Pendulums & Springs

What you will investigate

This pendulum virtual lab compares a swinging mass with a spring oscillator. Change one condition at a time to investigate period, displacement, velocity, acceleration, and energy. The course also includes mass-measurement and physical-pendulum extensions. These are related oscillation models, but their period dependencies are not interchangeable.

Measure repeated oscillations using the same definition of a complete cycle. A crossing interval can be part of a cycle rather than the whole period. Keep release angle, length, mass, planetary gravity setting, and measurement mode attached to the record so that a comparison isolates the intended change.

Small-angle pendulum references

For the small-angle ideal pendulum, T = 2π√(L/g). The following reference values use the course's Earth gravity setting and its small-angle expression. They are not the exact finite-angle period for every release amplitude.

Ideal small-angle reference length	Period
0.25 m	1.003544962 s
0.50 m	1.419226895 s
1.00 m	2.007089923 s

The relationship is proportional to the square root of length, not to length itself. A period-squared versus length graph can make the relationship linear. Keep release angle controlled, because the course's finite-angle model applies an amplitude correction when interpreting a gravity estimate.

Mass and amplitude depend on the model

Changing pendulum-bob mass does not change its ideal period at matching length, gravity, and release angle. Increasing a pendulum's release angle beyond the small-angle range can lengthen the period. These statements have different conditions; the first does not establish that every change in the pendulum setup leaves the timing unchanged.

For the ideal spring oscillator, T = 2π√(m/k). Mass affects this model's period, unlike the matching ideal pendulum comparison. The spring's stiffness must also be specified. Compare a static force–extension estimate of stiffness with a period-based estimate instead of assuming the label on a spring is an exact physical measurement in every run.

Motion graphs and energy

The course links displacement, velocity, and acceleration using the same coordinate. In the ideal spring model, acceleration is proportional to displacement and points toward equilibrium. Maximum displacement does not coincide with maximum speed. A graph-based explanation can test whether the force diagram, motion arrows, and timing tell a consistent story.

The energy view relates kinetic and potential energy through the cycle. Equilibrium position and turning points have different energy distributions. Changing amplitude changes the total oscillator energy even when the ideal spring period remains unchanged. It is therefore misleading to interpret an unchanged period as proof that the energy has remained unchanged.

Extensions and measurement limits

The inertial-balance comparison distinguishes inertia from a weight reading. The physical-pendulum extension considers mass distribution about a pivot rather than representing the entire object as a point bob. A meterstick pivot comparison therefore requires its own inertia and geometry information.

In Real mode, length error, actual release-angle variation, spring calibration, and timing resolution affect the analysis. Averaging repeated periods can improve a timing estimate, but it does not automatically remove a systematic length error. Fit and interpret a graph with the appropriate correction and uncertainty rather than forcing every line through an expected intercept.

Standards and classroom discussion

The course supports the catalog's AP Physics oscillation topics and selected energy relationships. Resonance and torsional-oscillator experiments are outside its stated experimental scope. Ask students to explain why mass affects the spring period but not the matching ideal pendulum period, then connect a motion graph with an energy description. Continue with energy on a track or Newton's second law.

Build an explanation from your records

Make a prediction before changing the conditions, then record a comparison that actually tests it. Keep the conditions attached to each row: a number without its setup can be misleading even when the arithmetic is correct. Explain what changed, what stayed fixed, and which observation supports your conclusion. If the result differs from your prediction, revise the explanation using the evidence rather than choosing a new answer without a reason.

The reference examples on this page describe the course model or its stated source data. They are not measurements collected from a physical laboratory by the reader. Compare like conditions, keep units visible, and distinguish rounded instrument displays from underlying model quantities. Use the investigation's stated limits when deciding whether a result can be generalized beyond the simulated setup. This guide supports preparation and interpretation; practical laboratory technique and safety training remain separate learning activities.