### Maximum frictional force (Mechanics)

#### Friction

Friction is a force which works in the opposite direction to the direction of motion when an object is on a rough surface. The maximum or limiting value of friction between two surfaces is $F_{\text{MAX} } = \mu R$ where $\mu$ is the coefficient of friction and $R$ is the normal reaction between the two surfaces. The coefficient of friction is a measure of the roughness of the surface - the rougher the surfaces the larger the value. In smooth surfaces there is no friction so $\mu=0$.

#### Worked Example: Finding the maximum frictional force

###### When a force is applied horizontally

Suppose that a particle of mass $8 \mathrm{kg}$ lies at rest on a rough horizontal surface. A force of magnitude $P\mathrm{N}$ is applied to the particle horizontally and the coefficient of friction between the particle and the surface is $\mu=0.8$. What magnitude of $P$ will force the particle to move from rest?

###### Solution

We can resolve $F = ma$ in the upwards direction, where acceleration is zero, in order to find the normal reaction, $R$. \begin{align} F & = ma, \\ R - mg & = ma, \\ R - \left(8 \times 9.8 \right) & = 8 \times 0, \\ R & = 78.4 \mathrm{N}. \end{align} From this we can find the maximum frictional force \begin{align} F_{\text{MAX} } & = \mu R, \\ & = 0.8 \times 78.4 \mathrm{N}, \\ & = 62.72 \mathrm{N}. \end{align} Therefore in order to move the particle $P$ must exceed $62.72 \mathrm{N}$. When $P = 62.72 \mathrm{N}$ the particle will remain at rest but is said to be in limiting equilibrium.

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