# Co – Efficient of Friction between a Body and Wedge

## Co – Efficient of Friction between a Body and Wedge

If we slide (or) try to slide a body over a surface, the motion is resisted by a bonding between the body and the surface. This resistance is represented by a single force and is called friction force. The force of friction is parallel to the surface and opposite to the direction of intended motion.

A body slides on a smooth wedge of angle $$\theta$$ and its time of descent is t. If the same wedge made rough the time taken by it to come down becomes n times more (i.e., nt)

The length of path in both the cases are same. For smooth wedge, $$S=ut+\frac{1}{2}a{{t}^{2}}$$,

$$S=\frac{1}{2}\left( g\,\sin \theta \right){{t}^{2}}…….\left( i\right)$$ $$\left[ As\,u=0\,and\,a=g\left( \sin \theta \right) \right]$$,

For rough wedge, $$S=ut+\frac{1}{2}a{{t}^{2}}$$,

$$\frac{1}{2}\left( g\,\sin \theta \right){{t}^{2}}=\frac{1}{2}g\left( \sin\theta -\mu \cos \theta \right){{\left( nt \right)}^{2}}…..(ii)$$ $$\left[As\,u=0\,and\,a=g\left( \sin \theta -\mu \cos \theta \right) \right]$$,

From equations (i) and (ii), we get:

$$\frac{1}{2}\left( g\sin \theta \right){{t}^{2}}=\frac{1}{2}g\left( \sin \theta -\mu \cos \theta \right){{\left( nt \right)}^{2}}$$,

$$\Rightarrow \sin \theta =\left( \sin\theta -\mu \cos \theta \right){{\left(n \right)}^{2}}$$

$$\Rightarrow \mu =\tan \theta \left[ 1-\frac{1}{{{n}^{2}}} \right]$$,

Therefore, the co – efficient the friction, $$\left( \mu \right)=\tan \theta \left[ 1-\frac{1}{{{n}^{2}}} \right]$$,

Friction always opposes the relative motion between any two bodies in contact it also the cause of motion.