There is a fly in my car

  

  One week after the Canadian Government declares the beginning of the quarantine I decided to use part of my time to read the books that I’ve bought since I arrived in Quebec. My routine was based on reading at least three hours per day. One hour after breakfast, forty minutes after lunch, and one hour and twenty minutes before going to bed. The good news is that I finish most of the books that I’ve bought (however, I bought new books to read... it's an endless loop). 

  Last week I started my routine period. I woke up to eat my breakfast and after that, I was about to read my book, but the owner came to me and asked me: “Rene, you study physics, don't you?” Sadly after my affirmative answer came a question hard to answer without some physics background. Notwithstanding, I said: “yes, I do. Are you interested in physics?” His question was the idea of this post, then I quote as far as I remember:

“Yesterday, driving my car I saw a fly inside it, but I was moving close to 95 km/h. I expected that the fly will go to the back to car and crash into the back window. However, the fly moved freely inside my car. Could you explain to me why?”

  That was a very interesting question, and the answer to this question is based on a tricky topic of physics, especially in mechanical physics, the so-called accelerating frames.

A brief context

Before starting with the explanation it’s important to understand what exactly physicists call “force”.  Since we are accustomed to the movement of bodies, our common sense leads us to think that a force is a kind of push. However, in a physical context, a force is an interaction between two or more bodies which can be by contact (like pushing a rock or lifting a book) or by long-range interaction (without direct contacts, like the attraction/repulsion between two magnets). Thus, we cannot define a force on an isolated body, the force is a patrimony between two or more bodies.  Having made this clear, let's move on to what Newton taught us in his second law of motion:

“The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.”  

Definition taken from: physicclass 

 

  The second Newtonian law tells us that the net force (the sum of all forces acting on a body) is related to the body’s acceleration and its mass. Once more, acceleration and mass are concepts widely used outside of physics. Therefore, it is important to clarify what acceleration and mass are in a physical context. First, the acceleration is a vector i.e. it is a quantity that has a value and a direction and is defined as the change of the velocity when the time passes. Since velocity is also a vector, a change can occur either if its value changes or its direction changes. To understand the concept think about what happens when you turn on a fan to a specific velocity, in the beginning, the blades of the fan move with a velocity that increases until the desired value is attained and after that, they remain at a fixed velocity (in value). However, since the blades move in a circle, they have a net acceleration even if their velocity is fixed in a constant value due to the change in the direction!!!  

  For the case of the mass, we can define it as a measure of how hard is to change movement of a body. Then, the more mass a body has, the harder it is to move it or stop it from moving. A small parenthesis... I remember that when I was in school my chemistry teacher said to me that mass is the quantity of matter that a body posses... This definition is completely useless in mechanics where the main goal is to describe the movement (end of parenthesis)! 

Now we only need to define the last concept. What is an inertial referential?

For the most part in physics’ books, the definition of inertial referential is referential where Newton’s laws are valid. Although this definition is quite accurate, it does not offer any information for someone who does not study physics. Thus, first, we define a referential as the place where we make the measurements, i.e. a laboratory, your room, the earth, or any place that you decide on using to make the measurements. This referential can be a static place like your home ground or a movable surface such as the aisle of a train. In the second case, it may or may not be accelerated. If we can choose a referential such as its velocity to be constant or null (zero acceleration) in relation to the body that we are interested in describe this referential as an inertial referential and then Newton’s laws are valid!

Galilean relativity

  Now let me introduce you two Nintendo characters Mr. Game&watch, and Princess Peach (this is my otaku moment!). Imagine that Mr. Game&watch has a velocimeter and he is capable of measuring the velocity. Since he is standing in the “street” we can consider him as an inertial referential (represented by the black axis). On the other side, Peach also has a velocimeter, therefore, she’s capable of measuring the velocity. Then, she’s a movable referential (represented by the red axis).  If both of them measure their own velocity and the velocity of the other, we found the following results:

The result of the measurements shows an interesting fact related to the velocity. Velocity is relative as to whom (which referential to be more accurate) makes the measurement. Thus, for Mr. Game&watch it is Peach who moves because she is driving her car at 90 Km/h. However, for Peach, she isn’t in movement because she’s sitting in the chair car and to her, Mr. Game&watch move to the left (minus signal, remember that velocity is a vector) to 90 Km/h. Consider now a third character that runs in the street, and Mr. Game&watch and Peach measure his velocity. 

