{"id":"a5cc8744-aba2-4663-a1bb-3f14339f9bfb","arxiv_id":"1908.04256","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A smartphone simultaneously records the rotating water surface and the angular velocity, and the fitted parabolic profiles agree with the predicted ω²/(2g) concavity and -L²/(24g) vertex height.","lead":"This experiment measures the parabolic surface of water in a rotating container using a smartphone's camera and gyroscope. The measured shapes match the standard physics prediction, giving a simple classroom experiment.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Vertex-height confirmation rests on excluding the lowest-omega point without per-point uncertainties; an all-point re-fit is needed to establish the claimed agreement.","rationale":"The central theoretical derivation is standard, and the concavity slope (20.16(4) versus 2g) provides independent support for the model. The reader's CONDITIONAL verdict already calls for addressing the post-hoc exclusion, per-point uncertainties, and unit notation. My stress-test elevates the post-hoc exclusion to a load-bearing check because it directly affects one of the two quantitative confirmations in the strongest claim. However, this does not warrant a stricter verdict: the concavity agreement stands, the paper is transparent about the exclusion, and a re-analysis of the existing data could resolve the concern. Therefore the reader's CONDITIONAL verdict remains appropriate. If the re-fit were to show a large shift in the vertex-height slope, the verdict would need to become REJECT; if the all-point fit agreed, it could become ACCEPT. Since the required data are not in the manuscript, CONDITIONAL is the correct standing verdict.","tokens_in":4317,"tokens_out":11076,"duration_ms":120777,"concrete_test":"Re-fit the vertex height z_v versus omega-squared using all measured frames, with per-point uncertainties propagated from the individual parabolic fits. Compute the slope both unweighted and inverse-variance weighted, and both with and without the excluded leftmost point. If the all-point slope differs from the model value -0.2655 mm*s^2/rad^2 by more than the combined uncertainty, the paper's vertex-height agreement is not established; otherwise the exclusion is benign. Also require an a priori rule for excluding outliers, such as a cutoff based on propagated uncertainty or studentized residuals.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Figure 6 caption states that the leftmost point, corresponding to small angular velocity and 'great uncertainty', was not taken into account in the linear fit of vertex height versus omega-squared. No per-point uncertainties or exclusion criterion are provided. Because the vertex-height slope (-0.27(1) mm*s^2/rad^2 versus the model -0.2655 mm*s^2/rad^2) is one of the two quantitative confirmations in the central claim, omitting a point post hoc could make the agreement appear better than it is. Without the excluded point's coordinates and uncertainty, the reader cannot judge whether the omission is justified or whether including it would shift the slope outside the model prediction. The concavity slope is separately fit and agrees, but both quantities come from the same parabolic fits, so the vertex-height confirmation is not fully independent. This is a load-bearing statistical concern, not an allegation of misconduct.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a simple experiment for an undergraduate fluid mechanics lab: a narrow rectangular container with dyed water is placed on a rotating table, and a smartphone fixed to the table records the free surface on video while its gyroscope measures the angular velocity. Assuming quasi-static rigid-body rotation, the authors derive the parabolic free surface z'(r) = H - (ω²/2g)(L²/12 - r²), with concavity ω²/(2g) and vertex height H - ω²L²/(24g). Fifteen video frames at different angular velocities are analyzed with Tracker; the fitted concavity and vertex height are plotted against ω², and linear fits give slopes 20.16(4) m·rad²/s² (compared with 2g) and -0.27(1) mm·s²/rad² (compared with -L²/(24g) = -0.2655(5) mm·s²/rad²), which the authors regard as good agreement.","tokens_in":4480,"tokens_out":8751,"duration_ms":86812,"significance":"If the two confirmations are robust, the paper provides a clean, low-cost demonstration of solid-body rotation and the parabolic free surface, with the pedagogical advantage of simultaneous smartphone camera and gyroscope measurements. The derivation is standard and free of fitted parameters (the constants g, L, H are external or measured), which is a strength. The use of a smartphone in a rotating frame to measure both the shape and angular velocity is a nice contribution to the physics-education literature.","major_comments":[{"comment":"The reported vertex-height slope -0.27(1) mm·s²/rad² is based on a linear fit that excludes the leftmost (smallest-ω²) point, justified only by the qualitative statement that this point has 'great uncertainty.' No