13Class 11 Physics

Oscillations formulas

Simple harmonic motion formulas for Oscillations. Everything follows from a = −ω²x, so identify ω first.

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Motion

Displacement: $\displaystyle x = A\sin(\omega t + \phi)$
Velocity: $\displaystyle v = \omega\sqrt{A^2 - x^2}$
Largest at the mean position, where v = Aω.
Acceleration: $\displaystyle a = -\omega^2 x$
Largest at the extremes, where |a| = Aω².
Angular frequency: $\displaystyle \omega = 2\pi\nu = \frac{2\pi}{T}$

Energy

Kinetic energy: $\displaystyle K = \tfrac12 m\omega^2(A^2 - x^2)$
Potential energy: $\displaystyle U = \tfrac12 m\omega^2 x^2$
Total energy: $\displaystyle E = \tfrac12 m\omega^2 A^2$
Stays constant.

Common systems

Spring and mass: $\displaystyle T = 2\pi\sqrt{\frac{m}{k}}$
Simple pendulum: $\displaystyle T = 2\pi\sqrt{\frac{L}{g}}$
For small angles.
Springs in series: $\displaystyle \frac{1}{k} = \frac{1}{k_1} + \frac{1}{k_2}$
Springs in parallel: $\displaystyle k = k_1 + k_2$

Where students slip up

Thinking mass changes a pendulum's period. It doesn't, though it does change a spring's.
Confusing ν and ω. ω = 2πν, and it's a common factor-of-2π slip.
Assuming energy oscillates at the same frequency as displacement. KE and PE oscillate at twice that frequency.

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