M061 · Oscillations / calibration

Mass–Spring Oscillator

A frictionless cart and a vertical suspension share a calibrated spring. Drag the mass, give it an initial kick, measure positive-going zero crossings and compare the measured period with the exact solution. Equal phase-space scales reveal what amplitude and gravity actually change.

Elapsed time $0\,\mathrm{s}$

Displacement from equilibrium $0\,\mathrm{m}$

Velocity $0\,\mathrm{m\,s^{-1}}$

Undamped reference period $0\,\mathrm{s}$

Mechanical energy relative to equilibrium $0\,\mathrm{J}$

Integrated dissipation $0\,\mathrm{J}$

Integrated drive work $0\,\mathrm{J}$

Energy ledger defect $0\,\mathrm{J}$

Position error against exact solution $0\,\mathrm{m}$

Endpoint change with half step $0\,\mathrm{m}$

Actual integration step $0\,\mathrm{ms}$

Period from positive-going crossings $\text{Collecting}$

Gravity-induced static extension $0\,\mathrm{m}$

Amplitude from the release state $0.3\,\mathrm{m}$

RELEASE / TIME / CALIBRATE

What sets the period?

Spring and gravity vectorsForces use a common instantaneous scale; arrow length is schematic.Moving mass

m

1.00\,\mathrm{kg}

Spring stiffness

k

20.00\,\mathrm{N\,m^{-1}}

Release displacement

x_0

0.30\,\mathrm{m}

Initial velocity

v_0

0.00\,\mathrm{m\,s^{-1}}

Vertical suspension: zero off, one on

I_{\mathrm{vertical}}

0.00\,

Local gravity

g

9.81\,\mathrm{m\,s^{-2}}

Maximum integration step

\Delta t_{\max}

20.00\,\mathrm{ms}

Inspect experiment time

t/t_{\mathrm{end}}

0.00\,\%

Drag the mass to set release displacement. Keyboard sliders cover every control. Blue is velocity divided by natural angular frequency; teal is displacement. The reference circle uses equal physical scales.

Point mass, massless Hookean spring and no friction; displacement is measured from equilibrium. Positive displacement is rightward on the track and downward in suspension. The vertical energy includes spring and gravity relative to equilibrium, up to a constant. The spring drawing is schematic. RK4 integration has an exact-solution and half-step audit; refinement is a diagnostic, not a certified error bound.

Physics tutorial

A spring as a clock

BackgroundA linear spring links force to displacement. Repeating motion provides a period standard that can calibrate stiffness from a known mass, or mass from a known spring. The horizontal cart and vertical suspension provide a direct test of what gravity changes.

Why it mattersOpenStax University Physics Volume 1, section 15.1 supplies the ideal simple-harmonic reference. This bench puts the exact reference beside event timing, phase geometry and an independently solved numerical trajectory.

Start with the essentials

Focus question: Does a larger amplitude, a heavier mass or a stronger gravity change the period?
One-sentence intuition: Measure events before comparing to the reference. Moving equilibrium changes the absolute position without changing the local linear stiffness.

Core mathematical model

Equilibrium and exact dynamics

\ell_{\mathrm{eq}}-\ell_0=\frac{mg}{k},\qquad m\ddot x+kx=0

Gravity shifts a vertical spring equilibrium; the displacement equation is unchanged. Positive displacement is down in suspension and right on the track.

Period and arbitrary release state

\omega_0=\sqrt{\frac{k}{m}},\quad T_0=\frac{2\pi}{\omega_0},\quad x(t)=x_0\cos(\omega_0t)+\frac{v_0}{\omega_0}\sin(\omega_0t)

Mass and stiffness set the period. Amplitude and release phase are independent of that period in the ideal linear model.

Energy circle

E=\frac12mv^2+\frac12kx^2=\frac12kA^2,\qquad x^2+\left(\frac v{\omega_0}\right)^2=A^2

Velocity is divided by natural angular frequency so both phase coordinates have units of length. Equal scales make a circle; a raw position–velocity plot generally makes an ellipse.

An event-based period measurement

\widehat T=\frac{t_{\uparrow,N}-t_{\uparrow,1}}{N-1},\qquad N\ge2

The clock uses linearly interpolated positive-going zero crossings of the numerical trajectory, not the known period. A stationary mass has no crossings and therefore no measured period.

Common difficulties

Amplitude is not stiffness

Typical misconceptionA larger release must oscillate more slowly.

Better mental modelCompare two amplitudes with fixed mass and stiffness; the periods agree in this linear model.

Equilibrium is not the natural length

Typical misconceptionThe vertical spring oscillates around its unstretched length.

Better mental modelStatic extension balances weight. Measure displacement around that shifted equilibrium.

No events means no measured period

Typical misconceptionThe clock should display the reference period even when nothing moves.

Better mental modelSet both release displacement and velocity to zero. The reference remains defined but the event instrument reports stationary.

Run the experiment

01
Time the reference
Measure eight periods with the horizontal preset.
What to observe: The event period agrees with the exact period; total energy stays constant.
02
Change scale and release phase
Double mass, then compare a larger release and an equilibrium kick.
What to observe: Mass changes period; amplitude and release phase change the orbit size or starting point.
03
Move equilibrium
Choose vertical suspension and vary gravity.
What to observe: Static extension changes while the period and relative energy law remain unchanged.
04
Audit the solve
Reduce maximum step and inspect exact-position error and the half-step endpoint change.
What to observe: The actual step may already be capped for stability. Refinement uses half of that actual step.