per-point uncertainties, error bars, or quantitative exclusion criterion are provided. Because the vertex-height comparison is one of the two central quantitative confirmations of the model, an unjustified exclusion could materially bias the agreement. Please report the coordinates and an uncertainty estimate for the excluded point, show the fit with all data points (or a weighted fit), and justify the exclusion statistically. If the all-point slope remains consistent with -L²/(24g), state that explicitly; if not, the claim of agreement needs to be qualified.","section":"Section IV, Figure 6 caption and results"},{"comment":"The theoretical derivation assumes steady, rigid-body rotation with negligible transients, while the experiment increases the angular velocity in 'small jumps.' The manuscript does not demonstrate that the fluid has reached a quasi-static state at the times the 15 frames were extracted. A non-zero Euler acceleration (dω/dt) during a jump produces an azimuthal pressure gradient and a surface that is not the simple parabola of Eq. (9); the gyroscope records the instantaneous ω, but the surface shape may lag. Please provide evidence that the fitted concavity and vertex height are stable within each constant-ω plateau, for example by analyzing consecutive frames at the same nominal ω or by estimating the spin-up time of the fluid (e.g., from the container width and water viscosity). This is needed to validate the comparison of the fitted parabolas with Eq. (9).","section":"Section II and Section III"}],"minor_comments":[{"comment":"The condition for the vertex to reach the bottom, ω ≥ √(24gH/L), is dimensionally inconsistent (the right-hand side has units of √(m/s²), not s⁻¹). From Eq. (10) with z_v = 0, the correct condition is ω ≥ (1/L)√(24gH).","section":"Section II, near Eq. (10)"},{"comment":"Please clarify the quantity plotted and the units of the slope 20.16(4) m·rad²/s². A direct fit of the concavity coefficient A to ω² should yield a slope 1/(2g) ≈ 0.051 s²/m; the stated value appears to be 2g, so the text, axes, and units need to be reconciled. Additionally, the difference between 20.16(4) and 19.62 is about 13 times the reported fit uncertainty; some discussion of systematic uncertainties is needed to support the statement of 'good agreement.'","section":"Section IV and Figure 5"},{"comment":"There are several typos and style issues: 'neglectd' (Section II) should be 'neglected'; 'parabole' should be 'parabola'; 'tipically' should be 'typically'; 'and data provided by theapp' in Section III has a doubled 'and'; the title word 'time-depend' should be 'time-dependent.'","section":"Throughout"},{"comment":"The intercept of the vertex-height fit (-7.72(3) cm) is compared with a direct image measurement (-7.6(2) cm); please state how the uncertainty of the direct measurement was obtained, since it is given with only one significant digit.","section":"Section IV, intercept comparison"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Take a look at this one when you get a chance. It's a teaching paper: the physics is the standard rotating-liquid parabola, and the new bit is using a smartphone's camera and gyroscope on the rotating table to measure the surface shape and the angular velocity at the same time. That's a genuinely nice trick for a student lab—cheap, simple, and it works. The derivation in Section II is textbook but clean, and the experimental results do confirm the predicted concavity and vertex-height dependence to within a few percent.\n\nThe main soft spot is statistical, and it's the one flagged by the stress-test. In Figure 6, the leftmost point (small omega) is excluded from the vertex-height fit because it has 'great uncertainty,' but the authors don't give per-point uncertainties, don't show the excluded point's coordinates, and don't specify any exclusion criterion. So the reader can't tell whether the agreement (slope -0.27(1) vs -0.2655) survives an all-point fit. For a small-omega point, the leverage on the slope is low, so the effect may be small—but the paper should show it. This is a legitimate review point, not misconduct.\n\nA second, smaller issue: the concavity slope is reported as 20.16(4) m·rad²/s² versus 2g = 19.62. The difference is about 2.7%, but it's many reported standard errors. That means either the fit uncertainty is underestimated or there is a systematic offset. The paper calls it 'good agreement' without comment. A sentence acknowledging systematic effects (or reporting a more honest uncertainty) would fix this.\n\nThere are also minor presentation issues: the unit notation 'm·rad²/s²' is confusing (rad is dimensionless), and the title promises time-dependent behavior but the analysis is quasi-static frames during a slow ramp—fine, but state it.\n\nOn the citation front, the self-citations are mostly about the smartphone-sensor methodology, which is the relevant literature, so no problem there. The derivation has a slight inconsistency (cylindrical coordinates for a rectangular container) but it's harmless because the short dimension is small.