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Introduction to Friction

We will take a look at equilibrium and friction

1: The friction box

The box with the mass mm kg is exposed to an external force PP according to the left figure. Doing a free body diagram and applying Newton's third law reveals a force FF acting opposite to PP along the contact surface. The magnitude of this friction force depends on the type of surface and the normal force NN. The relation between the normal force and friction force can be modelled using the Coulombs law:

FμsN\boxed{F\le \mu_s N}

where μs\mu_s is known as the static coefficient of friction. Coulomb developed a friction theory in the 1700s for dry friction. This model assumes a linear relationship between FF and NN and even if we can empirically show that this does not generally hold for an arbitrarily large NN, Coulombs law is taken to be "good enough" for most situations, but should be used with care.

2: Relation between the external force PP and friction force FF .

As the force PP exceeds the largest possible frictional force FmaxF_{\max } the box will start to slide and get an acceleration, thus the model transitions to what is known as kinetic friction and the opposing force is given by Fk=μkNF_k =\mu_k N, where μk\mu_k is the kinetic coefficient of friction. This is where statics ends and dynamics continues.

The coefficients of friction can be calculated from empirical observations using measurements of forces, positions or angles.

⚠ Note
Any list containing the coefficient will be approximate, good accuracy is application dependent and must be measured.

Here is a list for (a rough) reference

Materialμs\mu_sμk\mu_k
Steel on steel0.1 - 0.30.03 - 0.3
Steel on wood0.5 - 0.70.2 - 0.5
Leather on metal0.3 - 0.60.2 - 0.3
Break pad on cast iron0.40.3
Hemp rope on steel0.30.2
Teflon on Teflon0.050.05
Rubber on metal0.4 - 0.50.3 - 0.4
Rubber on asphalt0.7 - 1.00.5 - 0.8
Rubber on ice0.10.05

Note that Coulombs law is a very simple model in the world of tribology. Many applications need more sophisticated models of friction.

The static coefficient of friction can also be expressed as the static angle of friction, αs\alpha_s, defined by

tan(αs)=FN=μS\tan \left(\alpha_s \right)=\frac{F}{N}=\mu_S

This corresponds to the angle of a sloping plane and box that is just about to begin sliding, see Example 1.

⚠ Note

For any problem involving rough surfaces where frictional forces need to be taken into account one needs to determine if the bodies are at rest or sliding. In general there are several types of states in contact points on a body. If we have a friction coefficient μs\mu_s and have the situation

F<μsNF<\mu_s N then static equilibrium is possible since the surfaces can generate the needed friction force that is required for static friction.

F=μsNF=\mu_s N then static equilibrium is still possible. This is a limit, often we write Fmax=μsNF_{\max } =\mu_s N, to show that this is the maximum friction force, this limit is also known as fully established friction.

F>μsNF>\mu_s N then static equilibrium is impossible. The surfaces cannot produce the frictional force that is required to keep static equilibrium. The body begins to slide, accelerates and we get dynamic equilibrium F=μkNF=\mu_k N, at this point the problem is transitioned into a dynamic problem and we have ΣF=m  a\Sigma F=m\;a.

The basic rule for analysis with friction is that the frictional force-vector F\mathit{\mathbf{F} } is set to be in the direction that opposes potential movement.

Essentially there two types of problem statements

  1. All external forces except friction forces are known and we need to determine if the friction force can balance the external forces. This means that we need to check if FμsNF\le \mu_s N in the contact surface, otherwise we have sliding and the problem is a dynamic problem with F=μkNF=\mu_k N.

  2. Some external forces are unknown and we want to know how much more we can load the body until sliding occurs. This means that we add Fi=μsNiF_i =\mu_s N_i to all rough contact points and include these friction equations in the equilibrium equations. Solving for the unknown forces to answer the problem statement.

We show some typical examples involving friction.

Example 1

Determine the maximum angle θ\theta at which the box with mass mm starts sliding down.

Solution:

Free body diagram, where we chose the coordinate system such that most forces are aligned with the coordinate system, this minimizes expressing forces in terms of the angle.

