An object, previously at rest, slides 3 m down a ramp, with an incline of (3pi)/8 , and then slides horizontally on the floor for another 5 m. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?

Mar 6, 2016

${u}_{k} \cong 0 , 45$

Explanation: $\text{Total Energy at The Point of A: } {E}_{p} = m \cdot g \cdot 3 \cdot \sin \alpha$
$\text{The Work doing by The Friction Force along AC:}$
${W}_{A C} = 3. {u}_{k} \cdot m \cdot g \cdot \cos \alpha$
$\text{Total Energy at The Point of C:}$
${W}_{C} = {E}_{p} - {W}_{A C}$
${W}_{C} = m \cdot g \cdot 3 \cdot \sin \alpha - 3. {u}_{k} \cdot m \cdot g \cdot \cos \alpha$
$\text{The Work doing by Friction Force along CD:}$
${W}_{C D} = {u}_{k} \cdot m \cdot g \cdot 5$
$3 \cancel{m . g .} \sin \alpha - 3. {u}_{k} . \cancel{m . g .} \cos \alpha = 5. {u}_{k} . \cancel{m . g}$
$3. \sin \alpha - 3. {u}_{k} . \cos \alpha = 5. {u}_{k}$
$2 , 772 - {u}_{k} .1 , 148 = 5. {u}_{k}$
$2 , 772 = 5. {u}_{k} + 1 , 148. {u}_{k}$
$2 , 772 = 6 , 148. {u}_{k}$
${u}_{k} \cong 0 , 45$