Couple buoyancy stiffness to a bounded oscillation model
A partially immersed body can gain an incremental restoring force from buoyancy. The supplied 2 kg vertical cylinder has constant cross-section 0.01 m², height 0.3 m and equilibrium immersion depth 0.15 m. An ideal 20 N/m spring has a suitable static preload; displacement is measured upward from that equilibrium.
Learning outcome: Derive the hydrostatic stiffness from displaced volume, combine it with spring stiffness and test the full amplitude range before assigning a period.
Starting recap
Derive equations with their initial or boundary conditions, integrate distributed profiles, and check model domains independently of numerical outputs.
All lessons are open directly. Your saved experiments remain when you leave and return.
Try
For density 800 kg/m³, combined stiffness is 100 N/m and accepted small-amplitude frequency is sqrt(50) rad/s. The same frequency could arise from a different density and a compensating spring stiffness, so an independent spring calibration matters.
Open the experimentPredict
Why is an amplitude of 0.2 m rejected for the single partial-immersion oscillator?
Experiment
Derive the hydrostatic stiffness from displaced volume, combine it with spring stiffness and test the full amplitude range before assigning a period.
Couple hydrostatic stiffness to a bounded oscillator
Worked teaching model · synthetic values · A2 kg vertical cylindrical body of cross-section.01 m²,height.3 m is partially immersed at equilibrium depth.15 m and connected to an ideal20 N/m spring with suitable static preload. Liquid surface fixed,g=10, added-mass/drag/waves neglected. Accept the linear buoyancy branch only for amplitude strictly below.15 m; touching/leaving an end requires a piecewise model. Virtual only.
Reset keeps your prediction, captures and practice history.
Buoyancy change within partial-immersion domain
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: −rho g A x (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Buoyancy change within partial-immersion domain
−rho g A x
| upward displacement (m) | buoyancy change (N) |
|---|---|
| -0.15 | 15 |
| -0.14625 | 14.625 |
| -0.1425 | 14.25 |
| -0.13875 | 13.875 |
| -0.135 | 13.5 |
| -0.13125 | 13.125 |
| -0.1275 | 12.75 |
| -0.12375 | 12.375 |
| -0.12 | 12 |
| -0.11625 | 11.625 |
| -0.1125 | 11.25 |
| -0.10875 | 10.875 |
| -0.105 | 10.5 |
| -0.10125 | 10.125 |
| -0.0975 | 9.75 |
| -0.09375 | 9.375 |
| -0.09 | 9 |
| -0.08625 | 8.625 |
| -0.0825 | 8.25 |
| -0.07875 | 7.875 |
| -0.075 | 7.5 |
| -0.07125 | 7.125 |
| -0.0675 | 6.75 |
| -0.06375 | 6.375 |
| -0.06 | 6 |
| -0.05625 | 5.625 |
| -0.0525 | 5.25 |
| -0.04875 | 4.875 |
| -0.045 | 4.5 |
| -0.04125 | 4.125 |
| -0.0375 | 3.75 |
| -0.03375 | 3.375 |
| -0.03 | 3 |
| -0.02625 | 2.625 |
| -0.0225 | 2.25 |
| -0.01875 | 1.875 |
| -0.015 | 1.5 |
| -0.01125 | 1.125 |
| -0.0075 | 0.75 |
| -0.00375 | 0.375 |
| 0 | 0 |
| 0.00375 | -0.375 |
| 0.0075 | -0.75 |
| 0.01125 | -1.125 |
| 0.015 | -1.5 |
| 0.01875 | -1.875 |
| 0.0225 | -2.25 |
| 0.02625 | -2.625 |
| 0.03 | -3 |
| 0.03375 | -3.375 |
| 0.0375 | -3.75 |
| 0.04125 | -4.125 |
| 0.045 | -4.5 |
| 0.04875 | -4.875 |
| 0.0525 | -5.25 |
| 0.05625 | -5.625 |
| 0.06 | -6 |
| 0.06375 | -6.375 |
| 0.0675 | -6.75 |
| 0.07125 | -7.125 |
| 0.075 | -7.5 |
| 0.07875 | -7.875 |
| 0.0825 | -8.25 |
| 0.08625 | -8.625 |
| 0.09 | -9 |
| 0.09375 | -9.375 |
| 0.0975 | -9.75 |
| 0.10125 | -10.125 |
| 0.105 | -10.5 |
| 0.10875 | -10.875 |
| 0.1125 | -11.25 |
| 0.11625 | -11.625 |
| 0.12 | -12 |
| 0.12375 | -12.375 |
| 0.1275 | -12.75 |
| 0.13125 | -13.125 |
| 0.135 | -13.5 |
| 0.13875 | -13.875 |
| 0.1425 | -14.25 |
| 0.14625 | -14.625 |
| 0.15 | -15 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| Hydrostatic incremental stiffness | 100 | N/m |
| Combined small-displacement stiffness | 120 | N/m |
| Entire motion strictly partially immersed | 1 | flag |
| Proposed minimum immersion depth | 0.05 | m |
| Proposed maximum immersion depth | 0.25 | m |
| Accepted ideal angular frequency | 7.745967 | rad/s |
| Accepted ideal period | 0.8111557 | s |
| Initial quadratic oscillation energy | 0.6 | J |
The entire selected oscillation stays inside the linear partial-immersion branch.
Upward displacement reduces displaced volume by A x, so buoyancy changes by−rho g A x. Add spring restoring change−k_s x about the preloaded equilibrium: m xddot=−(k_s+rho g A)x. A period alone identifies this combined stiffness, not density separately if spring stiffness is unknown. Added mass, drag, radiated waves and end-crossing buoyancy are excluded, not assumed negligible in every real system.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
The entire selected oscillation stays inside the linear partial-immersion branch.
Upward displacement reduces displaced volume by A x, so buoyancy changes by−rho g A x. Add spring restoring change−k_s x about the preloaded equilibrium: m xddot=−(k_s+rho g A)x. A period alone identifies this combined stiffness, not density separately if spring stiffness is unknown. Added mass, drag, radiated waves and end-crossing buoyancy are excluded, not assumed negligible in every real system.
Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.
Understand
Apply
Public practice checks a specific claim. It does not establish independent mastery.
Which independent measurement separates density inference from an unknown spring stiffness?
Continue
Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.
Skipping retains experiments and awards no mastery.
Save a science backup or import one
Science work is stored locally in this browser, separately from mathematics and AI. It does not sync automatically. Use this science-only backup; this checkout's older all-subject export does not include science.
A legacy file is available only when the complete snapshot fits the 36 MB limit. For larger snapshots, prepare a complete file set below; never discard notebooks to make a smaller file.
Matching imported notebook drafts replace their current drafts. Omitted lessons and known first attempts remain. Export your current work first if you want to retain both drafts.
Complete file sets and course backups
A file set can hold more work than one science file. Keep its manifest and every part together. Each file is limited to 36 MB. This format needs this version of LearnAI or a compatible newer version; older versions can use only the legacy single-file backup above.
The set includes every saved science course and global continuation at this moment. Later edits need a new export. A conflicting tab exports its own visible work, which may differ from saved work in another tab.
0 file(s) selected. File order and filenames do not determine identity. Missing, changed or conflicting parts block the entire restore; no good part is applied by itself.
Science storage: loading. No account or remote computation is required.