The Wind Tunnel Reading That Does Not Fit
A student rig measures lift on a small wing in a wind tunnel at fixed angle of attack and fixed air density. Lift is expected to follow L = k v squared, where k depends on density, wing geometry and angle of attack but not on speed. Four runs are recorded:
| v (m/s) | 20 | 30 | 40 | 50 |
|---|---|---|---|---|
| L (N) | 96 | 216 | 300 | 600 |
The technician who took the 40 m/s reading writes in the log: "lower lift at 40 m/s confirms the wing stalled at that speed." The report is due tomorrow and the tunnel slot cannot be rebooked before then. The angle-of-attack setting was not changed between runs, and no other reading in the table is disputed.
1. Determine k from the consistent runs, and use it to state what the 40 m/s reading should have been and what lift is predicted at 45 m/s.
Read the worked answer
Test L/v² on each run:
- 20 m/s: 96 ÷ 400 = 0.240
- 30 m/s: 216 ÷ 900 = 0.240
- 50 m/s: 600 ÷ 2500 = 0.240
Three independent runs give k = 0.24 N per (m/s)², so the square law holds across a 2.5× speed range.
Expected at 40 m/s: 0.24 × 1600 = 384 N. The recorded 300 N is 0.78 of that, and implies its own k of 300 ÷ 1600 = 0.1875.
Prediction at 45 m/s: 0.24 × 2025 = 486 N.
2. Evaluate the technician's stall explanation. What alternative explanations must be excluded first, and what evidence in the table argues against a stall?
Read the worked answer
The stall claim is not supported by the data.
- The 50 m/s point sits exactly on the 0.24 trend. A genuine stall at 40 m/s would show a plateau or a fall in lift, and the wing would not spontaneously return to perfect square-law behaviour 25% faster at the same angle of attack.
- A stall is a property of angle of attack, not of speed. With the angle unchanged between runs, lift should rise monotonically with speed; a dip at one speed is not a stall signature.
- A single-point deviation while its neighbours fit is the signature of a measurement or transcription fault, not of aerodynamics.
Alternatives to exclude before writing anything:
- A misread or mis-transcribed value (e.g. 384 recorded as 300 on the manometer or in the log).
- A balance/load-cell calibration shift that was corrected again for the 50 m/s run.
- A tunnel speed error at that setting (actual speed lower than indicated - note 300 N corresponds to about 35.4 m/s, not 40).
- A change in density or temperature at that run (though a density change would shift every reading at that condition, not one).
Conclusion: the honest log entry is "40 m/s reading inconsistent with L proportional to v squared; cause unknown, under investigation" - not a stall.
3. Suppose the team later finds that the tunnel air density on the day was 0.9 times the calibration value. Does that explain the 40 m/s point? Justify with numbers, and state what it would do to the 45 m/s prediction.
Read the worked answer
It does not explain the anomaly.
Density scales k directly (k is proportional to density), so a 0.9 density factor gives:
- k = 0.24 × 0.9 = 0.216
- 20 m/s: 0.216 × 400 = 86.4 N (recorded 96)
- 30 m/s: 0.216 × 900 = 194.4 N (recorded 216)
- 50 m/s: 0.216 × 2500 = 540 N (recorded 600)
The 40 m/s point requires k = 0.1875, which is 0.78 of the calibration value - not 0.9. A density change moves all points by the same factor; the 40 m/s point is displaced by a different factor from its neighbours. So density cannot be the cause, and it cannot rescue the 300 N reading.
Effect on the prediction: if the 0.9 density figure is accepted, the whole curve must be re-based to k = 0.216, giving 0.216 × 2025 = 437 N at 45 m/s. The 40 m/s reading must still be re-taken or rejected, and any report must state which density value the quoted k refers to.