\n\nBottom line: this paper is for the physics-teaching community, not for people looking for new physics. The smartphone-based synchronization is the selling point. It deserves a serious referee, and a conditional recommendation: ask the authors to report the excluded point, run an all-point fit, and add a sentence on uncertainties. With those changes it would be a solid addition to the teaching literature. I'd bring it to a reading group for physics-education folks, though I wouldn't cite it in my own research.","headline":"A neat smartphone-lab variant on a textbook problem; the results support the model, but the paper needs to justify a post-hoc data exclusion and discuss its uncertainty budget.","tokens_in":4996,"tokens_out":6846,"would_cite":false,"duration_ms":64040,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The free surface of a rotating rectangular liquid is a parabola whose concavity equals ω²/(2g) and whose vertex drops as ω²L²/(24g), and a smartphone-based experiment matches this.","keywords":["free surface","rotating frame","parabolic surface","smartphone sensors","gyroscope","video analysis","fluid dynamics","solid-body rotation"],"falsifier":"Monitor the free surface continuously while the angular velocity is increased in one large step rather than many small ones; if the fitted concavity during the spin-up transient differs measurably from $\\omega^2/(2g)$ and converges only after a settling time, the quasi-static assumption that the analysis relies on is violated.","tokens_in":4128,"feed_emoji":"🌀","tokens_out":5819,"duration_ms":54636,"temperature":0.7,"pith_summary":"The paper derives the steady shape of the free surface of a liquid in a narrow rectangular container rotating about its vertical axis: the surface is a parabola with concavity $\\omega^2/(2g)$ and vertex height $H - \\omega^2L^2/(24g)$. The authors then test this prediction with an experiment that uses a smartphone fixed to the rotating table to record the surface on video and the angular velocity with the built-in gyroscope, so both quantities are measured on the same time axis. Analyzing selected frames with video software, they fit parabolic profiles and compare the fitted concavity and vertex height with the theoretical slopes. The measured concavity slope, 20.16(4) m·$rad^{2}$/$s^{2}$, is close to $2g = 19.62$ m/$s^{2}$, and the vertex-height slope, -0.27(1) mm·$s^{2}$/$rad^{2}$, is close to $-L^2/(24g) = -0.2655(5)$ mm·$s^{2}$/$rad^{2}$. The point of the work is to show that a simple, inexpensive smartphone setup can make meaningful measurements in fluid dynamics and test a standard theoretical result.","feed_headline":"A phone camera plus gyroscope verifies the rotating liquid parabola","feed_subtitle":"Concavity follows ω²/2g and vertex drop follows -L²/24g, measured with a smartphone and free video software.","key_machinery":"The central object is the parabolic free-surface equation $z'(r) = H - (\\omega^2/2g)(L^2/12 - r^2)$, obtained by integrating the hydrostatic pressure balance $\\nabla p = \\rho(\\vec{g} - \\vec{a})$ under rigid-body rotation ($\\vec{a} = -\\omega^2 r \\,\\hat{r}$) and fixing the integration constant with volume conservation. The experiment's other key element is the simultaneous measurement: a smartphone fixed to the rotating frame records the surface with its camera and the angular velocity with its gyroscope, and the frames are matched to the sensor data by a synchronization mark; the surface points are then fitted to a parabola in the Tracker video-analysis software.","core_discovery":"Under the assumptions that the fluid rotates as a rigid body with velocity $\\vec{u} = \\omega r \\,\\hat{\\theta}$ and that transients are negligible because the angular velocity is increased slowly, the pressure field integrates to $p(r,z) = p_{\\rm atm} + \\rho g(H-z) + \\rho\\omega^2(r^2 - L^2/24)/2$, and the free surface is the parabola $z'(r) = H - (\\omega^2/2g)(L^2/12 - r^2)$. The vertex is at $r=0$ with height $H - \\omega^2L^2/(24g)$, and two fixed nodal points at $r = \\pm L/\\sqrt{12}$ remain at height $H$ regardless of $\\omega$. The experiment confirms these predictions: the linear fit of concavity versus $\\omega^2$ has slope 20.16(4) m·$rad^{2}$/$s^{2}$, matching $2g$, and the vertex-height slope is -0.27(1) mm·$s^{2}$/$rad^{2}$, matching $-L^2/(24g)$. The intercept of the vertex-height fit gives the rest water level, 7.72(3) cm, consistent with the 7.6(2) cm measured directly.","pith_inferences":["The method could be extended to measure transient relaxation: by analyzing frames immediately after each voltage jump, one could extract the spin-up time of the fluid and test whether the quasi-static assumption holds for large jumps.","The same smartphone-sensor pairing (camera plus gyroscope) could be applied to other non-inertial-frame phenomena, such as the effective gravity in an accelerating or oscillating frame, without additional hardware.","Automated edge detection