# 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?

Jan 22, 2017

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: $m g h$

Frictional heating: $\mu {F}_{N} \Delta {d}_{1} + \mu {F}_{N} \Delta {d}_{2}$

where $\Delta {d}_{1}$ is the distance the object slides along the ramp, and $\Delta {d}_{2}$ 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 $\Delta {d}_{1}$ (its displacement along the ramp).

$h = \Delta {d}_{1} \sin \left(\frac{\pi}{8}\right)$

Also, we must note that the normal force on an incline is not equal to mg, but to $m g \cos \theta$, where $\theta = \frac{\pi}{8}$ in this case.

With all that looked after, our equation becomes

$- m g \Delta {d}_{1} \sin \left(\frac{\pi}{8}\right) + \mu m g \cos \left(\frac{\pi}{8}\right) \Delta {d}_{1} + \mu m g \Delta {d}_{2} = 0$

(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 $\Delta {d}_{1}$ and 3m for $\Delta {d}_{2}$ we get:

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

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

$14.09 \mu = 4.60$

$\mu = 0.326$