Why is the accelerometer value 9.8 m/s²?

You have probably noticed that when you select absolute acceleration and leave your cell phone stationary, the measurement displayed is 9.80 m / s². Why is the acceleration not zero?

The accelerometer has become an important sensor in our smartphones. When the iPhone was released in 2007, this sensor was only used to manage the automatic orientation of the screen, but over the years many other uses have been added such as motion detection. As we often put our cell phones in our pockets, the accelerometer makes it possible to analyze our physical activity very precisely, for example, to recognize if we are walking and how many steps we have taken, or if we are climbing stairs. It also detects if we fall suddenly and if we do not get up, the cell phone will alert emergency medical services.

How to calculate the acceleration of a smartphone? The usual way to determine the acceleration of an object is to calculate the changes in its velocity vector over short time intervals. Unfortunately we do not have a sufficiently precise way to measure the speed of movement of a smartphone: even outdoors, the accuracy of the GPS is at best only about a meter. It is therefore not feasible to calculate the acceleration by deriving the speed of the smartphone.

The most widely used method is to measure, not the acceleration of the laptop, but the acceleration to which a small mass located inside the laptop and connected to the frame of the smartphone by a spring is subjected. If this small mass has a sufficiently low weight and the spring has a significant stiffness, then the acceleration of the smartphone can be considered as equal to the acceleration of this mass (see footnote for more details).

Consider the components in the diagram on the left. If we move the smartphone, the small mass will initially remain in its position by inertia, and the length of the spring will change by a value denoted by x. This deformation of the spring creates a return force which is proportional to its elongation: F = kx with k the stiffness of the spring. According to Newton's second law this force creates an acceleration of the mass such that F = ma where a is the acceleration of the mass and m its weight. We deduce that kx = ma hence a, the acceleration of the mass: a = kx / m. If we know the deformation x of the spring, we can deduce the acceleration of the mass.

How to calculate this deformation with sufficient precision? There are different possibilities, and one of them involves the characteristics of the capacitors. A capacitor is made up of two conductive armatures separated by an insulator. Its main property is that it can store opposing electrical charges on its reinforcements. It turns out that the storage capacity of a capacitor is inversely proportional to the distance between the conductive plates. As there are many ways to electronically calculate the capacitance of a capacitor, if we connect one plate to the moving body and another to the mass connected to the spring, we can then estimate the spacing of the plates by calculating the capacitance of the capacitor.

Everything is easy on paper, but how do you do it in real life to make this measuring instrument fit in a laptop. This is where MEMS technology comes in, which stands for Micro Electro Mechanical System. A MEMS is a small integrated circuit in which we have mechanical parts and electronic parts fully integrated. The first MEMS were developed in the 1970s.

What does a MEMS accelerometer look like? This is the MEMS photo of an iPhone 4 (https://www.memsjournal.com/2010/12/motion-sensing-in-the-iphone-4-mems-accelerometer.html). We see in this photo the springs, the mass which surrounds the object, and the capacitors which are oriented in two directions, X and Y. These two series of capacitors are oriented at right angles to measure the acceleration in two directions. If we want to know the acceleration in the three directions, we must add a third accelerometer in the direction of the face of the smartphone. As these are generally thin, the engineers have changed the design and in the photo we see this sensor above the other two.

Now that we know how the accelerometer on our smartphones works, let's try to understand what they are measuring. When we place our vertical sensor, the mass of the accelerometers is attracted by gravity, and so the sensor will indicate a force and therefore an acceleration, that of g. This is the reason why the sensor displays gravity when it is at rest. If I drop my laptop, then the laptop is weightless for a short time, and the absolute acceleration is zero. You can check this with FizziQ by dropping your laptop on a (nice soft) bed and recording the absolute acceleration. This experiment will also allow you to calculate the gravity g by measuring the duration of the fall.

To summarize, the acceleration measured by the MEMS of our laptops is by construction affected by gravity, and therefore at rest the accelerometer will display the value equal to the acceleration of gravity, or 9.81 m / s². By using the different components of the absolute acceleration, we can determine the orientation of the laptop. If I select the absolute vertical acceleration, I will find the projection of the acceleration due to gravity on the vertical axis of my laptop. My flat laptop shows zero, but vertically the measurement is 9.81 m / s² ...

Well, not exactly. You will notice that the number displayed is not exactly equal to 9.81 m / s² but to an approximate value of this value. In fact, all laptops will show different values ​​because the sensors are not precisely calibrated to give this information. Is this a problem? Not really because an accuracy better than 1% is not necessarily necessary for usual motion recognition applications.

It would be a different story if we used these sensors to calculate our position like nuclear submarines do ...

Note: the calculation shows that the acceleration of the mass is equal to the sum of two terms: the acceleration of the laptop and an oscillation proportional to the acceleration of the laptop and whose period depends on the square root of (k / m ). Thanks to Daniel Rouan for his help.

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