We have equilibrium equations and an additional friction equation Fmax=μsNF_{\max } =\mu_s N. Then we just solve for as many variables as we have equations.

clear
syms F theta mg mu N

ekv = [mg*sin(theta) - F == 0
       -mg*cos(theta) + N == 0
       F == mu*N]
(mgsin(θ)F=0Nmgcos(θ)=0F=Nμ) \left(\begin{array}{c} \textrm{mg}\,\sin \left(\theta \right)-F=0\\ N-\textrm{mg}\,\cos \left(\theta \right)=0\\ F=N\,\mu \end{array}\right)
[mu,F,N] = solve(ekv, [mu,F,N])
mu =
sin(θ)cos(θ) \frac{\sin \left(\theta \right)}{\cos \left(\theta \right)}
F =
mgsin(θ) \textrm{mg}\,\sin \left(\theta \right)
N =
mgcos(θ) \textrm{mg}\,\cos \left(\theta \right)

We have θ=tan1(μs)\theta =\tan^{-1} \left(\mu_s \right), also known as the angle of friction.

Example 2

Two masses are rigged according to the figure. In what interval does m2m_2 need to be in to prevent sliding of m1m_1. Let μs=0.3\mu_s =0\ldotp 3 and α=20°\alpha =20\degree.

Solution:

We need to examine two cases, depending on if the box is just about to start gliding up or down. These cases result in two FBDs where the direction of the friction force is opposing the direction of movement. We chose the coordinate system to be aligned to the sliding surface, since just the weight force is acting vertically. Let Fmax=μsNF_{\max } =\mu_s N.

Case 1: First the edge case where m2m_2 if big enough to cause sliding just to occur upwards the surface. Thus FF is directed against this potential movement, i.e., downwards.

clear
syms m1 m2 F g mu N alpha

FG = [-m1*g*sind(alpha); -m1*g*cosd(alpha)];
FT = [m2*g; 0];
Ff = [-F;0];
FN = [0; N];

ekv = [FG+FT+Ff+FN==0
       F==mu*N]
ekv =
(gm2Fgm1sin(πα180)=0Ngm1cos(πα180)=0F=Nμ) \left(\begin{array}{c} g\,m_2 -F-g\,m_1 \,\sin \left(\frac{\pi \,\alpha }{180}\right) = 0\\ N-g\,m_1 \,\cos \left(\frac{\pi \,\alpha }{180}\right) = 0\\ F=N\,\mu \end{array}\right)
[F,N,m2] = solve(ekv,[F,N,m2])
F =
gm1μcos(πα180) g\,m_1 \,\mu \,\cos \left(\frac{\pi \,\alpha }{180}\right)
N =
gm1cos(πα180) g\,m_1 \,\cos \left(\frac{\pi \,\alpha }{180}\right)
m2 =
m1sin(πα180)+m1μcos(πα180) m_1 \,\sin \left(\frac{\pi \,\alpha }{180}\right)+m_1 \,\mu \,\cos \left(\frac{\pi \,\alpha }{180}\right)
alpha = 20; mu = 0.3;
vpa(subs(m2),3)
0.624m1 0.624\,m_1

Case 2: Then the case where m2m_2 is small enough for sliding just to occur downwards the surface. Thus FF is directed upwards.

syms alpha mu F N m2
Ff = [F;0];
ekv = [FG+FT+Ff+FN==0
       F==mu*N]
ekv =
(F+gm2gm1sin(πα180)=0Ngm1cos(πα180)=0F=Nμ) \left(\begin{array}{c} F+g\,m_2 -g\,m_1 \,\sin \left(\frac{\pi \,\alpha }{180}\right) = 0\\ N-g\,m_1 \,\cos \left(\frac{\pi \,\alpha }{180}\right) = 0\\ F=N\,\mu \end{array}\right)
[F,N,m2] = solve(ekv,[F,N,m2])
F =
gm1μcos(πα180) g\,m_1 \,\mu \,\cos \left(\frac{\pi \,\alpha }{180}\right)
N =
gm1cos(πα180) g\,m_1 \,\cos \left(\frac{\pi \,\alpha }{180}\right)
m2 =
m1sin(πα180)m1μcos(πα180) m_1 \,\sin \left(\frac{\pi \,\alpha }{180}\right)-m_1 \,\mu \,\cos \left(\frac{\pi \,\alpha }{180}\right)
alpha = 0; mu = 0.3;
vpa(subs(m2),3)
ans =
0.3m1 -0.3\,m_1
syms alpha
m2 = simplify(subs(m2)/m1)
m2 =
sin(πα180)3cos(πα180)10 \sin \left(\frac{\pi \,\alpha }{180}\right)-\frac{3\,\cos \left(\frac{\pi \,\alpha }{180}\right)}{10}
figure
fplot(m2,[0,90]);
hold on;
fplot(0,[0,90])
xlabel('\theta'); ylabel('m_2')
title('m_2 causing sliding downwards')
vpasolve(m2==0,20)
16.699244233993621840368847179984 16.699244233993621840368847179984

Results

The mass m2m_2 needs to be at least 0.06m10\ldotp 06m_1 to prevent sliding downwards and less than 0.62m20\ldotp 62m_2 to prevent sliding upwards.