Donkey kong (DK) is running in the same direction that Peach drives. The result of the measures are:

  This information shows us that for an inertial referential (Mr. G&W) DK moves toward the right direction with a magnitude of 30 Km/h, but for Peach, DK moves toward the left direction with a magnitude of -60 Km/h. Again the results are different because the velocity is a relative concept and depends on where we make the measurement. At this point I would like to ask you, how do you think that the result would change if DK would eat a magical banana and his velocity increases to 120 Km/h?

  In all previous cases, we are considering that Peach moves with a constant velocity (acceleration = 0) i.e. she’s an inertial referential and Newton’s laws hold their validity. Therefore, dynamical properties are the same independent of whom makes the measure. In mechanics, this is known as Galilean relativity postulate. Things change when the referential moves with a velocity close to light speed, and we need to reformulate the laws of mechanics, but this belongs to the realm of relativity theory.

  But what happens if Peach accelerates with a constant acceleration (this means that the velocity changes, but always changes at a fixed rate) for example 15 m/s2, it means that each second her velocity increased to 15 m/s. Thus, in 3s her velocity will have an increase in 45 m/s. In this case, the laws of Newton are not more valid, and the net force measure by Mr. G&W and Peach will be different. It would seem as some forces only appear in certain conditions.

The fictitious forces

To understand why Newton's laws are not valid in accelerating referential consider the next situation:

Kirby is inside of an elevator, angry because each time that the elevator goes up or down he felt that his weight increases or decreases, and he don’t know why (yes, I’m using super smash bros characters ^_^). Sadly Kirby doesn’t know that the Newton laws don’t work in a noninertial referential (as is the case of the elevator) and when he design a free body diagram (a plot that shows the forces acting on the interest body, in this case, himself) he obtains:

His result implies that the weight and the normal have the same value. Therefore, nothing in his result shows why his weight changes when the elevator goes ups or downs. The correct result needs the inclusion of a fictitious force that we call Fin. With this the correct expression for the net force is:

  In this result, we include an extra force Fin that appears because we are in a noninertial referential with a constant acceleration. Using this equation we figure out the problem of Kirby’s weight.

  Analyzing this equation we see that the normal force changes when the elevator goes ups or downs. If we define m(g-a) as the apparent weight. Then, The apparent Kirby’s weight changes according to if the elevator ups (increases) or downs (decreases).

  This term Fin does not represent an interaction between two bodies. Therefore, it isn’t a force is a term that appears as a result of choosing an accelerating frame to make the measures. If we’ll make the measure from an inertial frame where the Newton laws are valid this term doesn’t appear. To prove it consider the next situation:

Here Yoshi is in the ground and he can be considered as an inertial referential in relation to the elevator. For him the system elevator + Kirby is an only body and drawing the free body diagram he found:

  Now the acceleration is the resulting of the net force acting on the system (elevator + Kirby) divided by the mass m.

  The Fin that appears when Kirby tries measuring his weight is sometimes called translational fictitious force and appears when we leading with referential at which the value of the velocity changes with the time. Nonetheless, the acceleration also appears when the direction of the velocity changes (like in the blades of the fan). However, I prefer to let this for my next post (I wrote a lot in this one hahahahaha), but I would like to finish this one presenting the complete form of the fictitious force

The first one called Ftrans is that I explained during this post. The other threes called centripetal, Coriolis, and azimuthal, respectively. Appears when the measures are made in a rotating referential. These three fictitious forces have a very physical interest because our planet is a rotating referential. Thus, these “forces” explain interesting phenomena like the functioning principle of a centrifugal machine, the easterlies, or the tides. Then, if you want to know what these phenomena happen on our planet you need to read my next post.

Oh! before finishing with all this information you’re capable to answer the question that my owner asked me!!! please comment below he will be happy to read it ;)

Thank you for reading my post, I hope that you enjoy reading it as much as I enjoyed writing it… Feel free to leave your comment about what you think about fictitious forces or better than that call me to play super smash bros online, and follow me in the next posts :0).