Energy Partition in SHM
For an ideal linear oscillator, E = kA^2/2 is constant, U = kx^2/2, and K = k(A^2 - x^2)/2, provided x is measured from equilibrium.
Why this shows up in the exam
Energy ratios at a displacement · Energy-time and energy-position graphs · Amplitude after an instantaneous speed change
Learn the idea
Total energy stays fixed while kinetic and potential energies exchange twice during each displacement cycle. At an extreme all usable energy is stored; at equilibrium it is kinetic, and every intermediate position divides the same total between the two forms.
🧠 Memory hook: Square the displacement: energy repeats twice.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- E = kA²/2 — constant total mechanical energy
- U = kx²/2 — elastic potential energy relative to equilibrium
- K = k(A² - x²)/2 — kinetic energy at displacement x
- f_energy = 2f — kinetic or potential energy repeats twice per SHM cycle
How to approach it
- 1Write all energies in terms of A and x
- 2Cancel common factors before substituting
- 3Check centre and extreme limits
Common slip-ups that cost marks
- •Using x instead of x squared
- •Letting total energy vary with position
- •Forgetting that energy frequency is twice displacement frequency
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A mass of 1 kg is attached to a spring of force constant 100 N/m. Find the angular frequency of small oscillations.
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