Gravity, omnipresent in our everyday lives, remains a complex subject to grasp in its entirety. While Newton's portrayal of gravity as a force influencing all matter is widely recognized, it inevitably sparks a multitude of questions. **In this article, we outline seven experiments that can be conducted using just a smartphone** to delve deeper into the realm of gravity. From embarking on a Zero G flight to exploring the behavior of a pendulum on the moon, venturing to the equator, or even attempting push-ups at 33,000 feet, we're set for a thrilling journey. Are you on board?

**Table of content :**

__What is gravity?__ - __Using a smartphone to study gravity __- __Zero flight G__ -__ Absolute and Linear Acceleration__ - __Einstein Elevator Thought Experiment__ - __Let's estimate g!__ - __A pendulum on the Moon__ - __Trajectory of a fall __- __Is it harder to do push-ups at 33 000 feet altitude?__ - __Measuring g at the equator__ - __Conclusion__

## What is gravity?

**Gravity is a fundamental force in the universe that governs the interaction between bodies on a macroscopic scale**. It is responsible for the mutual attraction between all objects possessing mass. When we drop an object, it falls towards the ground due to the gravitational force exerted by the Earth.

**Galileo is the first to have studied in detail the phenomenon of gravity in a scientific way**. Galileo's experiment on the fall of bodies from the top of the tower of Pisa is probably a myth, but many thought experiments that he conceptualized allowed him to show that, contrary to preconceived idea, two objects of different masses fall at the same speed. He also studied the mathematical rule that governs the distance travelled by a ball rolling freely along an inclined plane and found that it increases with the square of the time elapsed. Today we know that: d = 1/2*g*t², where g is the acceleration of gravity.

**This law will be demonstrated a century later with the theory of gravity proposed by Sir Isaac Newton**. He postulates that every object in the universe attracts all other objects with a force proportional to their mass and inversely proportional to the square of the distance between them. According to Newton's law of universal gravitation, the gravitational force (F) between two objects of mass (m1) and (m2) separated by a distance (r) is given by the equation: F = G * (m1 * m2 ) / r², where G is the universal gravitational constant, which has a value of approximately 6.674 × 10^-11 N(m/kg)².

For an object on the surface of the earth, the gravitational force that applies to an object, also called the weight of the object, can be written simply: F = m * g, where g is the acceleration of the gravity, and m the mass of an object. The value of the acceleration due to gravity g is approximately 9.81 m/s². The value of g, depends on a number of factors, like altitude or latitude (since the Earth is not a perfect sphere).

Newton's theory can explain many phenomena such as the movement of the planets. But it however fails to resolve some phenomenon like the gravitational lensing effet, or the precession of the orbit of Mercury. Also it does not explain where the gravitational force comes from and why its action is instantaneous. **The theory of general relativity, formulated by Albert Einstein at the beginning of the 20th century, brings a revolutionary approach to gravity.** Contrary to the Newtonian view of gravity as a force of attraction at a distance, Einstein describes gravity as a distortion of spacetime caused by the presence of mass and energy. In this theory, objects move along geodesic curves in curved spacetime, creating the effect perceived as a gravitational force. Thus, according to the theory of general relativity, gravity is not a mysterious force of attraction, but rather a consequence of the curvature of space-time induced by the distribution of mass and energy.

## Using a smartphone to study gravity

No that we know a bit more on gravity, let's experiment with it! Easy to say but what scientific instruments do we have available to analyse this complex phenomenon ? Luckily everyone now has in his or her pocket the right instrument to study gravity : a smartphone. Smartphones, ubiquitous and integral to daily life, possess a unique capacity to transform how we approach education and scientific exploration, particularly in the study of fundamental forces such as gravity.

Smartphones are equipped with a variety of sensors that can measure acceleration, orientation, and motion. These include accelerometers, gyroscopes, and magnetometers, which can detect changes in movement and orientation with respect to Earth's gravitational field. Free apps like FizziQ provide a simple and direct access to all theses sensors and allows students and educators to conduct experiments to observe and quantify gravitational acceleration in real-time. This hands-on approach demystifies abstract concepts and allows learners to directly observe physical laws in action.

