When a body moves on a surface or through a viscous medium (e.g. air, water), there are forces of friction because the body interacts with its surroundings.

Force of static friction, f_{s} : the force that counteracts the applied force and keeps the object from moving .

Force of kinetic friction, f_{k} : the retarding frictional force on the object in motion.

f_{s} increases as the magnitude of the applied force increases, keeping the object in place.

When it is on the verge of moving, f_{s} is at maximum.

When the applied force exceeds (f_{s})_{max}, the object accelerates .

Once the object is in motion, the retarding frictional force becomes less than (f_{s})_{max}.

#### Empirical Laws Of Friction

The direction of the force of static friction between any two surfaces in contact is opposite the direction of any applied force.

$$f_{s} \leq \mu_{s} n$$

,where

μ_{s} is coefficient of static friction

The direction of the force of kinetic friction acting on an object is opposite the direction of its motion and is given by:

$$f_{k} = \mu_{k} n$$

, where

μ_{k} is coefficient of kinetic friction

#### Coefficients Of Friction

The values of μ_{s} and μ_{k} depend on the nature of the surfaces (0.05-1.5), but μ_{k} is generally < μ_{s}.

The coefficients of friction are nearly independent of the area of contact between the surfaces.

Although μ_{k} varies with v, we normally neglect this (cf. stick-and slip motion at low v).

$\mu_{s}$ | $\mu_{k}$ | |
---|---|---|

Steel on steel | 0.74 | 0.57 |

Aluminum on steel | 0.61 | 0.47 |

Copper on steel | 0.53 | 0.36 |

Rubber on concrete (Dry) | 1.0 | 0.8 |

Rubber on concrete (Wet) | 0.30 | 0.25 |

Zinc on cast iron | 0.85 | 0.21 |

Copper on cast iron | 1.05 | 0.29 |

Glass on glass | 0.94 | 0.40 |

Copper on Glass | 0.68 | 0.53 |

Teflon on Teflon | 0.04 | 0.04 |

Teflon on steel | 0.04 | 0.04 |

Synovial joints in humans | 0.01 | 0.003 |

#### Experimental Determination Of $\mu_{s}$ And $\mu_{k}$

Suppose a block is placed on a rough surface inclined relative to the horizontal. The incline angle is increased until the block starts to move. By measuring the critical angle, $\theta_{c}$ at which this slippage just occurs, we can obtain $\mu_{s}$.

Once the block starts to move, it accelerates down the incline. However, if $\theta$ is reduced to a value less than $\theta_{c}$, it may be possible to find an angle $\theta_{c}^{\prime}$ such that the block moves down the incline with constant speed. We can then obtain $\mu_{k}$

Let’s work out the details:

The horizontal forces acting on the block:

$$\begin{aligned} \sum F_{x} &= 0 \\ mg \sin{\theta}-f_{s} &= 0 \end{aligned}$$

The above equation will give the static frictional force on the block:

$$f_{s} = mg \sin{\theta}$$

The vertical forces acting on the block:

$$\begin{aligned} \sum F_{y} &= 0 \\ n-mg \cos{\theta} &= 0 \end{aligned}$$

The above equation will give the normal force on the block by the ramp:

$$n = mg \cos{\theta}$$

Hence, we have:

$$f_{s} = n \tan{\theta}$$

Since $f_{s} \leq \mu_{s} n$, at max angle $\theta_{c}$, we can calculate $\mu_{s}$:

$$\mu_{s} = \tan{\theta_{c}}$$

If moving at constant $\vec{v}$, from $f_{k} = \mu_{k} n$, we have:

$$\mu_{k} = \tan{\theta_{c}^{\prime}}$$