Short on time? This page is the recap. For graphs, bar charts and worked examples, use the full study guide. This is the algebra-based course: no calculus is needed.
Recap
- Name the system: spring–object or pendulum bob–Earth.
- Total energy E = K + U. With no friction and no outside work, it is constant.
- At a turning point, K = 0 (the smallest it can be) and U is largest.
- At equilibrium, U is smallest and K is largest.
- For a spring, all the energy is ½kA² at the turning point, so E = ½kA².
- Changing the amplitude changes the largest potential energy, and so the total energy. It does not change the period.
Key relationships
| Relationship | Symbols and units | Use it to |
|---|---|---|
| E = K + U | J | write the energy at any instant |
| K = ½mv² | J; m in kg; v in m/s | kinetic energy |
| U = ½kx² | J; k in N/m; x in m | spring potential energy |
| U = mgh | J; h above the lowest point | pendulum–Earth potential energy |
| E = ½kA² | J | total energy of a spring oscillator |
| v_max = A√(k/m) | m/s | top speed of a spring oscillator |
| ½mv² + ½kx² = ½kA² | J | speed at any displacement |
| v_max = √(2gh_max) | m/s | top speed of a pendulum bob |
Assumptions behind the numbers
- No friction or air resistance: the total energy stays constant.
- Ideal, massless spring; light, inextensible pendulum string.
- For a pendulum, h is measured from the lowest point; small angles are needed for SHM, but not for energy conservation.
Mistakes to avoid
- Putting the largest kinetic energy at the turning points.
- Saying energy is “lost” when the object stops.
- Assuming half the amplitude means half the energy is kinetic (it is ¾).
- Doubling the energy when the amplitude doubles (it is × 4).
- Thinking a heavier pendulum bob is faster at the bottom.
- Linking more energy to a longer period.
Quick self-check
- A spring with k = 128 N/m oscillates with amplitude 0.050 m. What is the total energy? (0.16 J)
- The object on that spring has mass 0.32 kg. What is its maximum speed? (1.0 m/s)
- At x = A/2, what fraction of the energy is potential? (One quarter)
Next: practice questions.