In the article we will use specifically the following instruments available in FizziQ :

Absolute Acceleration which provides the result of forces that are applied to the smartphone in all directions

Video motion analysis, a unique tool in FizziQ to analyze quickly videos and chronophotographie of falling objects

Sound intensity which provides an accurate measure of sound level and how it deviates. This measure will allow us to compute g precisely

Other sensors like magnetometer, gyroscope or linear acceleration can also be useful to complement the instruments described above

Another useful characteristics of apps like FizziQ is to provide a complete environment to gather, analyse and share data. The apps have chronometers or triggers to start or end measures automatically, they also include notebooks to organise the data, create graphs in different formats and add text and photos to record experiment settings. All the apps have the ability to share the data in PDF, Excel or even Python format so that further analysis can be conducted on raw information.

Smartphones and tablets are thus the ideal tool to conduct simple experimentation in science and especially to understand the concept of gravity. Ready to try our seven editing science experiments?

## Zero G flights

To train the astronauts in weightlessness, that is to say the absence of gravity, they are subjected to Zero G flights. The astronauts are placed in the empty cabin of the plane, the interior walls of the device are covered with mattresses. In these flights, the plane describes parabolas. During the end of the ascending phase, then the beginning of the descending phase, the astronauts can remove their safety belts and fly freely in the cabin as if they were weightless.

To better understand this phenomenon, let's do the following experiment. Let's put our smartphone on a table and then in FizziQ open the Absolute Acceleration measurement. We see the value of 9.81 m/s² is displayed. Now let's orient the smartphone differently in all directions. We will see that the value changes and then also stabilizes at the value 9.81 m/s². This therefore means that the smartphone is subjected to a force equivalent to an acceleration of 9.81 m/s². It is the acceleration of gravity. Yet the smartphone does not move, which means that the smartphone somehow measures the force created by the table in reaction to gravity.

To prove this, le'ts remove the table. The smartphone falls, but what is the acceleration detected by the accelerometer ? We place a mattress on the floor, then press the Record button and we launch our smartphone so that it describes a parabola and falls on the mattress. Then Stop recording. We note that during the entire period in the air, the acceleration of the smartphone is zero. Although the smartphone was in free fall, and therefore its vertical speed varied for an observer placed on the ground, the accelerometer does not seem to perceive any movement, it acts as if the smartphone was in zero gravity.

Let's go back to the case of flight Zero G. At the beginning of the ascending phase, the astronauts are subjected to an acceleration which will cause them to describe a parabolic movement, then the plane slows down and aligns its vertical and horizontal speed with that of the passengers. The astronauts have the sensation of being weightless, as the cabin in which they are falls at the same speed as them. However, to an observer outside the plane, the astronauts would be in free fall.

## Einstein Elevator Thought Experiment

A thought experiment is a hypothetical scenario used to explore the consequences of a principle or theory in the absence of actual, physical experimentation. It involves reasoning through a problem using only the imagination and knowledge of physical laws, without the need for empirical evidence or practical execution. Thought experiments have been employed in various fields, including physics, philosophy, mathematics, and ethics, serving as a powerful tool to conceptualize ideas, challenge existing notions, and stimulate intellectual exploration.

Albert Einstein, one of the most prominent users of thought experiments, utilized them extensively to develop his revolutionary theories in physics, including the special and general theories of relativity. Einstein's thought experiments allowed him to visualize complex problems and paradoxes in physics that were difficult or impossible to test with the technology available during his time.

One of the most famous thought experiments of Einstein is the "chasing a beam of light" scenario, which he conceived at the age of 16. This thought experiment led him to question the established notions of space and time, ultimately contributing to the development of the special theory of relativity. In this experiment, Einstein imagined what it would be like to ride alongside a beam of light. If he moved at the speed of light alongside the beam, he realized that the light would appear stationary to him, which contradicted Maxwell's equations that light always travels at a constant speed regardless of the observer's motion. This contradiction led him to further explore the relationship between time, space, and velocity, culminating in his groundbreaking work on relativity.

**Another thought experiment which is related to gravity is the Einstein's "elevator thought experiment". It was instrumental in developing the equivalence principle**, central to his general theory of relativity. In the experiment, Einstein imagined himself inside a closed elevator in deep space that's accelerating upwards, a dropped ball appears to fall towards the floor similarly to Earth's gravitational pull. In contrast, a stationary elevator near a planet like Earth experiences a similar effect due to gravity. The essence of this thought experiment is that within the confines of the elevator, one cannot distinguish between effects of gravity and pure acceleration.

