AcademyOscillations
Academy
Applications of Harmonic Motion
Level 1 - Physics topic page in Oscillations.
Principle
Stable systems become harmonic when displaced only slightly.
This page turns SHM from a spring-only model into a local model for many stable systems.
Notation
\(x_0\)
equilibrium position
\(\mathrm{m}\)
\(\eta\)
small displacement from equilibrium
\(\mathrm{m}\)
\(F(x)\)
force as a function of position
\(\mathrm{N}\)
\(k_{\mathrm{eff}}\)
effective spring constant
\(\mathrm{N\,m^{-1}}\)
\(\omega\)
small-oscillation angular frequency
\(\mathrm{rad\,s^{-1}}\)
\(m\)
oscillating mass
\(\mathrm{kg}\)
Method
Derivation 1: Expand the force near equilibrium
At equilibrium the net force is zero. For small displacement, the leading nonzero force term is the local slope times the displacement.
Equilibrium point
\[F(x_0)=0\]
Small displacement
\[\eta=x-x_0\]
Linear approximation
\[F(x_0+\eta)\approx F(x_0)+F'(x_0)\eta\]
Use equilibrium
\[F(x_0+\eta)\approx F'(x_0)\eta\]
Derivation 2: Identify the effective stiffness
A stable equilibrium pushes back toward the equilibrium point, so the force slope must have the restoring sign.
Restoring form
\[F\approx-k_{\mathrm{eff}}\eta\]
Compare slopes
\[k_{\mathrm{eff}}=-F'(x_0)\]
Stable condition
\[k_{\mathrm{eff}}>0\]
Derivation 3: Convert the local model to SHM
Once the force is linear and restoring, the simple harmonic motion result applies to the displacement from equilibrium.
Newton model
\[m\frac{d^2\eta}{dt^2}=-k_{\mathrm{eff}}\eta\]
SHM frequency
\[\omega=\sqrt{\frac{k_{\mathrm{eff}}}{m}}\]
Vertical spring shift
\[ky_0=mg\]
A constant force shifts equilibrium but does not change the local stiffness.
Rules
These are the compact results from the derivations above.
Equilibrium condition
\[F(x_0)=0\]
Effective stiffness
\[k_{\mathrm{eff}}=-F'(x_0)\]
Small-oscillation frequency
\[\omega=\sqrt{\frac{k_{\mathrm{eff}}}{m}}\]
Vertical spring shift
\[ky_0=mg\]
Examples
Question
A vertical spring stretches by \(y_0\) under mass \(m\). Find its oscillation frequency about equilibrium.
Answer
The static shift obeys
\[ky_0=mg\]
but oscillations about that shifted point still use the spring stiffness: \[\omega=\sqrt{\frac{k}{m}}\]
Checks
- Oscillation is about the shifted equilibrium.
- Stable equilibrium needs positive effective stiffness.
- The linear model fails for large displacement.
- Constant forces shift equilibrium but not stiffness.