Simple Harmonic Motion
Springs, pendulums, and oscillation
Simple Harmonic Motion (SHM) occurs when a restoring force is proportional to displacement: F = -kx. The period depends only on mass and spring constant, not on amplitude — a key insight that confuses many students. Energy constantly transforms between kinetic (maximum at equilibrium) and potential (maximum at extreme positions), with total mechanical energy conserved in the absence of damping.
If you double the amplitude of an ideal spring oscillator, does it take longer to complete a cycle?
Most people guess yes, but the answer is no for ideal simple harmonic motion — the period is independent of amplitude when the restoring force is exactly linear.
What is Simple Harmonic Motion?
Pluck a guitar string, push a kid on a swing, watch a tuning fork hum after you tap it on a desk — all of them trace out the same mathematical shape: a smooth sine wave moving back and forth around a balance point. Simple harmonic motion is the physics of any system where the force pulling you back toward equilibrium grows in proportion to how far you've strayed (F = -kx). That single rule produces the cleanest motion in physics: position is sinusoidal, the period is independent of amplitude, and energy sloshes between kinetic and potential without leaking out. This lab gives you a spring-mass system with sliders for k, m, amplitude, and optional damping. Watch the energy bar split between Ek and Ep as the mass races through equilibrium and freezes at the turning points, and try to break the period formula by changing amplitude — you can't.
Parameters explained
- Amplitude (A)(m)
- Amplitude A is the starting stretch from equilibrium, measured in meters. Increasing it gives the mass farther to travel and stores more elastic potential energy because E = 1/2 kA^2. It also raises maximum speed by the same factor, so the period of an ideal spring does not change when only A changes. Use this slider with damping at 0 first: compare a small and large amplitude while watching the period readout. Then add damping and notice that the amplitude envelope shrinks over time as mechanical energy leaves the visible oscillation.
- Spring Constant (k)(N/m)
- Spring Constant k controls how strongly the spring pulls back for each meter of displacement. A larger k means a steeper restoring force, so the angular frequency rises as omega = sqrt(k/m) and the period gets shorter. Keep Mass and Amplitude fixed, then move only k to isolate this relationship in the live period readout. This is the cleanest way to connect Hooke's Law, F = -kx, to the motion you see: stiff springs snap the mass back quickly, while soft springs produce slower, wider-looking oscillations.
- Mass (m)(kg)
- Mass m is the inertia attached to the spring. A heavier mass resists acceleration more, so the same spring takes longer to complete each cycle. The period follows T = 2pi sqrt(m/k), which means doubling mass increases period by sqrt(2), not by 2. Hold Spring Constant and Amplitude steady, then move only Mass to make this square-root relationship visible. The gravitational weight of the mass is not what sets the oscillation timing here; the model measures motion around equilibrium, where inertia and spring stiffness dominate.
- Damping Coefficient (b)
- Damping sets how strongly the model removes mechanical energy from the oscillator as it moves. At 0, the Undamped SHM preset keeps tracing a steady phase-space ellipse and the total energy stays constant. Higher values oppose motion more strongly, so the amplitude shrinks and the phase path spirals inward. Compare Lightly Damped with the Critically Damped button before changing the slider manually. The HTML control uses a normalized 0 to 0.5 range, so treat it as a visual damping-strength comparison rather than a measured damping coefficient in kg/s.
Common misconceptions
MisconceptionIf I double the amplitude, the oscillation takes twice as long because the mass has farther to travel.
CorrectThe mass does travel farther, but it also moves faster — its maximum speed scales with A. The two effects cancel exactly. Period stays at T = 2π√(m/k) for any amplitude, which is why a clock pendulum keeps the same time even as its swing shrinks.
MisconceptionThe mass is moving fastest at the extreme positions because the spring is pulling hardest there.
CorrectAt the extreme positions the mass is momentarily at rest — that's where it turns around. Force is largest there, but velocity is zero. Maximum speed happens at the equilibrium point, where force is zero but kinetic energy is at its peak.
MisconceptionWhen damping kicks in, the energy is destroyed.
CorrectEnergy is conserved — it just leaves the visible oscillation. Damping converts kinetic and elastic potential energy into thermal energy in the spring, the surrounding air, and any bearings. Total energy of the universe is unchanged; the oscillator just gives up its share to heat.
