simple harmonic motion

Tags

phys/wave

cegep/3

Word count

233 words

Reading time

2 minutes

Oscillatory motion whose restoring force is directly proportional to displacement
Creates sinusoidal waves
One-dimensional projection of uniform circular motion
Abbr. SHM

$$\begin{array}{rl}x(t)& =A\cos (\omega t+{\varphi }_0)\\ v(t)& =-A\omega \sin (\omega t+{\varphi }_0)\\ a(t)& =-A{\omega }^2\cos (\omega t+{\varphi }_0)\\ \\ \omega & =2\pi f=\frac{2\pi }{T}\\ & =\sqrt{\frac{k}{m}}\\ \varphi & =\omega t+{\varphi }_0\\ F& =-kx\\ \\ \frac{{\mathrm{d}}^{2}x}{\mathrm{d}{t}^2}+\frac{g}{L}\theta & =0\\ E=K+U& =\frac{1}{2}m{v}^2+\frac{1}{2}k{x}^2\\ & =\frac{1}{2}m{v}_{\text{max}}^2\\ & =\frac{1}{2}k{A}^2\end{array}$$

Oscillation around the equilibrium is symmetrical even for vertical SHM.

No work is done by non-conservative forces => mechanical energy is conserved

Examples

A 0.500-kg cart connected to a light spring for which the spring constant is 20.0 N/m oscillates on a frictionless, horizontal air track.

1. Using energy, calculate the maximum speed of the cart if the amplitude of the motion is 3.00 cm.

$$\begin{array}{rl}E=\frac{1}{2}k{A}^2& =\frac{1}{2}m{v}_{\text{max}}^2\\ \frac{1}{2}20\cdot 0.03& =\frac{1}{2}0.5v_{\text{max}}^2\\ v_{\text{max}}& =\sqrt{1.2}\frac{\mathrm{m}}{\mathrm{s}}\end{array}$$

2. Compute the kinetic and potential energies of the system when the position of the cart is 2.00 cm.

$$\begin{array}{rl}U& =\frac{1}{2}k{x}^2\\ & =\frac{1}{2}20\cdot {0.02}^2\\ & =0.004\mathrm{J}\\ \\ E=\frac{1}{2}k{A}^2& =K+U\\ \frac{1}{2}20\cdot {0.03}^2& =K+0.004\\ K& =0.005\mathrm{J}\end{array}$$

Contributors

Ajitani Hifumi

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