Angle of Friction : In situation of limiting friction, the angle between normal reaction R and resultant P of normal reaction R and limiting friction f(s), is called the angle of friction. In figure it is shown by α.
In figure, a block A is shown placed on a rough surface and an external force F is acting on it to move it. As a reaction of F, there will be two forces.
1. Normal reaction R which balances the weight mg of the block.
2. Limiting friction force f(s) = µ(s). R parallel to surface of contact which is equal and opposite to applied force \(\vec F\). Resultant of both these reactions is P which makes an angle a with normal reaction R.
\(\therefore\ \tan\alpha = \frac{PR}{OR} = \frac{f(s)}{R} = \frac{\mu_s(R)}{R}\)
\(\tan\alpha =\mu_s\)
∴ Angle of friction
\(\alpha = \tan^{-1}(\mu_s)\)
Coefficient of Static Friction:
\(\therefore\ \mu_s = \tan\alpha\)
Therefore, the tangent of angle of friction is called coefficient of static friction.
Magnitude of resultant force : This force is known as contact force. The value of contact force shown in figure is,
It magnitude of resultant force
\(P = \sqrt{f_{(s)}^{2} + R^2\because \vec R\bot\overrightarrow{f_{(s)}}}\)
\(=\sqrt{(\mu_sR)^2 + R^2}\)
\(P = R\sqrt{1 + \mu_s^{2}}\)
Friction on an Inclined Plane-Angle of Repose:
The angle between inclined plane and horizontal, when any block placed on the plane remains at rest, is called angle of repose.
In figure, a block of mass M is placed on an inclined plane of angle of inclination f3. The forces acting on the block are shown in the diagram.
Free body diagram of block
Equations of motion,
Mg cos ß = R .......(1)
and Mg sin ß = µs.R .......(2)
On dividing equation (2) by (1), we have
\(\frac{Mg\sin\beta}{Mg\cos\beta} = \frac{\mu sR}{R}\)
or tan ß = µs ......(3)
but tan α = µs ........(4)
where α is angle of friction.
From equations (3) and (4), we have
ß = α
i. e, Angle of repose = Angle of friction
If the angle of inclination of the plane ≤ß, then the block placed on the plane will remain at rest. But when angle of inclination of the plane > ß, then the block will start slipping on the plane in downward direction.
