An object, previously at rest, slides #1 m# down a ramp, with an incline of #pi/3 #, and then slides horizontally on the floor for another #3 m#. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?

This problem is most easily done by conservation of energy.
Two energy forms are involved:

A change in gravitational potential energy: #mgh#

Frictional heating: #muF_NDelta d_1+muF_NDeltad_2#

where #Deltad_1# is the distance the object slides along the ramp, and #Deltad_2# is the distance is slides along the horizontal surface.

Before we can continue, we need to be aware of a couple of complications

We must express #h# (the height the object descends) in terms of #Deltad_1# (its displacement along the ramp).

#h=Deltad_1sin(pi/3)#

Also, we must note that the normal force on an incline is not equal to mg, but to #mgcostheta#, where #theta=pi/3# in this case.

With all that looked after, our equation becomes

#-mgDeltad_1sin(pi/3)+mumgcos(pi/3)Deltad_1+mumgDeltad_2=0#

(The first term is negative because the potential energy decreases.)

Notice that we can divide every term by mg, (including the right side of the equation)

So, inserting 1 m for #Deltad_1# and 3m for #Deltad_2# we get:

#-1(0.866)+(mu)(0.50)1+mu(3)=0#

#-0.866+0.50mu+3mu=0#

#3.5mu=0.866#

#mu=0.247#