An object, previously at rest, slides #12 m# down a ramp, with an incline of #pi/8 #, 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?

1 Answer
Jan 22, 2017

Answer:

The coefficient of kinetic friction will be 0.326

Explanation:

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/8)#

Also, we must note that the normal force on an incline is not equal to mg, but to #mgcostheta#, where #theta=pi/8# in this case.

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With all that looked after, our equation becomes

#-mgDeltad_1sin(pi/8)+mumgcos(pi/8)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 12 m for #Deltad_1# and 3m for #Deltad_2# we get:

#-12(0.383)+(mu)(0.924)12+mu(3)=0#

#-4.60+11.09mu+3mu=0#

#14.09mu=4.60#

#mu=0.326#