} = -\Delta U\) and \(W_\text{net} = \Delta K\) combine into \(\Delta(K + U) = W_\text{non-cons}\). Deeper still (Noether's theorem, for later courses): energy conservation is the consequence of physics being the same today as tomorrow — time-translation symmetry.
Worked example: roller coaster loop
A cart starts from rest at height \(h\) and must maintain contact at the top of a loop of radius \(R\). Minimum \(h\)?
At the loop top, gravity alone must supply the centripetal force: \(mg = mv^2/R \Rightarrow v^2 = gR\). Energy from start to loop top (height \(2R\)):
No forces along the track ever entered the calculation — that's the power of the method.
Worked example: friction included
A 2 kg block slides from rest down a ramp of height 1.5 m, arriving at the bottom at 4 m/s. How much energy went to heat?
\(E_\text{thermal} = mgh - \tfrac12 mv^2 = 2(9.8)(1.5) - \tfrac12(2)(16) = 29.4 - 16 = 13.4\,\text{J}\) — the books always balance.
When to use energy vs Newton
| Question asks about... | Best tool |
|---|---|
| speed at a position | energy (path-independent) |
| time, or force at an instant | Newton / kinematics |
| direction of motion at a point | Newton (energy is a scalar — it forgot direction) |
| systems with friction over known distance | energy with \(f_k d\) term |
Common mistakes
- Using \(E\) conservation across friction without the heat term. Check for non-conservative forces before writing \(K_i + U_i = K_f + U_f\).
- Expecting energy methods to give direction or time. Energy is a scalar; it yields speeds, not velocity vectors or durations.
- Double counting: if you include a force via potential energy, don't also add its work.
- Mixing zero-points mid-problem. Choose where \(U = 0\) once.
Related concepts
- Potential energy & work–KE theorem — the ingredients
- Collisions — where kinetic energy may vanish but momentum survives
- First law of thermodynamics — energy conservation with heat promoted to a first-class citizen
- Bernoulli's equation (fluids) — energy conservation per unit volume of fluid
Knowledge graph position
Prerequisites: Work & kinetic energy, Potential energy. Leads to: Collisions, SHM, thermodynamics, Bernoulli.
Quiz
Q1 (computational). A pendulum is released from rest with the string horizontal (length \(L\)). Speed at the bottom?
Answer
Drop height \(= L\): \(v = \sqrt{2gL}\). The string tension does no work (⊥ motion), so pure energy conservation applies.
Q2 (conceptual). A ball bounces, each bounce reaching 80% of the previous height. Where does the energy go?
Answer
Into thermal energy and sound during each inelastic contact (deforming the ball and floor). Mechanical energy shrinks by 20% per bounce; total energy is conserved throughout.
Q3 (multiple choice). Two ramps, one steep and one gentle, connect the same two heights (frictionless). A block slides down each. At the bottom: (a) steep ramp gives higher speed (b) equal speeds, different times (c) equal speeds and equal times
Answer
(b). Same \(\Delta U\) ⇒ same speed. The steep ramp is quicker, though — time is Newton's department, not energy's.
Q4 (conceptual). Why does the loop-the-loop answer \(h = \frac52 R\) not depend on the cart's mass?
Answer
Every term in the budget (\(mgh\), \(\frac12 mv^2\), \(mgR\)) is proportional to \(m\) — gravity accelerates all masses alike, so \(m\) cancels. The same cancellation behind Galileo's tower experiment.