Winter Olympics 2018: The Physics of Blazing Fast Bobsled Runs
I don’t know very much about bobsleds—but I know quite a bit about physics. Here is my very brief summary of the bobsled event in the winter Olympics. Some humans get in a sled. The sled goes down an incline that is covered in ice. The humans need to do two things: push really fast to get the thing going and turn to travel through the course. But from a physics perspective, it’s a block sliding down an incline. Just like in your introductory physics course.
So here is a block on a low friction inclined plane—see, that’s just like a bobsled.
You can see that there essentially just three forces acting on this box (bobsled). Let’s take a quick look at each of these forces.
In this situation, the gravitational force is the simplest because it doesn’t change. When you are near the surface of the Earth, the gravitational force (also called the weight) just depends on two things: the gravitational field and the mass of the object. The gravitational field actually decreases as you get farther away from the center of the Earth—but even the top of the tallest mountain isn’t that far away, so we say this value is constant. This gravitational field has a value of about 9.8 Newtons per kilogram and points straight down (and we use the symbol g for this). When you multiply the gravitational field by the mass (in kilograms), you get a force in Newtons. Simple.
The next force is the force with which the inclined plane pushes up on the box. But wait! It’s not really pushing up, it’s pushing perpendicular to the surface. Since the force is perpendicular, we call this the normal force (the geometry definition of normal). However, there’s still a small problem—there is no equation for normal force. The normal force is a force of constraint. It pushes with whatever magnitude it needs to to keep the box constrained to the surface of the plane. So really the only way to find the magnitude of this normal force is to assume the acceleration perpendicular to the plane is zero. That means that this force has to cancel the component of the gravitational force that is also perpendicular to the plane. In the end, the normal force will decrease as the angle of the incline increases (a block on a vertical wall would have zero normal force).
The last force is the frictional force. Like the normal force, this force is also an interaction between the box and the plane. But this frictional force is parallel to the surface instead of perpendicular. If the block is sliding, we call this kinetic friction. In the most basic model, the magnitude of this frictional force depends on just two things: the types of surfaces interacting (we call this the coefficient of friction) and the magnitude of the normal force. The harder you push two surfaces together, the greater the frictional force (but you already knew that).
Now we are ready for the important part—the relationship between force and acceleration. The magnitude of the total force on the object in one particular direction is equal to the product of the object’s mass and acceleration. For the x-direction, this would look like this:
The key here is that the acceleration of the object depends on both the total force and the mass of the object. If you keep the force constant but increase the mass, the object would have a smaller acceleration. Now let’s put this all together. I will set the x-axis along the same direction as the plane. This means there are two forces that will influence the acceleration down the inclined plane: part of the gravitational force and the frictional force. The gravitational force obviously increases with mass—but so does the frictional force since it depends on the normal force. What we have are two forces that increase with mass. So the mass of the block doesn’t matter for the acceleration down the incline. It only depends on the inclination angle and the coefficient of friction. In a race, a big block and a small block would end in a tie (assuming they started with the same speed).
If mass doesn’t matter, then why would a four person bobsled be faster than a two person one? Obviously, there must be some other force involved—one that doesn’t depend on the mass of the object. This other force is the air resistance force. You already know about it: Whenever you stick your hand out of a moving car window, you can feel this air resistance force. In the basic model, it depends on several things: the density of air, the size and shape of the object, and the speed of the object. As you increase the speed, this air resistance force also increases. But notice that this does not depend on the mass.
Let me show the impact this has on a bobsled with the following example. Suppose I have two blocks sliding down identical inclines and traveling at the same speed. Everything is identical except for the mass. Box A has a small mass and box B has a large mass.
Although they have the same air force and same speed, the heavier box (box B) will have the greater acceleration. This same air resistance force will have a smaller impact on its acceleration because it has a larger mass. So mass does indeed matter in this case. Actually, the air drag matters quite a bit. That’s why bobsled teams are also very concerned about the aerodynamics of their vehicle. When competing in the Olympics, every little bit matters.