# An object, previously at rest, slides 3 m down a ramp, with an incline of (pi)/3 , and then slides horizontally on the floor for another 25 m. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?

Feb 23, 2016

0.098

#### Explanation:

By conservation of energy, all the gravitational potential energy converted by friction to heat.

Delta "GPE" = mg (sin(pi/3)*3"m")

"Work done by friction" = Sigma ("Friction" * "distance")

$= m g \cos \left(\frac{\pi}{3}\right) \mu \left(3 \text{m") + mg mu(25"m}\right)$

So, equating them gives

$m g \left(\sin \left(\frac{\pi}{3}\right) \cdot 3 \text{m") = mgcos(pi/3) mu(3"m") + mg mu(25"m}\right)$

sin(pi/3) * 3"m" = mu(cos(pi/3)*(3"m") + (25"m"))

mu = frac{sin(pi/3) * 3"m"}{cos(pi/3)*(3"m") + (25"m")}

 = frac{sqrt3/2*3"m"}{1/2*(3"m") + (25"m")}

$\approx 0.098$