could replace manual labeling of the eight points, making the measurement faster and possibly more precise, and would allow tracking the surface continuously rather than at discrete frames.","For containers with larger width $d$, the assumption $L \\gg d$ and the two-dimensional prismatic flow may break down; testing the predicted nodal points with wider tanks would delimit the model's range of validity."],"forward_implications":["The same setup yields a measurement of $g$: the concavity-versus-$\\omega^2$ slope is $2g$, so a student can extract the gravitational acceleration from a linear fit, with precision comparable to the given 20.16(4) value.","The two nodal points at $\\pm L/\\sqrt{12}$ provide a built-in check: they stay at the rest height $H$ for any angular velocity, so a misaligned fit can be spotted.","The model sets an upper limit: for $\\omega \\ge \\sqrt{24gH/L}$ the vertex reaches the container bottom, so beyond that the parabolic formula no longer applies; the experiment stops before that.","Because the angular velocity is time-dependent and slowly stepped, the same video contains many quasi-static states, so one run produces a full calibration curve rather than a single point."],"supporting_citations":[{"why":"Supplies the fluid-mechanics assumption that rigid-body rotation makes viscous stresses null and gives the pressure-gradient relation $\\nabla p = \\rho(\\vec{g} - \\vec{a})$.","marker":"[1]"},{"why":"Provides the prior rotating-liquid-mirror experiment that measured $g$ from the parabolic surface, which the present setup extends to simultaneous smartphone measurement.","marker":"[3]"},{"why":"Earlier apparatus for studying uniform circular motion in a liquid, the baseline the new setup improves upon.","marker":"[2]"},{"why":"The Tracker video-analysis software used to extract surface points and perform the parabolic fits.","marker":"[12]"},{"why":"Demonstrates the smartphone-based approach to fluid experiments in accelerated frames, which this paper adapts.","marker":"[6]"}],"fun_headline_variants":["Smartphone gyroscope and camera verify rotating liquid parabola","Phone sensors confirm the classic spinning fluid parabola","Rotating liquid parabola measured with a smartphone","Gyroscope plus camera check the rotating fluid surface","Smartphone experiment matches theory for spinning liquid shape"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The fluid has reached steady solid-body rotation by the time each frame is recorded, so the slowly stepped angular velocity can be treated as quasi-static and transient effects neglected.","fun_headline_variants_meta":{"raw":{"variants":["Smartphone gyroscope and camera verify rotating liquid parabola","Phone sensors confirm the classic spinning fluid parabola","Rotating liquid parabola measured with a smartphone","Gyroscope plus camera check the rotating fluid surface","Smartphone experiment matches theory for spinning liquid shape"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000428,"raw_usage":{"total_tokens":2178,"prompt_tokens":924,"completion_tokens":1254,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":540,"completion_tokens_details":{"reasoning_tokens":1196}},"tokens_in":540,"tokens_out":1254,"duration_ms":9522,"temperature":1.0,"reasoning_tokens":1196,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:45:48.788831+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Monitor the free surface continuously while the angular velocity is increased in one large step rather than many small ones; if the fitted concavity during the spin-up transient differs measurably from $\\omega^2/(2g)$ and converges only after a settling time, the quasi-static assumption that the analysis relies on is violated.","supporting_citations":[{"cited_title":"Fluid mechanics, 2003","cited_arxiv_id":null,"evidence_quote":"Supplies the fluid-mechanics assumption that rigid-body rotation makes viscous stresses null and gives the pressure-gradient relation $\\nabla p = \\rho(\\vec{g} - \\vec{a})$."},{"cited_title":"Measuring g using a rotating liquid mirror: enhancing laboratory learning","cited_arxiv_id":null,"evidence_quote":"Provides the prior rotating-liquid-mirror experiment that measured $g$ from the parabolic surface, which the present setup extends to simultaneous smartphone measurement."},{"cited_title":"Apparatus for the study of uniform circular motion in a liquid","cited_arxiv_id":null,"evidence_quote":"Earlier apparatus for studying uniform circular motion in a liquid, the baseline the new setup improves upon."},{"cited_title":"Tracker: Free video analysis and modeling tool for physics education, June 2014","cited_arxiv_id":null,"evidence_quote":"The Tracker video-analysis software used to extract surface points and perform the parabolic fits."},{"cited_title":"Understanding coffee spills using a smart- phone","cited_arxiv_id":null,"evidence_quote":"Demonstrates the smartphone-based approach to fluid experiments in accelerated frames, which this paper adapts."}],"review_version":1}