The Zero G flight gives a good example of what happens inside Einstein elevator. The smartphone in free fall is equivalent to an elevator which falls with the same acceleration as gravity. Inside the smartphone, the accelerometer cannot detect wether it is in free fall or whether the gravity is zero.

This thought experiment was crucial for Einstein because it led him to realize that gravity and acceleration are locally indistinguishable and that gravity could be thought of as the curvature of spacetime caused by mass. In general relativity, gravity is not a force in the traditional sense but a result of masses moving along the curves in spacetime created by the presence of mass and energy.

## Absolute and Linear Acceleration

The analysis of the Zero G flight can shed some light on what is happening to our smartphone and help us answer the question : w**hy is acceleration null ?** The accelerometer is a tiny electro-mechanical device made of a small weight connected to the frame by a spring. If the frame moves, the weight reacts with a delay and by measuring the change of distance between the frame and the weight, one can compute the acceleration of the frame __(see our article on the subject)__. If we come back to our experiment, **when the smartphone falls, all the parts of the smartphone fall together (the frame and the weight), and hence there is no relative movement of the weight versus the frame. As a consequence the measurer acceleration is zero. **On the contrary when we hold the case, the weight is pulled down by gravity but the case is held fixed by our arm. So there is a relative move of one versus the other. **The accelerometer then detects the reaction force of our arm, equal to gravity.**

If you look at FizziQ accelerometer menu, you will also see another sensor than Absolute Acceleration, it is called Linear Acceleration. How different is it from the other one ? Linear acceleration measures the acceleration of the smartphone, excluding gravity. Select in FizziQ linear acceleration, you will see that the acceleration is zero if you do not move. If you move in the right direction, the sensor will indicate the acceleration of the smartphone only, excluding gravity.

How is this possible ? In fact linear acceleration is not a sensor per say, it cannot be directly measured by a single sensor; instead, it's calculated using data from three types of sensors found in most smartphones: the accelerometer, gyroscope, and magnetometer. These sensors, functioning as part of a Micro-Electro-Mechanical Systems (MEMS) setup, work together to provide a comprehensive picture of the smartphone's movement and orientation.

How does it work ? The gyroscope measures the rotation speed of the device, allowing for the determination of changes in orientation from a resting state. The magnetometer measures the Earth's magnetic field, providing a stable reference point for orientation. While both sensors can help in calculating linear acceleration, the gyroscope is generally more reliable, except when influenced by external magnetic fields, which can disrupt the magnetometer's accuracy.

By integrating data from the accelerometer, gyroscope, and magnetometer, it's possible to calculate a smartphone's linear acceleration, effectively excluding the influence of gravity. This intricate process highlights both the capabilities and limitations of current technology in accurately measuring motion, humorously suggesting that perhaps space aliens have a better solution for completely isolating gravity's effect.

## Let's estimate g!

One of the essential parameters of the theory of gravitation is the acceleration due to gravity, g. Depending on whether this value is high or low, we feel heavy like on Earth or light like on the Moon. We have seen that the accelerometer made it possible to estimate this value, but scientists in the 16th century did not have this instrument. **Can we calculate g without using the accelerometer?**

For this we are going to do the same experiment as Galileo and measure the time it takes for a body to fall from a certain height. We know that the relationship between the total duration t of the fall of an object and the height h at which this object is dropped is h = 1/2*g*t². To calculate g we therefore only need to measure the duration of the fall of a given object from a known height.

Let's choose an object, for example a ball, and let's drop it from a certain height h, for example from the first floor of a building. With a smartphone stopwatch, we measure the duration of the fall. For example, by placing myself on the first floor of a building and dropping a ball, I obtained the value of 0.95 s of fall for 3.5 m, which gives a g value of 7.75 m/s² .

This value is not very precise, because it is difficult to start and stop the timer at the exact moments when the ball is released and when it lands. This is because of this that Galileo used inclined ramps to conduct his analysis on falling balls, as they fall less quickly. An error of 10% on the duration results in an error of more than 20% on the measurement of g. Triggering the stopwatch by hand creates a lot of uncertainty, so we need to use a more precise method. In FizziQ we have the ability to record the sound volume over a given period. We will therefore create a device in which a sound is created when the object begins to fall and another when the object hits the ground. It will then not suffice for us to measure the time interval between these two events.

In this experiment, we drop a bolt from a shelf in such a way as to generate a noise when it begins its fall and another when it ends. Let's place the bolt at the very edge of the shelf then with a tool, we give a sharp blow to the bolt to push it into the void, creating a small characteristic noise.

