Friction

Friction is a force (measured in Newtons) acting along the surfaces of two objects in physical contact that impedes the relative motion of these objects (the motion of one with respect to the other). As such, it can be said that friction always acts opposite the motion of the object or opposite the force applied.

There are two main types of friction:

1. Static friction: the frictional force that opposes any attempt to move a stationary object along a surface.

2. Sliding friction: (sometimes called "kinetic" friction): this frictional force opposes the sliding motion of two surfaces rubbing together.

A third type of friction, rolling friction, is a special case of static friction and kinetic friction - which will be discussed later.

Physically, friction arises from surface roughness and attractive forces between the surfaces of two objects in contact.

The frictional force depends on how hard the surfaces are pushed together: usually that quantity is calculated by determining the surface ‘normal’ force (F_n).

The harder the surfaces are pushed together, the more surface contacts are made, increasing friction.

Interestingly, friction does not depend on the surface area of the objects in contact. This is because as the area gets larger, the force per unit area gets smaller, decreasing the number of points in physical contact. Some of the characteristics of static and kinetic friction are:

Static friction: -proportional to F_n(surface normal force)

-independent of area

-reaches a maximum value (which depends on the surface materials) in preventing motion between surfaces, then drops to the lower value of sliding friction as the object begins to move.

-designated as f_s

Sliding friction: -proportional to F_n

-independent of area

-independent of speed of surfaces

-designated as f_k

-always less than static friction f_k < f_s (meaning it’s easier to push an object once it’s moving)

Let’s examine the relationship between these two forces and the applied force that creates them (they can be thought of as "reaction" forces). If you observe the graph below, it shows the static frictional force increasing to a maximum with the application of a force ("applied force") then dropping off sharply to a lesser value (kinetic, or sliding, friction) once the object starts moving. This diagram is typical of many contact surfaces.

We can quantify this force in the following way: since friction is proportional to the force pressing the surfaces together (Fn), we can write,

f a F_n

which means that,

f / F_n = constant

This constant we call the coefficient of friction: m (the Greek letter ‘mu’). Thus we can write the equation as,

f = (m)(F_n)

Since static friction and kinetic friction are different, there is a m for each:

m_s = coefficient of static friction

m_k = coefficient of kinetic (sliding) friction

so that we can write for each type of friction,

Static: f_s < m_sF_n

Kinetic: f_k = m_kF_n

Note that static friction is expressed as an inequality. This is because it varies from zero to a maximum. At the maximum value, and only at the maximum value (just before the object moves), the static frictional force is exactly equal to m_sF_n, or

f_{s max} = m_sF_n

We can find the actual values of m_s and m_k in the following way:

• • for m_s: a block is placed on a variable incline that is slowly raised from the horizontal until the block just begins to slide....

    

    This is called the "critical angle", q_c. At this angle, the static friction has reached its maximum value and we can write SF_x = 0 (the max value of static friction occurs the instant before the block moves), so:

    • SF_x = 0 .................................SF_y = 0

    • -f_s + mgsinq_c = 0................. F_n - mgcosq_c = 0

    • -m_sF_n + mgsinq_c = 0 ........................... F_n = mgcosq_c

    and using the value of F_n from the y-equation we get,

    -m_smgcosq_c +mgsinq_c = 0

    m_s = mgsinq_c / mgcosq_c

    m_s = tanq_c

    And so the coefficient of static friction is simply defined as the tangent of the critical angle (note that as the incline is raised higher and higher, the force due to gravity, what we call the weight, increases down the incline and the force pressing the block against the incline, F_n, decreases)

    for m_k: we perform the same experiment as above but now the block is moving down the incline at some acceleration.

    

    Newton’s 2nd law becomes:

    SF_x = ma

    -f_k + mgsinq = ma

    -m_k + mgsinq = ma

    but F_n = mgcosq (from before), so that,

    - m_k(mgcosq) + mgsinq = ma

    -m_k(gcosq) +gsinq = a

    m_k = (gsinq - a) / (gcosq)

    Which defines the coefficient of kinetic friction. Note that if friction is zero, you can set m_k = 0 and the equation simplifies to .... a = ______?

Rolling friction: -a form of static friction:

If the tires slide, such as when the wheels lock on braking, the sliding friction is less than static friction and so the car will take a longer time to slow down. This is why anti-lock brakes are useful: they prevent the wheels from locking and creating sliding friction.