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walking

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Rewritten friction chapter of Tipler & Mosca:

1. An opposing force is a force which opposes relative motion between two surfaces.

2. There are two types of opposing forces: frictional and drag.

3. Friction is an electromagnetic attractive force which opposes the relative motion of two objects in sufficiently close contact.

4. Our first proposition regarding friction is that it is independent of the surface area of the objects in contact.

5. The argument for this is as follows. At the microscopic level, two objects only actually contact at widely spaced prominances called asperities. As the friction and hence the em attraction between the objects increases, the asperities are more squashed together increasing the microscopic contact area, thus friction is proportional to microscopic contact area. Having established this, we observe that when an object lying on a side of greater surface area is switched to a side of lesser surface area, the microscopic contact area decreases while the force at each asperity increases due to the greater force per unit area, creating greater squashing. These two effects cancel each other out and the result is that the microscopic contact area remains unchanged, and hence also the friction is unchanged since as we showed already, friction is proportional to microscopic contact area. This proves the proposition.

6. Now there are three types of friction:

-Static friction

-Kinetic friction

-Rolling friction

7. Static friction is the friction which occurs between two objects at rest relative to each other.

8. The mathematical description of static friction is ##f_s\le \mu_s F_n## where ##\mu_s## is a constant called the coefficient of static friction and depends on the nature of the surfaces in contact, and ##F_n## is the normal contact force.

9. It is actually found that frictional forces are generally of the form ##f=\mu F_n##. The argument for this is that as friction and hence em forces between two surfaces increase, the surfaces must be pushing each other with more force, meaning that the normal contact force between them is also higher. Thus, friction is proportional to normal contact force, and he constant of proportionality, ##\mu##, is simply found by experiment for each particular kind of surface.

10. An interesting result regarding static friction is that the angle at which maximum static friction occurs is given by ##tan \theta = \mu_s##. The argument for this doesn't contain anything new and can be easily proved using the previous results and basic dynamics.

11. Kinetic friction is the friction between two objects in relative, sliding motion.

12. The formula for it is ##f_k=\mu_k F_n##.

13. Kinetic friction is less than maximum static friction.

14. Rolling friction is the friction between two objects in relative, rolling motion, such as when a wheel rolls along the ground.

15. The formula for it is ##f_r=\mu_r F_n##.

16. A good example of the three types of friction is a car wheel. The accelerating force of a car wheel at low speed is static friction. At higher speeds this becomes kinetic friction. Alongside either of these, there is also rolling friction opposing the direction of the car's motion.

17. The explanation for these is as follows. Static friction is because the wheel tries to slide backwards and this relative motion is opposed by the ground, keeping the wheel static relative to the ground. At higher speeds however, the wheel actually slides and so the accelerating force becomes kinetic friction instead. Finally, rolling friction occurs throughout because the wheel is always peeling away from the ground, and the em attraction opposes this relative motion between wheel and ground, thus causing a friction opposing the wheel's tendency to peel forwards.

18. We now come to drag forces, which are forces between an object and a fluid, rather than two objects. (A "fluid" is simply any liquid or gas, such as water or air.)

19. The formula for drag forces is ##f=bv^n##, where b is a constant depending on factors such as the shape of the object, with objects of greater surface area having a higher value of b.

20. A parachute is an application of this to reduce the terminal speed of a skydiver.

21. The mathematics of this is that as skydiver falls at terminal speed, their weight and drag force are equal. Then as the parachute opens, the drag force increases and thus becomes greater than the weight. This causes the skydiver to accelerate upwards, thus reducing their speed. As the speed reduces, ##bv^n## reduces until it again becomes equal to the weight, but now at a new, lower terminal speed. This low terminal speed continues until the diver reaches the ground.

I have omitted a few parts such as the thresh-hold breaking system which was described in this chapter. I have also omitted the non friction parts of the chapter, such as numerical integration, because I was only trying to write an exposition of friction. Also, I was only aiming at using the frictional information in this chapter rather than elsewhere in the book or in other books.