AcademyOscillations
Academy
Energy in Oscillators
Level 1 - Physics topic page in Oscillations.
Principle
Energy in SHM trades between kinetic and potential forms.
The SHM force model fixes the motion; energy gives a second way to find speeds and turning points.
Notation
\(K\)
kinetic energy
\(\mathrm{J}\)
\(U_s\)
spring potential energy
\(\mathrm{J}\)
\(E\)
total mechanical energy
\(\mathrm{J}\)
\(A\)
amplitude
\(\mathrm{m}\)
\(k\)
spring constant
\(\mathrm{N\,m^{-1}}\)
\(m\)
oscillating mass
\(\mathrm{kg}\)
Method
Derivation 1: Build the energy terms
Kinetic energy depends on speed. Spring potential energy comes from the work needed to stretch or compress the spring from equilibrium.
Kinetic energy
\[K=\frac{1}{2}mv^2\]
Spring force
\[F_s=-kx\]
Stored spring energy
\[U_s=\int_0^x ks\,ds=\frac{1}{2}kx^2\]
Derivation 2: Use the turning point
At maximum displacement, the oscillator is momentarily at rest. That fixes the total energy in terms of amplitude.
Conserved total
\[E=K+U_s\]
Turning point
\[x=\pm A,\qquad v=0\]
Total energy
\[E=\frac{1}{2}kA^2\]
Derivation 3: Solve for speed at a position
Away from the turning points, the energy gap between total and potential energy is kinetic energy.
Energy balance
\[\frac{1}{2}mv^2+\frac{1}{2}kx^2=\frac{1}{2}kA^2\]
Isolate speed
\[v^2=\frac{k}{m}(A^2-x^2)\]
Use frequency
\[v=\pm\omega\sqrt{A^2-x^2}\]
The sign depends on which way the oscillator is moving.
Rules
These are the compact results from the derivations above.
Kinetic energy
\[K=\frac{1}{2}mv^2\]
Spring potential
\[U_s=\frac{1}{2}kx^2\]
Total energy
\[E=\frac{1}{2}mv^2+\frac{1}{2}kx^2=\frac{1}{2}kA^2\]
Speed from position
\[v=\pm\omega\sqrt{A^2-x^2}\]
Examples
Question
A spring oscillator has
\[k=80\,\mathrm{N\,m^{-1}}\]
and \[A=0.20\,\mathrm{m}\]
Find total energy.Answer
\[E=\frac{1}{2}kA^2=\frac{1}{2}(80)(0.20^2)=1.6\,\mathrm{J}\]
Checks
- Total energy scales as amplitude squared.
- Speed is maximum at equilibrium.
- Potential energy is maximum at turning points.
- Energy conservation assumes no damping or driving.