# An object, previously at rest, slides 8 m down a ramp, with an incline of pi/4 , and then slides horizontally on the floor for another 12 m. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?

Apr 26, 2016

$\mu = 0 , 28$

#### Explanation:

$\text{Total Energy for System at the Point of A is:}$

${E}_{A} = m \cdot g \cdot h$

$\text{where h=} 8 \cdot \sin \left(\frac{\pi}{4}\right) = 8 \cdot 0 , 707 = 5 , 66 m$

$\textcolor{g r e e n}{{E}_{A} = m \cdot g \cdot 5 , 66 J}$

$\text{This Energy is spent by friction force along the path A,B,C}$

W_"AB"=color(red)(F_f*bar(AB) " changed heat or any other energy" )

W_"AB"=color(red)(mu*m*g*cos(pi/4)*8

W_"BC"=color(blue)(F_f*bar(BC)" changed heat or any other energy")

${W}_{\text{BC}} = \textcolor{b l u e}{\mu \cdot m \cdot g \cdot 12}$

$\textcolor{g r e e n}{\cancel{m \cdot g} \cdot 5 , 66 = \textcolor{red}{\mu \cdot \cancel{m \cdot g} \cdot 0 , 707 \cdot 8}} + \textcolor{b l u e}{\mu \cdot \cancel{m \cdot g} \cdot 12}$

$5 , 66 = 8 , 49 \mu + 12 \mu$

$20 , 49 \mu = 5 , 66$

$\mu = \frac{5 , 66}{20 , 49}$

$\mu = 0 , 28$