# Road Resistance

[To understand the material of these documents, I assume the reader have a firm understanding of Newtonian Kinematics.]

### Introduction

When we try to slide something on the floor, we know that it takes strength to do it. This is because there is friction between the object and the floor. Similarly, cars are connected to the road by the tires that can't move unless we apply a force that is greater than the frictional force to the tires. It turns out this frictional force affects not only the straight line performance of the car but also the handling characteristics of a car.

Here are some characteristics of the frictional force:

1. It points against the force of action because it is a reaction force defined by Newton's Third Law.
2. The amount of frictional force is shown experimentally to be directly proportional to the force acting perpendicular to the contact surface. In the case of moving an object of the floor, you can express the frictional force as μmg where μ is the friction coefficient, m is the mass of the object and g is gravity.
3. μ depends on the how the two surfaces interact, it is usually determined experimentally.
4. Frictional force has two types: static friction and kinetic friction.

### Static Friction

Static friction an reaction force that keeps an object from sliding against a surface. If you keep increasing the force that applies to the object, you will meet an equal amount of static friction to keep the object at rest. However, up to a certain point, the static friction can't increase any more. The amount of force at that point is called limiting friction. According to the experiments conducted by the physicists, limiting friction is directly proportional to the weight of the object (actually the force acting perpendicular to the contacting surfaces, when we are on the ground, this is the same as the object weight). It turns out it is possible to deduce the limiting friction coefficient by looking at the skidpad results quoted by the car magazines. The wayto calculate it will be demonstrated later in this page.

When the force applied to object exceeds the limiting friction, the object starts to move. When the object is moving, frictional force starts to drop significantly and then reach another plateau that is called kinetic friction - the frictional force acting against the moving direction of an object.

### Kinetic Friction

Kinetic Friction is the amount of force acting against a moving object that is sliding (as oppose to rolling) against a surface. When an object is sliding, the surface of it that is in contact with the ground is always the same. It is best illustrated by moving box on a flat floor. As we can see, when we slide the box around, the bottom is always the surface that touches the floor during the motion.

Physicists found out the kinetic friction is also directly proportional to the weight of the moving object. Kinetic friction coefficient is always a fraction of the limiting friction coefficient.

### Rolling Resistance

Rolling Resistance is a completely different beast from static or kinetic friction. It is actually a type of energy loss. Rolling Friction applies to the case when a circular deformable objects. When such an object moves on the floor, it is always deformed at the bottom. The energy spent on deforming the a rolling object can be translated to a frictional force that is the rolling resistance. It turns out rolling resistance is also directly proportional to the weight of the object but the rolling resistance coefficient is several magnitude smaller than the kinetic friction coefficient.

Rolling resistance comes into play when a car is moving forward or backward by rotating/rolling the tires by running the engine. It comes into play in our acceleration formula outlined in the Introduction section.

### Calculting Limiting Friction from Skidpad result

When I started to read car magazine, I was always baffled by the skidpad. The car magazines always tout about how many g's a car can pull. They claim that the more g's a car can pull the better "handling" the car has.

After some research of my own, I finally understand what it means. Skidpad is a circular race track with a 300 feet radius (For metric system places, they use 100m radius). The test driver drives the car around the skidpad and slowly increase the driving speed. He/she drives the car until the tires at the outer circle starts to slip/skid. At that point, the test driver obtains the maximum speed the car can travel in a 300 feet radius skidpad. By using this speed, the car magazine can deduce how many g's a car can pull.

When a car is moving in a perfect circle, due to Newton's First Law, there must be a force acting perpendicular to the car's motion pointing away from the circle because of the inertia of an object to maintain a straight line movement. This perpendicular force is called the centripetal force and it is defined by mv2/r where r is the radius of the circle. Number of g's is recorded as v2/gr. It is originally defined as how many times of the earth gravity you can feel when a car is turning.

Here is a brief description of the physical phenomenon surrounding a skidpad test. Note that when a car is moving in a circle at a constant speed, the tires are rolling. The rolling action ensures that the tires are momentarily at rest. Thetires are subjected to static friction. However, as v increases, the centripetal force acting on the tires increases. When it exceeds the limiting friction of the car, the tires will be locked and no longer rolling. It will now begin to slip/skid towards the direction point away from the center of the circle. If you are slowly increasing the speed, you will find that you cannot maintain running in a 300 feet skidpad, your car must now turn in a curve that has a radius greater than 300 feet. At the point, when the test driver can't maintain the 300 feet radius circle, the speed is recorded and the number of g's is deduced.

An easy way of deducing limiting friction coefficient is to equate μlfmg and mv2/r. This will imply the limiting friction of your tires is actually equal to the number of g's your car can pull in a skidpad. For example, our Skyline pulls 0.94g on a 300 feet skidpad, so μlf is also 0.94.

The real number is higher than 0.94 (how much higher depends on how fast you are running the skidpad). This is because the car develops yaw angle (ie the angle between the direction of the chassis and the direction the car is traveling) while turning, so the tires loses grip. Please refer to the Traction Circle and Yaw Angle sections for more details on this topic.

First Draft: November 7th, 2003
1.0 Published: November 14th, 2003
1.1 Published: December 16th, 2003
Yee Man Chan with input from Chi Yung Ng