MisconceptionA heavier mass on the same spring oscillates faster because gravity pulls it harder.
CorrectHeavier mass means slower oscillation. The gravitational pull just shifts the equilibrium position lower; it doesn't enter the period. The dynamics around equilibrium follow T = 2π√(m/k), so more mass means a longer period — even though gravity is pulling harder.
MisconceptionEnergy is greatest when the mass is moving fastest.
CorrectTotal mechanical energy is constant in undamped SHM (E = ½kA² = ½mv²_max). What changes is the split: at equilibrium, energy is 100% kinetic; at maximum displacement, it's 100% potential; everywhere else, it's a mix that adds to the same total.
How teachers use this lab
- Amplitude-independence test: have pairs predict whether doubling A changes the period, then run two trials with the Amplitude slider while keeping Spring Constant, Mass, and Damping fixed. Students compare the live T value and explain why greater distance does not mean a longer ideal-SHM period.
- Mass versus stiffness station: assign one group to vary only Spring Constant and another to vary only Mass. Each group records T values, identifies whether period rises or falls, and connects the evidence to T = 2π√(m/k).
- Energy split lab: use the Undamped SHM preset, pause mentally at x = 0, ±A/2, and ±A, and ask students to predict how KE and PE should trade places while the total energy readout stays constant.
- Damping comparison: run Undamped SHM, Lightly Damped, and Critically Damped in sequence. Students describe the visible motion, energy trend, and phase-space path for each preset, treating the buttons as damping-strength cases before touching the Damping slider.
- Misconception probe: ask when the mass is moving fastest before students touch the controls. Then have them watch the velocity readout and energy panel as the mass crosses equilibrium and reaches each turning point.
Frequently asked questions
Why is the period independent of amplitude in SHM?
Because the restoring force is exactly linear in displacement (F = -kx). When you double the amplitude, you double both the maximum force and the maximum distance to travel. Larger force gives larger acceleration; larger distance demands more travel time. The two effects scale identically and cancel out in the equations of motion. This linear restoring force is the defining property of SHM, and it's what makes the period formula T = 2π√(m/k) so clean — no A in sight.
How are kinetic and potential energy related in SHM?
Total mechanical energy E = ½kA² is conserved when damping is zero. At maximum displacement, all of E sits as elastic potential energy ½kx² with x = A; the mass is momentarily at rest. At equilibrium (x = 0), all of E sits as kinetic energy ½mv², and v = v_max = Aω. Everywhere in between, the two add to the same total. Watching the energy panel cycle between KE and PE as the mass moves makes this visible in real time.
What does damping actually do to the motion?
Damping is a velocity-dependent effect that opposes motion and drains mechanical energy into heat. In this HTML simulation, the Damping slider is a normalized visual control from 0 to 0.5, not a lab-measured coefficient in kg/s. At 0, the oscillator keeps cycling with a steady phase-space ellipse. As damping increases, the amplitude decays and the phase path spirals inward. Use the Undamped SHM, Lightly Damped, and Critically Damped buttons as quick comparison cases for increasing damping strength.
How does this connect to AP Physics 1 standards 3.B.3 and 5.B.3?
AP Physics 1 standard 3.B.3 expects students to predict the motion of an object under a restoring force proportional to displacement — exactly the F = -kx setup here. Standard 5.B.3 layers in energy conservation, expecting students to track the kinetic-potential exchange and recognize total mechanical energy is preserved in undamped SHM. NGSS HS-PS3-2 reinforces the energy side: explain that visible oscillation is a transformation between two energy stores, with damping converting orderly motion into thermal energy.
Why does the same equation describe springs, pendulums, and atoms in a crystal?
All three have a stable equilibrium and a smooth potential energy curve near that equilibrium. Whenever you zoom in close enough to a smooth U(x) minimum, it looks parabolic — and a parabolic potential energy gives a linear restoring force, which is the SHM equation. Springs are designed parabolic, pendulums approach parabolic for small angles, and the bonds between atoms in a crystal lattice are parabolic for small displacements. That's why SHM is one of the most reused models in all of physics, from clocks to acoustic waves to phonons to molecular vibrations.