Friction

What is friction?

Although two surfaces that are in contact may seem very smooth, at a microscopic scale, there are many peaks and troughs that result in friction. In practice, it is impossible to create an object that has an absolutely smooth surface. According to the law of the conservation of energy, no energy in a system is ever destroyed. In this case, friction produces heat energy, which is dissipated through the medium and the objects themselves.

Friction Results from Interatomic Electric Forces

Friction is a type of contact force, and as such, it results from interatomic electric forces. On a microscopic scale, the surfaces of objects are not smooth; they are made of minuscule peaks and crevices. When the peaks slide against and run into each other, the electron clouds around the atoms of each object try to push away from each other. There could also be molecular bonds that form between parts of the surfaces to create adhesion, which also fights against movement. All these electric forces put together compose the general friction force that opposes sliding.

Static frictional force

As the name suggests, this is the friction force (fs) that is in action when the objects in interaction are static. As the frictional force is a force like any other, it is measured in Newtons. The direction of the friction force is in the opposite direction to that of the applied force. Consider a block of mass m and a force F acting on it, such that the block remains at rest.

There are four forces acting on the object: the gravitational force mg, the normal force N, the static frictional force fs, and the applied force of magnitude F. The object will remain in equilibrium until the magnitude of the applied force is bigger than the frictional force. The frictional force is directly proportional to the normal force on the object. Hence, the lighter the object, the less the friction.

\[f_s \varpropto N\]

To remove the sign of proportionality, we have to introduce a proportionality constant, known as the coefficient of static friction, here denoted as μ_s.

However, in this case, there will be an inequality. The magnitude of the applied force will increase to a point after which the object will start moving, and we no longer have static friction. Thus, the maximum value of static friction is μ_s⋅N, and any value less than this is an inequality. This can be expressed as follows:

\[f_s \leq \mu_s N\]

Here, the normal force is \(N = mg\).

Kinetic frictional force

As we saw earlier, when the object is at rest, the frictional force in action is static friction. However, when the applied force is greater than the static friction, the object is no longer stationary.

At the point where the applied force exceeds the static frictional force, kinetic friction comes into action. As the name suggests, it is associated with the motion of the object. Kinetic friction does not increase linearly as the applied force is increased. Initially, the kinetic frictional force decreases in magnitude and then remains constant throughout.

Kinetic friction can further be classified into three types: sliding friction, rolling friction, and fluid friction.

All three types of kinetic friction can be determined using a general theory of kinetic friction. Like static friction, kinetic friction is also proportional to the normal force. The proportionality constant, in this case, is called the coefficient of kinetic friction.

\[f_k = \mu_k N\]

Here, μ_k is the coefficient of kinetic friction, while N is the normal force.

The values of μ_k and μ_s depend on the nature of the surfaces, with μ_k being generally less than μ_s. Typical values range from 0.03 to 1.0. It is important to note that the value of the coefficient of friction can never be negative. It may seem that an object with a greater area of contact will have a bigger coefficient of friction, but the weight of the object is evenly spread and so does not affect the coefficient of friction. See the following list of some typical coefficients of friction.

Surfaces
Rubber on concrete	0.7	1.0
Steel on steel	0.57	0.74
Aluminium on steel	0.47	0.61
Glass on glass	0.40	0.94
Copper on steel	0.36	0.53

The geometric relation between static and kinetic friction

Consider a block of mass m on a surface and an external force F applied parallel to the surface, which is constantly increasing until the block starts moving. We have seen how static friction and then kinetic friction come into action. Let us represent the frictional forces graphically as a function of the applied force.

As discussed earlier, the force applied is a linear function of static friction, and it increases to a certain value, after which kinetic friction comes into action. The magnitude of kinetic friction decreases until a certain value is attained. The value of friction then remains almost constant with the increasing value of external force.

Friction Force Calculation

Friction is calculated using the following formula, with \(\mu\) as the coefficient of friction and F_N as the normal force:

\[|F_f| \leq \mu |F_n|\]

Each force has units of Newtons, N. This formula shows that the magnitude of the friction force depends on the coefficient of friction, as we discussed above, as well as the magnitude of the normal force. As the coefficient of friction or normal force increases, the friction force increases. This intuitively makes sense - when we push a box, it's harder to push when the surface is rougher and when the box is heavier.

