An object, previously at rest, slides $12 m$ $12 m$ down a ramp, with an incline of $\frac{π}{8}$ $pi/8$ , and then slides horizontally on the floor for another $3 m$ $3 m$ . If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?

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Answer 1 · 2017-01-22T02:51:54

The coefficient of kinetic friction will be 0.326

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

A change in gravitational potential energy: $m g h$ $mgh$

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:

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

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

$14.09mu=4.60$

$mu=0.326$

An object, previously at rest, slides 12 m12m12 m down a ramp, with an incline of pi/8 π8pi/8 , and then slides horizontally on the floor for another 3 m3m3 m. If the ramp and floor are made of the same material, what is the material's kinetic friction coefficient?