Angle and Cone of friction

The relationship between the Angle and Cone of Friction

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Description

Engineers often refer to either the coefficient of friction or the angle of friction. This section describes their relationship and goes on to explain the Cone of friction:

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Let R be the normal reaction at a point of contact O and F the frictional force acting in a direction perpendicular to R. Then the total force at O is given by

$\sqrt[]{R^2 + F^2}$

(2)

acting in a direction making an angle

$tan^{-1}\left( \frac{F}{R} \right)$

(3)

with the normal reaction.

If friction is limiting, $\inline F = \mu\:R$ and the action at O makes an angle of $\inline tan^{-1} \mu$ with the normal reaction. This angle is denoted by $\inline \lambda$ .

Thus

$\mu = tan\:\lambda$

(4)

and the magnitude of the limiting friction can be found if either $\inline \mu$ or $\inline \lambda$ is known.

If the direction in which the body tends to move is varied the the force of limiting friction will always lie in the plane through O perpendicular to the normal reaction and the direction of the total action at O will always lie 0n a cone with it's vertex at O and axis along the line of the normal reaction. The semi vertical angle will be $\inline \lambda$

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This cone is called the cone of friction. If friction is not limiting, the angle made by the total action at O will be less than $\inline \lambda$ . Hence whether the friction be limiting or not, the direction of the total action at O must be inside or on the cone of friction. It follows that if P is the resultant of the other forces acting on the body which is in equilibrium, the direction of P must lie inside or on the cone o friction, since P must balance the total action at O