Friction on an inclined plane

Thus far, we have focused on objects on a horizontal surface. Now, let us consider an object at rest on an inclined plane, which forms an angle θ with the horizontal.

Considering all the forces acting on the object, we find that the gravitational force, friction, and the normal force are all the forces that need to be taken into consideration. As the object is in equilibrium, these forces should cancel each other out.

We can consider our Cartesian axes anywhere to make our calculations convenient. Let us imagine the axes along the inclined plane, as shown in figure 4. First, gravity is acting vertically downwards, so its horizontal component will be mg sinθ, which balances the static friction acting in the opposite direction. The vertical component of gravity will be mg cosθ, which is equal to the normal force acting on it. Writing the balanced forces algebraically, we get:

\[f_s = mg \sin \theta_c\]

\[N = mg \cos \theta\]

When the incline angle is increased until the block is on the verge of slipping, the force of static friction has reached its maximum value μ_sN. The angle in this situation is called the critical angle θ_c. Substituting this, we get:

\[\mu_s N = mg \sin \theta _c\]

The normal force is:

\[N = mg \cos \theta_c\]

Now, we have two simultaneous equations. As we are looking for the value of the coefficient of friction, we take the ratio of both the equations and get:

\[\frac{\mu_s N}{N} = \frac{mg \sin \theta_c}{mg \cos \theta_c} \qquad \mu_s = \tan \theta_c\]

Here, θc is the critical angle. As soon as the angle of the inclined plane exceeds the critical angle, the block will start moving. So, the condition for the block to stay in equilibrium is:

\[\theta \leq \theta_c\]

When the incline exceeds the critical angle, the block will start accelerating downwards, and kinetic friction will come into action. It can thus be seen that the value of the coefficient of friction can be determined by measuring the angle of the inclination of the plane.

How friction affects movement

If an object is moving on a surface, it will begin to slow down due to friction. The greater the frictional force is, the more quickly the object will slow down. For instance, there is a very small amount of friction acting on the skates of ice skaters, allowing them to glide easily around an ice rink without significant deceleration. On the other hand, there is a very large amount of friction acting when you try to push an object over a rough surface - such as a table across a carpeted floor.

It would be extremely difficult to move without friction; you probably know this already, because when you try to walk over ground covered in ice and attempt to push off against the ground behind you, your foot will slip from beneath you. When you walk, you push your foot against the ground in order to propel yourself forwards. The actual force pushing you forwards is the frictional force of the ground on your foot. Cars move in a similar way, the wheels push back on the road at the point on the bottom where they are in contact with it and the friction from the road surface pushes in the opposite direction, causing the car to move forward.

Heat and friction

If you rub your hands together, or against the surface of a desk, you will experience a frictional force. If you move your hand fast enough you will notice it becomes warm. Two surfaces will become heated as they are rubbed together and this effect will be greater if they are rough surfaces.

The reason that two surfaces become heated when they experience friction is that the frictional force is doing work and converting energy from the kinetic energy store in the movement of your hands to the thermal energy store of your hands. As the molecules that make up your hand rub together, they gain kinetic energy and begin to vibrate. This kinetic energy associated with the random vibrations of molecules or atoms is what we refer to as thermal energy or heat.

Air resistance can also cause objects to become very hot due to the thermal energy released. For instance, space shuttles are covered in heat-resistant material in order to protect them from burning up. This is due to the large increases in temperature as a result of air resistance they experience when they travel through the Earth's atmosphere.

Damaged surfaces and friction

Another effect of friction is that it can cause two surfaces to become damaged if they are easily deformed. This can actually be useful in some cases:

Terminal velocity

One of the interesting effects of drag is terminal velocity. An example of this is an object falling from a height down to the earth. The object feels the gravitational force due to the earth and it feels an upwards force due to air resistance. As its speed increases, the frictional force due to air resistance also increases. When this force becomes large enough so that it is equal to the force due to gravity, the object will no longer be accelerating and will have reached its maximum speed - this is its terminal velocity. All objects would fall at the same rate if they did not experience air resistance.

The effects of air resistance can also be seen in the example of the top speed of cars. If a car is accelerating with the maximum driving force that it can produce, the force due to air resistance will increase as the car moves faster. When the driving force is equal to the sum of the forces due to air resistance and friction with the ground, the car will have reached its top speed.