# An object, previously at rest, slides 6 m down a ramp, with an incline of (3pi)/8 , and then slides horizontally on the floor for another 8 m. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?

Jun 6, 2016

$\mu = 0.409301$

#### Explanation:

Here ${d}_{1} = 6$[m], ${d}_{2} = 8$[m] and $\alpha = 3 \frac{\pi}{8}$ and $\mu$ is the kinetic friction coefficient.

The initial potential energy is

$m g {d}_{1} \sin \left(\alpha\right)$

The total dissiped work is given by

${\tau}_{1} + {\tau}_{2}$

where ${\tau}_{1} = m g \sin \left(\alpha\right) \mu {d}_{1}$ and ${\tau}_{2} = m g \mu {d}_{2}$

so equating

$m g {d}_{1} \sin \left(\alpha\right) = m g \sin \left(\alpha\right) \mu {d}_{1} + m g \mu {d}_{2}$

Solving for $\mu$ gives

$\mu = \frac{{d}_{1} \sin \left(\alpha\right)}{{d}_{1} \sin \left(\alpha\right) + {d}_{2}} = 0.409301$