When the bolt hits the ground, it makes another crash sound. With FizziQ we measure the time elapsed: we select the sound volume, then we start recording and we do the experiment described precisely. When the bolt hits the ground, we stop recording. By studying the data in the workbook, we can accurately determine the first and second shock, and therefore have an accurate measurement of the fall time.

The photo shows the device used and the graph, the measurement made. The floor has a height of 1.28 m and the measured duration is 0.51 s, which gives a value of g of 9.84 m/s².

To improve the measurement, you can use a sound stopwatch __as for the measurement of the speed of sound__. You can also use a __trigger with acceleration__.

## A pendulum on the moon

In a NASA video taken during the Apollo14 mission on the surface of the Moon, we see that the period of a box that oscillates like a pendulum is significantly greater than that which this pendulum would have had on earth: __https:// history.nasa.gov/alsj/a14/Apollo14SEQ_BayPendulum.mpg__. Can we explain this phenomenon?

Galileo, the first to have carried out in-depth experiments on the movement of pendulums, he showed in 1632 that the period of the pendulum for weak oscillations does not depend on its mass, nor on the amplitude of the oscillations but only on its length. This remark will be the basis of clock movements that use pendulums. Huygens in 1659 determines the exact expression of the period of a pendulum for weak oscillations: T=2π*√(l/g), where g is gravity.

The period depends on the length l of the pendulum but also on a fundamental terrestrial parameter: the acceleration of gravity, g. This allowed scientists for the first time to accurately determine the constant g, or rather the length of a pendulum that had a period of one second. In 1690, in his Discourse on the cause of gravity, Huygens indicates that the length of the pendulum beating the second in Paris is 3 feet 8.66 lines or 0.9941 m, which corresponds to a gravity in Paris of 9.812 m/ s² (with our units). The pendulum becomes at the time the instrument for measuring gravity.

We see that this formula the period is inversely proportional to the square root of g. On the moon, the period of a pendulum would therefore be 2.5 times greater than on the earth, verified by the calculations that scientists have made by analyzing the video of Apollo 14 and detailed on the NASA website: __https: //history.nasa.gov/alsj/a14/a14pendulum.html__.

As we are not on the Moon, we will check Galileo's formula on earth. We will use the luxmeter provided by the FizziQ application for smartphones to accurately calculate the period of a pendulum. We suspend a pendulum made with a fairly heavy ball at the end of a thread in such a way that the ball obscures the photoelectric cell of a smartphone when it is in the low position. The photocell of Android devices is usually placed to the right of the camera. It can be located by measuring the brightness with the FizziQ Illumination instrument and placing your finger where you think it is.

By measuring the illumination, we can very precisely determine the period of the pendulum which corresponds to the difference between two peaks of luminosity. We then verify Galileo's law on pendulums. We can also use this measurement to do more precise calculation of g.

## Trajectory of a fall

When watching a basketball game, we are used to seeing beautiful parabolas described by the balls when thrown from a distance. Can we model this curve? For this we use one of the most obvious assets of smartphones and tablets: the camera which allows the physicist to make precise videos of the movements he or she is studying. Thanks to the video analysis tools of the FizziQ application, you can also analyze in detail the kinematics of these movements, trace their trajectories and export the characteristics of the movements to a spreadsheet.

The user can either create his video of an object in free fall, or use one of the videos available in the application. The video library of the FizziQ application contains many videos, especially on sports, which can be used to study the kinematics of free fall: https://www.fizziq.org/cinematique.

For example, let's use the video of a bullet drop from the kinematics module. You can find in the following tutorial on Youtube how to conduct video analysis using FizziQ: __https://www.youtube.com/watch?v=sZdndmHefH8__.

After having carried out the analysis of the fall, we place the data of the trajectory in the notebook of experiments. By plotting the graph for the vertical position of the ball as a function of time, we see that this trajectory is a parabola, thus confirming Galileo's result.

What is the equation of this parabola? By pressing down on the interpolation functions, we choose the quadratic interpolation which gives the equation of the trajectory. In graph 3, the equation of the interpolated function is f(x) = 4.72x²-1.48x+2.05. This analysis also makes it possible to find the acceleration of weightlessness g = 2*4.72 = 9.44 m/s².

Galileo's intuition on the dependence of the position of a falling ball on time was therefore correct!

## 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).

## Measuring g at the equator

Following 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 smartphone 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.

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