Conservation of Energy
Introduced| English | Meaning |
|---|---|
| conservation of energy/ˌkɒnsəˈveɪʃn ɒv ˈenədʒi/ | The principle that energy cannot be created or destroyed, only transferred or transformed. |
| thermal energy/ˈθɜːml ˈenədʒi/ | Energy associated with the random motion and interactions of particles in a substance. |
A roller coaster with no engine — yet it races on
- After the first climb, a roller coaster's motor switches off. Still it races through loops and hills.
- Its energy is never created or destroyed — only passed between forms.
- At the top it is mostly potential energy; at the bottom, mostly kinetic.
- This is the conservation of energy 能量守恒, one of physics' deepest laws.
Mechanical energy transforms
- Mechanical energy is the sum of kinetic and potential energy: $E_k + E_p$.
- With no friction, this total stays constant as motion unfolds.
- Going down, $E_p$ falls and $E_k$ rises by the same amount; going up, the reverse.
- Energy is simply shifted from one account to the other.
Watch PE turn into KE
Drop from a height and watch potential energy convert to kinetic energy; add friction to send some to heat.
Mechanical energy is the sum of kinetic energy and ____ energy.
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Mechanical energy $= E_k + E_p$ — kinetic plus potential.
On a frictionless track, where is the kinetic energy greatest?
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At the lowest point the PE is smallest, so the most energy is kinetic — the fastest point.
Where friction sends the energy
- Real tracks have friction, so mechanical energy seems to "disappear".
- It hasn't — it becomes thermal energy 热能 (heat) in the surfaces.
- Counting heat too, the total energy is still perfectly conserved.
- The universe keeps an exact ledger: energy only changes form.
When friction slows a sliding block, that energy is destroyed.
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It is not destroyed — it becomes thermal energy (heat). The total energy is conserved.
On a real (rough) slide, where does the "missing" mechanical energy go?
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Friction converts mechanical energy into thermal energy. Counting heat, the total is conserved.
Using conservation to solve problems
- Set the energy at the start equal to the energy at the end: $E_{k,i} + E_{p,i} = E_{k,f} + E_{p,f}$.
- For a drop from rest through height $h$: $mgh = \tfrac12 m v^2$, so $v = \sqrt{2gh}$.
- No need to track the messy forces along the way — just compare start and end.
- This shortcut is one of the most powerful tools in mechanics.
A ball is dropped from rest at $5\ \text{m}$ ($g = 10\ \tfrac{\text{m}}{\text{s}^2}$, no air resistance). What is its speed at the bottom, in $\tfrac{\text{m}}{\text{s}}$?
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$mgh = \tfrac12 mv^2 \Rightarrow v = \sqrt{2gh} = \sqrt{2\times10\times5} = 10\ \tfrac{\text{m}}{\text{s}}$.
Select all correct statements about a frictionless pendulum swing.
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Energy is conserved, not created: PE peaks at the top of the swing and becomes KE at the bottom.
Energy is never destroyed — when it seems to vanish, it has usually turned into heat through friction or air resistance. "Energy lost" really means "energy transferred to a form we are no longer counting".
A ball is dropped from rest at a height of $5\ \text{m}$ ($g = 10\ \tfrac{\text{m}}{\text{s}^2}$, no air resistance).
- All the PE becomes KE: $mgh = \tfrac12 m v^2$, so $gh = \tfrac12 v^2$.
- $v = \sqrt{2gh} = \sqrt{2 \times 10 \times 5} = \sqrt{100} = 10\ \tfrac{\text{m}}{\text{s}}$.
Conservation of energy: energy is never created or destroyed, only transformed. Without friction, mechanical energy $E_k + E_p$ stays constant, so $E_{k,i} + E_{p,i} = E_{k,f} + E_{p,f}$. Friction turns mechanical energy into heat, but the total is always conserved.
Write from a landing speed of 20 m/s
- A stone is dropped from rest through 20 m. Use g = 10 and ignore air resistance.
- Find the speed at the bottom.
A stone drops from rest through 20 m. Use g = 10 m/s squared and ignore air resistance. What is its speed at the bottom?
2 times 10 times 20 is 400, and the square root of 400 is 20.
What is the speed at the bottom?
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The speed squared is 400, so the speed is 20 m/s.