Sedentary Shoes

The topics learned in the first few weeks of Physics 111 conjured up a funny memory that I have from my childhood. My dad, a NYC fireman, was on his way to work and was dropping me off at school on the way. We had traveled a few blocks from home when we stopped at a red light, and a woman gave us a very funny glance, pointed at the roof of our car, and then motioned for us to pull our windows down. She then exclaimed to us, "There's a pair of boots on your roof!" Pulling over, my dad realized that he had forgotten to place his work boots back inside the car: carrying multiple things in his hands, he had temporarily placed them on the roof so that he could unlock the car and put his coffee in the cup holder. Being in a rush he had forgotten about them and left them up there. We both laughed about how we could have driven without them falling off. Now, I can understand that physics played an instrumental role in keeping his shoes stationed at the top of our car.

Newton's laws of motion played a role in allowing the shoes to stay stationary. Specifically, Newton's second law of motion, which states that “the acceleration of an object is directly proportional to the net force acting on it, and inversely proportional to its mass. The direction of acceleration is in the direction of the net force acting on the object” (Class notes). 

                                                 ΣF = ma 

I have made the following free body diagram to consider the forces acting on the car and the boots.  

It is important to note that the force of friction acting on the boots is in the same direction as the car's travel. This is due to the fact that, lying on top of the car, the boots would have been jerked backwards if they had ever begun to move. Because friction is a resistive force, it is thus acting in the +x direction.

The question to consider, then, is why didn't the boots fall off of the car as the car accelerated and traveled during its route to school? The important factors in answering this question are friction and acceleration. It is clear that the car accelerated when it went from rest, in our driveway, to driving through my neighborhood. Because the boots were stationed on top of the car, the acceleration of the car is the same acceleration of the boots. However, while the car was accelerating, the boots must have been carried along on top of the car in order for them to stay secured to the car during travel. This stationary aspect is thus a result of the force of friction between the car roof and the boots. 

Considering Newton's second law, it is apparent that the boots will not move (accelerate) without a force acting on it to cause this motion. The only force acting on the boots in the x direction is the force of friction, thus the equation for the boots becomes:

FFr = mbootsacar

Because the boots never slid, the relevant friction force is static friction:

The static friction coefficient depends on the surfaces of the situation: in this case the boots and the roof. If a force was applied that was greater than this force of static friction, the boots would have started moving, transitioning into the realm of kinetic friction. Even though the boots were in motion as the car moved (they were carried along on the top), the part of the boots in contact with the roof were stationary relative to the roof, so it was static rather than kinetic friction. 

Summing all of these pieces of information together: for the boots to stay on the roof, the force of static friction was equal to the force of acceleration acting on the boots (the acceleration of the car).  In other words, the total force applied when the car began to move was not enough to move the boots, thus "canceling out" the effect of the car's movement. If the force of the car's acceleration would have increased dramatically, the static friction would have increased until the maximum possible static friction was achieved: it would then transition to kinetic friction, be able to overcome the force of the car's acceleration, and also move. For all of this to be true, I assume that the acceleration of the car must not have been very large: if there was a very large acceleration it would have most likely surpassed the force of static friction and enabled the boots to move. This is reasonable because we were driving slowly through my neighborhood, with no major roads or highways: the speed of the car was relatively slow and there were no major changes in speed which would have tossed the boots off. Additionally, there may (or may not) have also been a large coefficient of static friction (allowing the boots to stay stationary), for the rubber soles of the boots on the roof- since the boots were textured and had grooves in them, despite the smoother roof. A larger coefficient would mean that the acceleration could have been larger and still been equal to a force of friction that lies in the static range: thus, the boots would not move. Despite these assumptions, it is certain that the acceleration was equal to the force of friction, sparing my dad the issue of being shoe-less at work. 

So, next time you forget to put your coffee, your phone, or your work boots back inside your car, don't worry! Physics may just work in your favor and keep your objects intact.