# An object, previously at rest, slides 7 m down a ramp, with an incline of (pi)/3 , and then slides horizontally on the floor for another 6 m. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?

Feb 27, 2016

0.64

#### Explanation:

It can be solved by applying conservation of mechanical energy.

The initial PE of the body = $m g l \sin \theta$,where m= mass of the object,l is the length of the inclined ramp i.e.7m, $\theta = \frac{\pi}{3} ,$angle of inclination of the ramp.and g = acceleration due to gravity.

The frictional force exerted by the ramp ${F}_{r} = \mu m g \cos \theta$, where$\mu =$coefficient of kinetic friction
Again the frictional force exerted by horizontal floor ${F}_{f} = \mu m g$
By law of conservation of energy
Work done against friction = Initial PE

$\therefore {F}_{r} \cdot 7 + {F}_{f} \cdot 6 = m g l \sin \theta$
$\implies \mu \cancel{m g} \cos \left(\frac{\pi}{3}\right) \cdot 7 + \mu \cancel{m g} \cdot 6 = \cancel{m g} \cdot 7 \sin \left(\frac{\pi}{3}\right)$
$\implies \mu \left(3.5 + 6\right) = 7 \cdot \frac{\sqrt{3}}{2}$
$\mu = 7 \cdot \frac{\sqrt{3}}{9.5 \cdot 2} = 0.64$