# An object, previously at rest, slides 9 m down a ramp, with an incline of (pi)/6 , and then slides horizontally on the floor for another 24 m. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?

Feb 20, 2016

$k \cong 0 , 142$

#### Explanation:

$\frac{\pi}{6} = {30}^{o}$
${E}_{p} = m \cdot g \cdot h \text{ Potential Energy of Object}$
${W}_{1} = k \cdot m \cdot g \cdot \cos 30 \cdot 9$
$\text{Lost energy because friction on inclined plane}$
${E}_{p} - {W}_{1} \text{: energy when object on ground}$
${E}_{p} _ {W}_{1} = m \cdot g \cdot h - k \cdot m \cdot g \cdot \cos {30}^{o} \cdot 9$
${W}_{2} = k \cdot m \cdot g \cdot 24 \text{ lost energy on the floor}$
$k \cdot \cancel{m \cdot g} \cdot 24 = \cancel{m \cdot g} \cdot h - k \cdot \cancel{m \cdot g} \cdot \cos {30}^{o} \cdot 9$
$24 \cdot k = h - 9 \cdot k \cdot \cos {30}^{o}$
"using "cos 30^o=0,866 ;h=9*sin30=4,5 m
$24 \cdot k = 4 , 5 - 9 \cdot k \cdot 0 , 866$
$24 \cdot k + 7 , 794 \cdot k = 4 , 5$
$31 , 794 \cdot k = 4 , 5$
$k = \frac{4 , 5}{31 , 794}$
$k \cong 0 , 142$