Mass–Spring Oscillator

What sets the period of a mass bouncing on a spring? Mass and spring stiffness set the period: four times the mass doubles it. Amplitude does not change it. Hanging the spring vertically only moves the equilibrium point down; gravity leaves the stiffness, and with it the period, unchanged.

Drag a spring mass and compare energy exchange, the phase-space ellipse and rotating phasor. Time noisy camera crossings and try vertical suspension.

Kinetic energy $28.8\,\mathrm{mJ}$

Potential about equilibrium $28.8\,\mathrm{mJ}$

Camera crossing period $1.57\,\mathrm{s}$

Ideal model period $1.57\,\mathrm{s}$

Model assumptions

A point mass and massless linear spring move without friction. Displacement is rightward on the track and downward in hanging mode, measured from equilibrium. Gravity shifts the hanging equilibrium but not the natural frequency. The location control translates the support and equilibrium together. The displayed potential is spring plus gravitational energy relative to equilibrium; separate changes remain in the exported record. Camera frames have exact timestamps and independent Gaussian position noise. Rising crossings are interpolated after a negative excursion; a straight-line fit of crossing time against crossing index estimates the period. Nine ideal periods set the recording window; inference uses camera samples, not the model period. Standard errors are conditional on this detector and noise model. Coil shape, carriage dimensions and supports are schematic. No wire mass, nonlinear elasticity, collisions or material limits are modeled.

Focus question: Which changes alter the period, and which only move the starting point on the orbit?
One-sentence intuition: Mass and stiffness set the period. Amplitude sets the size of the orbit; release phase chooses where the motion starts.

Core mathematical model

Motion relative to equilibrium

m\ddot q+kq=0,\quad\omega_0=\sqrt{k/m},\quad T_0=2\pi/\omega_0

Position is positive rightward on the track and downward when hanging. The equilibrium-location control translates the entire support.

Release amplitude and phase

q=A\cos(\omega_0t+\varphi),\qquad v=-A\omega_0\sin(\omega_0t+\varphi)

A quarter-turn phase starts at equilibrium with a kick. Neither phase nor amplitude changes the ideal period.

The ellipse and the circular phasor

E=\frac{p^2}{2m}+\frac{kq^2}{2},\quad p=mv,\quad q^2+(-v/\omega_0)^2=A^2

Momentum against displacement makes an ellipse. The phasor uses negative velocity divided by frequency and equal length scales, so it turns counterclockwise.

The camera clock

t_{\uparrow,j}=a+j\widehat T+r_j

Interpolate rising crossings in the camera samples after a resolved negative excursion. Fit crossing time against index; the slope is the period, with a conditional regression standard error.

Hanging does not change the linear restoring law

\Delta\ell_{\rm static}=mg/k,\quad\Delta U_s=\tfrac12kq^2+mgq,\quad\Delta U_g=-mgq

Gravity shifts equilibrium. Its energy change cancels the elastic linear term, leaving the potential plotted above.

Common difficulties

Force and velocity do different jobs

Typical misconceptionThe spring force is largest where the carriage is fastest.

Better mental modelAt a turning point speed is zero and restoring force is largest. At equilibrium the spring’s net restoring force is zero while speed is largest.

Hanging equilibrium is stretched

Typical misconceptionGravity must make the clock slower.

Better mental modelIt stretches the spring to a new equilibrium. For this ideal linear law the stiffness about that point stays unchanged.

Run the experiment

01
Follow one quarter turn
Pause, then move Replay time through a quarter period.
What to observe: Elastic potential becomes kinetic energy while the phase-space point moves along the ellipse.
02
Change the clock
Compare A heavier carriage with A quicker spring.
What to observe: More mass lengthens the period; more stiffness shortens it. The camera and model periods track the same change.
03
Rotate the apparatus
Choose Hang it vertically and adjust gravity in the details.
What to observe: Static extension changes, but the period at fixed mass and stiffness stays the same.