# An object, previously at rest, slides 7 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?

May 14, 2017

${\mu}_{k} \approx 0.933$

#### Explanation:

Warning: this is potentially a bit of a tricky problem, so I've included a detailed explanation. You may skip to the bottom of the answer to see the derivation if you have a solid understanding of the concepts.

We are given the angle of incline, displacement of the object down the ramp, and displacement of the object between the bottom of the ramp and its stopping point. We can find the coefficient of kinetic friction using the work-energy theorem.

The first step is to draw a force diagram.

where ${f}_{k}$ represents the force of kinetic friction, $n$ represents the normal force, and ${F}_{g}$ represents the force of gravity (both $x$- and $y$-components).

Work-Energy Theorem:

• When the object is at rest at the top of the ramp, it possesses only gravitational potential energy, given by $\textcolor{b l u e}{{U}_{g} = m g h}$.

• If this were a friction-free system, we would expect all of that potential energy to be converted into kinetic energy as the object slides down the ramp and onto the floor, moving off in that direction forever at at a constant speed (provided there are no other outside forces acting upon it). Because this isn't a friction-free system, we will not see conservation of mechanical energy. The reason the object eventually comes to a stop on the floor is due to friction; this is what energy is lost to as the object slides down the ramp and across the floor.

Work is defined as the mechanical transfer of energy.

• When the object comes to a stop on the floor, it no longer has either gravitational potential or kinetic energy, but by the Law of Conservation of Energy, we know that the energy had to go somewhere.
• In this case, if we treat friction as the only outside force energy can be lost to, that means that all of the potential energy stored in the object at the top of the ramp is accounted for in the work done by friction. In other words, the energy lost is equal to the work done by friction. This means that $\textcolor{b l u e}{\Delta U = {W}_{f}}$.

• Force of kinetic friction: $\textcolor{b l u e}{{f}_{k} = {\mu}_{k} n}$

• Work done by a constant force: color(blue)(W=F(Δr)cos(theta))

Note: In the work equation, $\theta$ is the angle between the force and displacement vectors. In this case, the angle where friction is concerned is ${180}^{o}$ (the opposite of the motion but on the same plane/parallel).

• Because $\cos \left({180}^{o}\right) = - 1$, we get negative work, which we might have guessed, given that kinetic friction works opposite the direction of displacement. I will omit this part of the calculation below to avoid confusion where $\cos \left(\theta\right)$ might come up in other equations and simply write any work done by friction as being negative.

Note: I will use Δr for the ramp and Δs for the floor to avoid confusion.

The potential energy stored in the object at the top of the ramp is equal to the total work done by friction. This includes both that done on the ramp and the floor (the point where all of the potential energy has been converted is when the object comes to a stop on the floor).

${U}_{g f} - {U}_{g i} = - {W}_{F}$

The final height is $0$, so this becomes

$- {U}_{g i} = - {W}_{F}$

Let's cancel the negative.

${U}_{g i} = {W}_{F}$

$\implies \textcolor{b l u e}{{U}_{g i} = {W}_{F \text{ ramp")+W_(F" floor}}}$

Using the equation above for gravitational potential energy, we get:

$\textcolor{b l u e}{m g {h}_{i} = {W}_{F \text{ ramp")+W_(F" floor}}}$

Looking at our force diagram above, we can generate sum of forces statements for when the object is on the ramp and the floor.

Ramp:

$\textcolor{b l u e}{\sum {F}_{x} = {\left({F}_{g}\right)}_{x} - {f}_{k} = m {a}_{x}}$

$\textcolor{b l u e}{\sum {F}_{y} = n - {\left({F}_{g}\right)}_{y} = \cancel{m {a}_{y}} = 0}$

Note no acceleration vertically due to dynamic equilibrium.

Therefore, $\textcolor{b l u e}{n = {\left({F}_{g}\right)}_{y}}$.

We can use basic trigonometry to find the parallel and perpendicular components of the force of gravity. Note that the angle between the force of gravity and the vertical is equal to the angle of incline. Therefore:

$\sin \left(\theta\right) = \text{opposite"/"hypotenuse}$

$\implies \sin \left(\theta\right) = {\left({F}_{g}\right)}_{x} / {F}_{g}$

$\implies \textcolor{b l u e}{{\left({F}_{g}\right)}_{x} = {F}_{g} \sin \left(\theta\right)}$

Similarly, we find that $\textcolor{b l u e}{{\left({F}_{g}\right)}_{y} = m g \cos \left(\theta\right)}$, and so our equation for kinetic friction on the ramp becomes $\textcolor{b l u e}{{f}_{k} = {\mu}_{k} m g \cos \left(\theta\right)}$.

Floor:

$\textcolor{\mathrm{da} r k b l u e}{\sum {F}_{x} = - {f}_{k} = m {a}_{x}}$

$\textcolor{\mathrm{da} r k b l u e}{\sum {F}_{y} = n - {F}_{g} = m {a}_{y} = 0}$

Therefore, $n = m g$, and so our equation for kinetic friction on the floor becomes $\textcolor{\mathrm{da} r k b l u e}{{f}_{k} = {\mu}_{k} m g}$.

For our potential energy, we will need to find $h$. This is the height of the ramp. We can do this using basic trigonometry:

$\sin \left(\theta\right) = \frac{h}{\Delta r}$

$\implies h = \Delta r \sin \left(\theta\right)$.

We now have:

color(blue)(m)color(darkblue)(g)Deltarsin(theta)=mu_kcolor(blue)(m)color(darkblue)(g)cos(theta)Δr+mu_kcolor(blue)(m)color(darkblue)(g)Δs

Both the mass, $m$, and $g$ are present in each term on both sides and therefore cancel

=>cancelcolor(blue)(m)cancelcolor(darkblue)(g)Deltarsin(theta)=mu_kcancelcolor(blue)(m)cancelcolor(darkblue)(g)cos(theta)Δr+mu_kcancelcolor(blue)(m)cancelcolor(darkblue)(g)Δs

=>Deltarsin(theta)=mu_kcos(theta)Δr+mu_kΔs

We can rearrange to solve for ${\mu}_{k}$:

Deltarsin(theta)=mu_k(cos(theta)Δr+Δs)

=>color(crimson)(mu_k=(Deltarsin(theta))/(cos(theta)Δr+Δs))

We are given:

• $\mapsto \Delta r = 7 \text{m}$
• $\mapsto \Delta s = 3 \text{m}$
• $\mapsto \theta = \frac{\pi}{3}$

${\mu}_{k} = \frac{7 \sin \left(\frac{\pi}{3}\right)}{7 \cos \left(\frac{\pi}{3}\right) + 3}$

$= \frac{\frac{7 \sqrt{3}}{2}}{\frac{13}{2}}$

$= \frac{7 \sqrt{3}}{13}$

$\approx 0.933$

$\therefore \textcolor{b l u e}{{\mu}_{k} \approx 0.933}$

Hope that helps!