Seven Experiments on Gravity with your Smartphone (part 2)
Updated: Sep 13
In this second part, we continue our quest to better understand the concept of gravity with experiments that are easy to perform with a smartphone. Follow this link to review the first part.
Is it harder to du push-ups at an altitude of 33 000 feet?
A question that should interest every athlete is the following: is it easier to do push-ups at an altitude of 30,000 feet than at sea level?
Flying by plane is not ecological, but if you nevertheless fly abroad, why not try this little experiment to answer the previous question? Before takeoff and when the plane is at rest, place your smartphone on the tablet, then in the FizziQ application, record the absolute acceleration for 10 seconds, then add this value to the experiment notebook. In the statistics at the bottom of the graph, you will find the average value over the period. You have to be careful not to move the smartphone when you press the record button.
When the plane reaches its cruising altitude, and when its flight is stable without turbulence, repeat the measurement for about ten seconds, then note the average absolute acceleration. The use of the average makes it possible to erase the small variations due to the micro-turbulences of the cabin. What value do you get? What was the change in the acceleration due to gravity? Taking a value for your weight, what is your new weight at altitude?
In the screenshot opposite, we obtain the value for g of 9.78 m/s2, a difference of about 3%. The weight of an athlete, P = m*g, is therefore 3% lower at an altitude of 10,000 meters than at sea level. If you usually do 30 push-ups, maybe you can do some 31? 😁 However, not enough to break a record! 💪
This trip is also an opportunity to confirm Newton's formula on universal gravitation. The formula which gives the value of g according to the altitude h can be deduced directly: g(h)=g(0)R²/(R+h)² with R = 6400 km and h in km. We tested this protocol during a trip between Paris and Copenhagen. The altitude at which we made the measurement was 10,300 meters. The calculation gives the following value: g(0) = 9.81 and h = 10.3 km, g(h) = 9.78 m/s², i.e. equal to the value we obtained (screenshot below -above).
How to get rid of gravity?
In a previous experiment, we measured acceleration using the FizziQ app. We have seen that if we measured the absolute acceleration of a stationary smartphone on a table, the absolute acceleration was not zero: it was equal to the acceleration of gravity.
Although useful, this measurement is not suitable for most applications that need to calculate the movement or position of the smartphone in the user's frame of reference. We then need a particular measurement: the linear acceleration. Linear acceleration measures the movement of the user without the component of gravity.
To discover this measurement, in FizziQ, open the list of Instruments and select the Y linear acceleration. Hold your laptop vertical. At rest you find that the value is zero. Now move the laptop up and down and you will know the acceleration of the smartphone. Compare these results with those obtained using the absolute acceleration Y. This integrates the gravity vector well.
Unlike absolute acceleration, there are no sensors to measure linear acceleration. It is the result of calculations using a combination of sensors: the accelerometer, the gyroscope and the magnetometer. It is the combination of the information given by these three sensors which makes it possible to know the linear acceleration.
Acceleration is measured by a MEMS system,a small integrated circuit that has mechanical parts and fully integrated electronic parts. It consists of a small mass connected to the frame of the device by a spring. When the smartphone moves, the small mass deviates from the frame at a distance that will depend on the characteristics of the spring. By measuring this distance, we calculate the acceleration that the smartphone has undergone. For more details, see our blog which describes in detail how the accelerometer works.
As the small mass is subjected to the force of gravity, the acceleration measured by the accelerometer includes the acceleration of gravity g. To measure the linear acceleration, we must therefore subtract the gravity vector, but for this we must know the orientation of the smartphone with respect to the gravitation vector. There are two other sensors in most laptops that can give this information: the magnetometer and the gyroscope. These two sensors are also MEMS.
The gyroscope is a sensor that allows us to calculate the rotation speed of our smartphone in the three directions. It allows us to calculate at any time how the mobile has rotated with respect to its initial position. Thanks to the gyroscope, we can determine at any time how the orientation of the laptop has changed from its initial state of rest. By applying these changes to the initial vector calculated for the acceleration of gravity, we can then deduce from the observed absolute acceleration its component, and thus determine the linear acceleration.
The magnetometer can also be used to calculate linear acceleration. It allows to calculate the magnetic field to which our laptop is subjected. In the absence of any other magnetic field (such as a magnet or a ferromagnetic object), the magnetometer gives the coordinates of the earth's magnetic field, which makes it possible to know the north for example. This field is very stable and can therefore be used as an absolute reference. Since we know the magnetic field at the initial instant, we can know the orientation changes of the laptop by comparing the vector of the magnetic field at any instant, and therefore adjust the gravity component of the acceleration to determine the acceleration absolute. With a limit however: if a magnetized or iron-magnetic object is close to the sensor, its measurement will be affected and the reference frame will be false. This explains why the gyroscope is a better sensor to calculate the linear acceleration than the magnetometer.
By combining the information from different sensors, we have managed, at least by the calculus, to get rid of gravity. Unfortunately, that's much harder to do in the real world ... may be space alien have a solution ?
The mystery of g at the equator
After Huygens' work on the pendulum in 1659, scientists are confident that they finally have an accurate measurement for the acceleration of gravity, g. However, against all odds, the astronomer Richer made a crucial discovery in 1672. While on a mission to Cayenne to measure the parallax of Mars, he noticed that the pendulum that beats the seconds was shorter in Cayenne than in Paris, suggesting that gravity varies with latitude. This experiment revives a competition between Newton and Huygens to determine the reason for this discrepancy and to obtain an equation which will make it possible to determine g at any place on Earth.
If you have the chance to travel between a country close to the equator and a destination further north, why not recreate Richer's experiment and study the variation of the acceleration of gravity according to latitude?
To do this with FizziQ, record the normal acceleration of the laptop placed on a table in the experiment notebook before your flight. Then at the new destination, also record the normal acceleration at rest. What difference do you get?
The acceleration of gravity is less strong at the equator due to two main factors: the effect of the Earth's rotation and the flattening of the Earth:
The Earth's rotation creates an outward-directed centrifugal force, which is greatest at the equator due to the greater distance from the axis of rotation. This force opposes Earth's gravity, slightly reducing the acceleration of gravity at the equator relative to the poles. Thus, gravity is less at the equator due to the rotation effect.
On the other hand, the Earth is not a perfect sphere, but rather an oblate ellipsoid at the poles. In other words, the diameter of the Earth measured from pole to pole is slightly shorter than the diameter measured at the equator. Since points at the equator are farther from the center of the Earth than points at the poles, the gravitational force exerted by the Earth on an object at the equator is slightly weaker than at the poles.
These two factors combined cause the acceleration due to gravity to be slightly lower around the equator than at other regions of the Earth. The general formula is: g(θ)=g(0)⋅(1+k⋅sin²(θ)) with k ≈ 0.00527 with g(0), the value of g at the equator: g(0) = 9.78 m/s².
Using this approximate formula, do you get the same value for the acceleration where you are?
Conclusion
We presented seven experiments to work alone or in groups on the notion of gravity. The seemingly simple concept study opens up many pedagogical paths in middle and high school and allows everyone to ask themselves fascinating questions about our universe and how it works.
References :
https://history.nasa.gov/alsj/a14/a14pendulum.html
https://planet-terre.ens-lyon.fr/ressource/pendule-pesanteur-latitude.xml