OSCILLATIONS & ENERGY

Mass–Spring Laboratory

Pull the mass, let go, and follow the energy.

Pull. Release. Repeat.

Ready

Force & motion vectors

Drag the mass and release from rest. Keyboard: ← / → to pull; Enter to release. Drag range ±2 m. Use the initial velocity control for a moving release.

Speed

DISPLACEMENT

VELOCITY

ACCELERATION

TIME

Step = 0.05 simulation seconds. Forces share one arrow scale; velocity and acceleration use separate scales. Zero vectors have no arrow.

Motion & energy

Last 30 s · time (s) →

K · kineticU · elasticE · mechanical (dashed)Q · dissipated

Follow the energy

Kinetic K

Elastic U

Dissipated Q

Bars share the initial energy scale. E = K + U. Damping and surface friction dissipate mechanical energy: dE/dt = −bv² − μmg|v| while sliding. Static friction does no work.

K = ½mv² · U = ½kx²
K + U + Q = E₀

Why?

Why does force point toward equilibrium?

F_spring = −kx. Stretching (x > 0) pulls left; compression (x < 0) pushes right. At x = 0, the spring force is zero.

Why is speed greatest at equilibrium?

Without damping or surface friction, K + U stays constant. At equilibrium, U = ½kx² is zero, leaving all mechanical energy as kinetic energy. At each turning point, v = 0 and all energy is in the spring.

Why does a heavier mass move more slowly?

The same spring force gives a heavier mass less acceleration: a = F/m. The ideal period T = 2π√(m/k) increases with √m; quadrupling m doubles T.

Why does a stiffer spring oscillate faster?

A larger k gives more restoring force for the same displacement. T decreases as 1/√k; quadrupling k halves the period.

Why does damping reduce amplitude?

F_damping = −bv always opposes velocity and removes energy at rate bv². Less mechanical energy means smaller excursions. It is velocity damping, not sliding friction.

Why doesn’t amplitude change the ideal period?

A larger release travels farther, but the linear spring produces proportionally larger forces and speeds. These effects balance: the ideal undamped period depends only on m and k.

Why can friction stop the mass off-center?

While sliding, friction opposes motion with magnitude μmg. At rest it balances the spring force whenever |kx| ≤ μmg, so the mass can stay still away from equilibrium. One coefficient controls both static and sliding friction here.

When does oscillation disappear?

With μ = 0, critical damping is b = 2√(mk). Critical and overdamped systems return toward equilibrium without sustained oscillation. From rest, critical damping returns fastest without overshooting; overdamping returns more slowly. An initial velocity may carry either system past equilibrium once. With surface friction, static holding can stop the return early.

F_spring = −kx
F_damping = −bv
F_friction = −μmg sign(v) (sliding)
ma = −kx − bv + F_friction

TRY A CHALLENGE

Find a two-second rhythm.

With a fixed 1 kg mass, no damping and no surface friction, adjust the spring stiffness for a period of 2.0 seconds.

The period is measured between two equilibrium crossings in the same direction. Aim for 2.00 ± 0.05 s.

Show a hint

Rearrange T = 2π√(m/k) to get k = 4π²m/T². A period that is too short needs a softer spring.