The proliferation of smartphones with internal sensors has led to some interesting opportunities to make large-scale measurements of a variety of physical phenomena.

The iOS app ‘*Gravity Meter’* and its android equivalent have been used to make measurements of the local surface acceleration, which is nominally 9.8 meters/sec^{2}. The apps typically report the local acceleration to 0.01 (iOS) or even 0.001 (android) meters/sec^{2 }accuracy, which leads to two interesting questions: 1)How reliable are these measurements at the displayed decimal limit, and 2) Can smartphones be used to measure expected departures from the nominal surface acceleration due to Earth rotation? Here is a map showing the magnitude of this (centrifugal) rotation effect provided by The Physics Forum.

As Earth rotates, any object on its surface will feel a centrifugal force directed outward from the center of Earth and generally in the direction of local zenith. This causes Earth to be slightly bulged-out at the equator compared to the poles, which you can see from the difference between its equatorial radius of 6,378.14 km versus its polar radius of 6,356.75 km: a polar flattening difference of 21.4 kilometers. This centrifugal force also has an effect upon the local surface acceleration by reducing it slightly at the equator compared to the poles. At the equator, one would measure a value for ‘g’ that is about 9.78 m/sec^{2} while at the poles it is about 9.83 m/sec^{2}. Once again, and this is important to avoid any misconceptions, the total acceleration defined as gravity plus centrifugal is reduced, but gravity is itself not changed because from Newton’s Law of Universal Gravitation, gravity is due to mass not rotation.

Assuming that the smartphone accelerometers are sensitive enough, they may be able to detect this equator-to-pole difference by comparing the surface acceleration measurements from observers at different latitudes.

**Experiment 1 – How reliable are ‘gravity’ measurements at the same location?**

To check this, I looked at the data from several participating classrooms at different latitudes, and selected the more numerous iOS measurements with the ‘Gravity Meter’ app. These data were kindly provided by Ms. Melissa Montoya’s class in Hawaii (+19.9N), George Griffith’s class in Arapahoe, Nebraska (+40.3N), Ms. Sue Lamdin’s class in Brunswick, Maine (+43.9N), and Elizabeth Bianchi’s class in Waldoboro, Maine (+44.1N).

All four classrooms measurements, irrespective of latitude (19.9N, 40.3N, 43.9N or 44.1N) showed distinct ‘peaks’, but also displayed long and complicated ‘tails’, making these distributions not Gaussian as might be expected for random errors. This suggests that under classroom conditions there may be some systematic effects introduced from the specific ways in which students may be making the measurements, introducing complicated and apparently non-random, student-dependent corrections into the data.

A further study using the iPad data from Elizabeth Bianchi’s class, I discovered that at least for iPads using the Gravity Sensor app, there was a definite correlation between when the measurement was made and the time it was made during a 1.5-hour period. This resembles a heating effect, suggesting that the longer you leave the technology on before making the measurement, the larger will be the measured value. I will look into this at a later time.

The non-Gaussian behavior in the current data does not make it possible to assign a normal average and standard-deviation to the data.

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**Experiment 2 – Can the rotation of Earth be detected?**

Although there is the suggestion that in the 4-classroom data we could see a nominal centrifugal effect of about the correct order-of-magnitude, we were able to get a large sample of individual observers spanning a wide latitude range, also using the iOS platform and the same ‘Gravity Meter’ app. Including the median values from the four classrooms in Experiment 1, we had a total of 41 participants: Elizabeth Abrahams, Jennifer Arsenau, Dorene Brisendine, Allen Clermont, Hillarie Davis, Thom Denholm, Heather Doyle, Steve Dryer, Diedra Falkner, Mickie Flores, Dennis Gallagher, Robert Gallagher, Rachael Gerhard, Robert Herrick, Harry Keller, Samuel Kemos, Anna Leci, Alexia Silva Mascarenhas, Alfredo Medina, Heather McHale, Patrick Morton, Stacia Odenwald, John-Paul Rattner, Pat Reiff, Ghanjah Skanby, Staley Tracy, Ravensara Travillian, and Darlene Woodman.

The scatter plot of these individual measurements is shown here:

The red squares are the individual measurements. The blue circles are the android phone values. The red dashed line shows the linear regression line for only the iOS data points assuming each point is equally-weighted. The solid line is the predicted change in the local acceleration with latitude according to the model:

G = 9.806 – 0.5*(9.832-9.78)*Cos(2*latitude) m/sec^{2}

where the polar acceleration is 9.806 m/sec^{2} and the equatorial acceleration is 9.780 m/sec^{2}. Note: No correction for lunar and solar tidal effects have been made since these are entirely undetectable with this technology.

Each individual point has a nominal variation of +/-0.01 m/sec^{2} based on the minimum and maximum value recorded during a fixed interval of time. It is noteworthy that this measurement RMS is significantly smaller than the classroom variance seen in Experiment 1 due to the apparently non-Gaussian shape of the classroom sampling. When we partition the iOS smartphone data into 10-degree latitude bins and take the median value in each bin we get the following plot, which is a bit cleaner:

The solid blue line is the predicted acceleration. The dashed black line is the linear regression for the equally-weighted individual measurements. The median values of the classroom points are added to show their distribution. It is of interest that the linear regression line is parallel, and nearly coincident with, the predicted line, which again suggests that Earth’s rotation effect may have been detected in this median-sampled data set provided by a total of 37 individuals.

The classroom points clustering at ca +44N represent a total of 36 measures representing the plotted median values, which is statistically significant. Taken at face value, the classroom data would, alone, support the hypothesis that there was a detection of the rotation effect, though they are consistently 0.005 m/sec^{2} below the predicted value at the mid-latitudes. The intrinsic variation of the data, represented by the consistent +/-0.01 m/sec^{2} high-vs-low range of all of the individual samples, suggests that this is probably a reasonable measure of the instrumental accuracy of the smartphones. Error bars (thin vertical black lines) have been added to the plotted median points to indicate this accuracy.

The bottom-line **seems to be that it may be marginally possible to detect the Earth rotation effect**, but precise measurements at the 0.01 m/sec^{2} level are required against what appears to be a significant non-Gaussian measurement background. Once again, some of the variation seen at each latitude may be due to how warm the smartphones were at the time of the measurement. The android and iOS measurements do seem to be discrepant with the android measurements leading to a larger measurement variation.

**Check back here on Wednesday, March 29 for the next topic!**

That is very cool.

I remember using a gravimeter in college for a number of projects (determining the density of our science building, mapping a bedrock river channel covered by glacial drift, etc.).

One thing we had to correct for was the change in tidal forces. We plotted changes in G at a specific location over time and then had to correct measurements made throughout the period of our observations. I believe those tidal values changed on the order of milli-gals so perhaps, at least on the Android devices, that might be a factor at play, yes?

Hi Cush! I looked into this and convinced myself that this effect is not significant at the 10 gal (0.01 m/sec2) level that the smartphone sensors can detect in a one-shot measurement. The variance that I see in the measurements from other causes is quite a bit higher than this and may have to do with how long the smartphone was in operation before the measurement. It seems like the heating of the accelerometer over time is an important effect for iPad measurements that I have looked at. It may actually be a good idea to purchase a professional-grade gravimeter and calibrate the smartphone systems that way in absolute terms.