An object, previously at rest, slides #3 m# down a ramp, with an incline of #pi/6 #, 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?

Conservation of energy is best for doing this problem.

Here, there are two forms of energy that must be considered - potential energy and frictional heating. The fact that the object starts at rest, and ends the problem at rest means we can ignore kinetic energy changes during the motion!

Conservation of energy will involve three terms

#Delta U + Delta E_(ramp) + Delta E_(floor) = 0#

Note I have two terms for frictional heating. This is necessary because the normal force changes when the object goes from ramp to floor, and so, the force of friction also changes, (although the coefficient of friction remains the same).

Let #d_1# be the distance of travel on the ramp, and #d_2# the distance along the floor. Remember also, that the normal force while the object is on the incline is #mgcostheta# and not simply #mg#

Then, the equation is

#-mgd_1sin(pi/6) + mumgcos(pi/6) d_1 + mu mg d_2=0#

where the height the object descends on the ramp is #h=d_1sin(pi/6)#

Getting the numbers in place, after first canelling "mg" from every term

#-(3)(.5) + mu(0.866)(3) + mu (8)=0#

(-1.5) + 10.6mu=0#

#mu=(1.5)/10.6 = 0.142#