Check out this 736 horsepower Volkswagen Golf. Yes, most traditional muscle cars are rear-wheel drive instead of front-wheel drive. What is the difference? Of course there is an issue when the front wheels are used for both steering and power. But there is something else - traction.

Fake Forces ———–

The best way to look at frictional forces on the tires for a car is to consider fake forces. Trust me that this is the best way to go. But what is a fake force? First, what is a "real" force. Real forces are interactions between two objects. Some examples are friction, gravity and the normal force (the force between two surfaces pushing on each other). With real forces, we can say the total force on some object is equal to the time rate of change of momentum. Of course this force-momentum relationship only works in an inertial reference frame (one that is not accelerating).

A fake force is a force that we need to add to an object that is in an accelerating frame of reference so that we can once again use the force-momentum relationship (also called the momentum principle). People like to use fake forces all the time. When you are in a car and you turn to the left, it feels like there is a force pushing you to the right. Or when you are in a car speeding up, it feels like there is a force pushing you back into the seat. These are both fake forces but they feel real. Well, the truth is that according to Einstein's equivalence principle, we can't tell the difference between gravitational forces and fake forces from acceleration.

But how do you use fake forces? In general, we can look at an object two ways. First, we could look at the object from an inertial frame and look at all the real forces. Second, we could use an accelerating frame and add a fake force. The fake force would have this value:

Yes, the fake force is a vector. Don't forget about that.

Equilibrium ———–

If we are looking at an accelerating car in the frame of the car, then it is at equilibrium. I know that seems weird, but frames of reference can be weird. For an object in equilibrium, two things must be true. The net force (vector force) must be zero (vector) and the net torque about any point must also be zero (technically also a vector).

With the definition of torque, I can write these conditions as:

A couple of important points about torque. You can pick any point about which to calculate the torque. The r is a distance from the point that the force is applied to torque point and θ is the angle between r and the force.

Friction ——–

One last thing and then we can get to the car. Friction. The most common model for the frictional force says that the friction force is proportional to normal force. This can be written as:

The less than or equal sign is there because the frictional force is whatever it has to be to make the two surfaces not slide relative to each other. Of course there is some maximum static frictional force - that's what the equal part is for.

An Accelerating Car ——————-

What force makes the car accelerate? The frictional force from the front wheels (since it is a front wheel drive car). Here is a diagram of the forces on the car including the fake force.

This might look complicated, but it's not too bad. Let me just point out a few things. The location of the force matters. For both the gravitational force and the fake force, they are not contact forces so they don't act at one point. We can pretend like it acts at one point called the center of mass. Here is a post where I show how to calculate this center of mass, but for this post I just picked a reasonable looking location. I'm not completely certain, but I think the "center of fake force" would be at the same location as the center for gravity. The other point is the forces on the tires. I labeled the force on the front tire as N 1 and the rear as N 2 .

Now what about the forces? Remember that in this frame, the total vector force is the zero vector. I can write this as the following two equations.

Just from these two equations, we know that the sum of the two normal forces has to equal the total weight of the car. However, we don't know how much goes to the front and how much goes to the rear wheels. Looking at the horizontal forces, we can see that the maximum frictional force depends on the normal force on the front wheels.

Now what about the torque? Let's look at the net torque as calculated about the back wheel. I will use the following values for distances:

b = the distance from the back wheel to the front wheel.

s = the horizontal distance from the rear wheel to the center of mass.

h = the vertical distance from the ground to the center of mass.

If I consider torque in the CCW direction to be positive, then I get the following:

Now I can use two things. The definition of friction force (coefficient times the normal force) and the fake force is mass times acceleration. From this, I can solve for the maximum acceleration.

How can you get the highest acceleration? Well, you could increase the gravitational field (g) - but let's assume that we stay on the same planet. The other two things you could do would be to lower the center of mass (h) and/or move the center of mass closer to the front wheel.

If the acceleration is too high, the torque from the fake force alone would be greater than the torque from gravity. This would cause the car to do a "wheelie" where the front tire isn't in contact with the ground. No contact means no frictional force and no acceleration.

What if you had a rear-wheel drive car? If you wanted to look at a similar calculation, everything would look the same except for the torque equation. You wouldn't want to calculate torque around the back wheel because the normal force on the back wheel wouldn't be in the equation. You would find that with increased acceleration, there is an increased normal force on the rear wheels.

There is one other very similar situation. What about braking? All you would need to do is to change the direction of the fake force. In this case, a braking acceleration increases the normal force on the front wheels. This is why you need to change your front brake pads more often than your rear brake pads.