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Energy in oscillations

In a harmonic oscillation the energy alternates between kinetic and potential form, while the total energy stays constant without friction. The speed and the acceleration are greatest at opposite points of the motion: the speed is greatest at the equilibrium position, the acceleration greatest at the turning points.

v_{max} = A\omega

maximum speed, at the equilibrium position

a_{max} = A\omega^2

maximum acceleration, at the turning points

E = \tfrac12 kA^2 = \tfrac12 mv_{max}^2

total oscillation energy

Symbols

$v_{max}$	maximum speed	m/s
$a_{max}$	maximum acceleration	m/s²
$E$	total oscillation energy	J

Example

An oscillation has $A = 0.10$ m and $\omega = 20$ rad/s. Maximum speed: $v_{max} = 0.10\cdot 20 = 2.0$ m/s, at the equilibrium position.

At the turning points the speed is zero and the acceleration is greatest – the opposite of the equilibrium position.

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Part of Physics: Oscillations and waves.