# Laws of Friction, Application in Pulleys, Brakes and Wedges

• ## The Origin of Friction

Friction is defined as the resistive force to motion, offered by the contacting surfaces. Another common observation is generation of heat from the contacting surfaces. The applied kinetic energy is converted to heat. The work done is $0$, till the friction is overcome.

The origin of friction is in the nature of surfaces, surface irregularities, roughness, deformation of surface etc. It ultimately boils down to the electromagnetic forces between charged particles. However, it is impractical as well as uneconomical to compute the magnitude of frictional force by this method. Hence, empirical methods are common in studying friction.

Tribology is a subject under mechanical engineering, where friction is studied extensively.

• ## Terms

I) Dry Friction – Surfaces are dry, magnitude of frictional force is generally high.

II) Fluid Friction – Relatively smaller magnitudes, used for lubrication. Exists between a solid and a liquid as well as between 2 liquids. Drag is a kind of fluid friction, where a fluid tries to move across a solid.

III) Limiting Friction – The maximum resistance offered due to friction. If the applied force is more than limiting friction, the bodies will move relative to each other.

IV) Static Friction – Friction between stationary surfaces

V) Kinetic Friction – Friction between moving (sliding/ rolling) surfaces

• ## Analysis of Friction

We will consider the case of limiting friction on a block under the influence of self-weight. Let $P$ be the applied force, $R$ the normal reaction, $W$ the weight and $F_{fr}$ the limiting frictional force. The body will be in equilibrium under these forces.

The coefficient of static friction, $\mu_s$ is defined by

${\mu_s = \frac {F_{fr}}{R}}$

If the applied force leads to motion, we consider the coefficient of kinetic friction. The kinetic friction is generally lower than static friction.

${\mu_d = \frac {F_{fr}^{kinetic}}{R}}$

• ## Angle and Cone of Friction

The angle of friction $\phi$ is the inverse tangent of the coefficient of friction. The cone of friction is an imaginary cone with $\phi$ as the semi-cone angle, normal reaction $R$ as height and the resultant of $R$ and $F_{fr}$ as the slant height.

The significance of cone of friction is as follows:

Let $P$ be the resultant of all external forces acting on the body, excluding the reaction and friction. If the body is in static equilibrium, then $P$ must lie inside the cone. If it doesn’t, the body will no longer be in equilibrium.

• ## Laws of Friction

These are generalized statements, based on observations:

1) Frictional force acts in a direction opposite to the movement of the body.

2) For static friction, the maximum resistive force is given by $\mu_s R$

3) For kinetic friction, the maximum resistive force is given by $\mu_d R$

4) The frictional force is independent of the area in contact, however, it depends on the nature of surfaces.

Now we will see few applications of friction with corresponding mathematical analyses. These are

I) Belt Drives

II) Band Brakes and

III) Wedges

• ## Belt Friction

Pulleys and belts are often used to transmit power from one point to another. (In flour mills, you must have seen a belt connecting the mill and the motor). In the absence of friction, the rotation of pulleys will not lead to movement of belt. Hence, friction is very important in power transmission.

Depending on the applications, the belts are chosen. The problems involve calculating the tensions on tight $(T_1)$ and slack $(T_2)$ sides. The relationship between $T_1$ and $T_2$ depends on the coefficient of friction $\mu$ \$between the belt material and pulley and the angle of lap $\beta$. The angle of lap is the angle made by an arc along the pulley, where the belt is lapped on the pulley. It is measured in radians (Always!).

### I) Flat Belts

The cross-section of a flat-belt is rectangular. For flat belts, the relationship between $T_1$ and $T_2$ is given by

${\frac {T_1}{T_2} = e^{\mu \beta}}$

### II) V Belts

V belts have trapezoidal cross-section. If $\alpha$ is the angle made by the imaginary triangle at its vertex (see the figure), then we have

${\frac {T_1}{T_2} = e^{\big( \frac {\mu \beta}{sin \ \alpha} \big)}}$

$\alpha$ is also known as the angle of groove.

• ## Brakes (Band Brake)

Friction can be used as a power-transmitting agent (in belts-pulleys) as well as a power-absorbing agent (in band brakes). The idea is, the band resists the motion of rotating wheel through friction. The braking torque is obtained by the relations

${\frac {T_1}{T_2} = e^{\mu \beta}}$

and

${Torque \ = \ (T_1 - T_2) \times \ radius \ of \ wheel}$

In this case, the values of $T_1$ and $T_2$ depend on the applied force $P$ as well. Drawing the free body diagram of lever and pulley is necessary.

• ## Wedge and Block Friction

A wedge is a triangular tool and an inclined plane, used for

I) producing large forces or

II) giving small displacements

When a force applied to its blunt end, the force gets divided into forces, normal to its inclined surfaces. (See the figure below)

While analyzing the wedge action, the weight of wedge is generally neglected.