Visualizing the beauty in physics and mathematics
๐ง This ball_on_spring.js uses Three.js
๐ง Note that the energy oscillates with a higher frequency than the position, namely twice as fast.
๐ The bal goes up/down with frequency ฯ
๐ The energy oscillates with frequency 2ฯ
Consider an ideal harmonic vibration. The position is given by:
\[y(t) = A \cos(\omega t)\]The velocity is given by:
\[v(t) = -A \omega \sin(\omega t)\]๐ด The kinetic energy is:
\[E_k = \tfrac12 m v^2 = \tfrac12 m A^2 \omega^2 \sin^2(\omega t)\]where we have used the trigonometric identity: \(\sin^2(\omega t) = \tfrac12 (1 - \cos(2\omega t))\)
From this it follows that:
โก๏ธ Frequentie = 2ฯ
๐ต The potential spring energy is given by:
\[E_p = \tfrac12 k y^2 = \tfrac12 k A^2 \cos^2(\omega t)\]where
\[\cos^2(\omega t) = \tfrac12 (1 + \cos(2\omega t))\]โก๏ธ Also 2ฯ
โช The total energy
\[E = E_k + E_p = \text{constant (zonder demping)}\]With damping:
The ball:
The energie:
โก๏ธ So two peaks per cycle
| Entity | Frequency |
|---|---|
| Position | ฯ |
| Velocity | ฯ |
| Energy (KE/PE) | 2ฯ |
| Total energy | ~ constant (or slowly diminishing) |
โ๏ธ Exactly in accordance with theory!
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