Feasibility analysis: At close to 90°90\degree the frictional force is so small that we get m1=m2m_1 =m_2. At zero degrees the model does not work since we would get negative masses. Case 2 does not exists below slopes of ca 17°17\degree, i.e., the box cannot slide downwards if the slope is less than 17°17\degree.

Example 3

Determine the least possible μs(θ)\mu_s \left(\theta \right) at AA such that the homogenous rod with a mass of mm does not slide. There is no friction at OO. Disregard the radius of the roller at O. Draw a graph of μs(θ)\mu_s \left(\theta \right) for valid angles.

Solution: We'll start with a coordinate system according to the FBD below. Create a FBD of the rod, inserting positions vectors as needed as well as force vectors.

Define the angle using trigonometry.

tan  θ=xL/2=sincos\tan \;\theta =\dfrac{x}{L/2}=\dfrac{\sin }{\cos }

We start by defining the position vectors.

clear
syms L positive 
syms theta positive
% assume(theta<=90)
normalize = @(v)v/norm(v);

OA = [-L/2*sind(theta)/cosd(theta), -L/2, 0];
eAO = -normalize(OA); 
OG = OA + eAO*L/2;

The we define the force vectors acting on the rod. FO{\mathit{\mathbf{F} } }_O is acting orthogonal to the rod. We can get its direction by using the unit circle and rotating clock-wise (adding negative angles) until we reach the desired direction. We can double check the direction by inserting values for theta. If unsure, draw the direction!

e(theta) = [cosd(theta),sind(theta),0];
syms F_O F_G F_Ax F_Ay mg
FO = F_O*e(-90-theta-90);
FA = [F_Ax, F_Ay,0];
FG = mg*[0,-1,0];

We add the equations.

ekv = [FO+FA+FG, cross(OG,FG)+cross(OA,FA)].'
(FAx+FOcos(σ3)FAymgFOsin(σ3)000mg(Lσ12σ2L2σ14σ2L24+L2σ124σ22)+FAxL2FAyLσ12σ2)where    σ1=sin(πθ180)    σ2=cos(πθ180)    σ3=π(θ+180)180 \begin{array}{l} \left(\begin{array}{c} F_{\textrm{Ax} } +F_O \,\cos \left(\sigma_3 \right)\\ F_{\textrm{Ay} } -\textrm{mg}-F_O \,\sin \left(\sigma_3 \right)\\ 0\\ 0\\ 0\\ \textrm{mg}\,{\left(\frac{L\,\sigma_1 }{2\,\sigma_2 }-\frac{L^2 \,\sigma_1 }{4\,\sigma_2 \,\sqrt{\frac{L^2 }{4}+\frac{L^2 \,{\left|\sigma_1 \right|}^2 }{4\,{\left|\sigma_2 \right|}^2 } } }\right)}+\frac{F_{\textrm{Ax} } \,L}{2}-\frac{F_{\textrm{Ay} } \,L\,\sigma_1 }{2\,\sigma_2 } \end{array}\right)\\ \mathrm{}\\ \textrm{where}\\ \mathrm{}\\ \;\;\sigma_1 =\sin \left(\frac{\pi \,\theta }{180}\right)\\ \mathrm{}\\ \;\;\sigma_2 =\cos \left(\frac{\pi \,\theta }{180}\right)\\ \mathrm{}\\ \;\;\sigma_3 =\frac{\pi \,{\left(\theta +180\right)} }{180} \end{array}
⚠ Note
Note that if the expressions are not equations, solve() treats them automatically as if the expressions are equal to zero!
[F_O, F_Ax, F_Ay] = solve(ekv,[F_O, F_Ax, F_Ay]);

simplify(F_O)
mgsin(πθ180)cos(πθ180) \textrm{mg}\,\sin \left(\frac{\pi \,\theta }{180}\right)\,\left|\cos \left(\frac{\pi \,\theta }{180}\right)\right|
simplify(F_Ax)
mgsin(πθ90)cos(πθ180)2 \frac{\textrm{mg}\,\sin \left(\frac{\pi \,\theta }{90}\right)\,\left|\cos \left(\frac{\pi \,\theta }{180}\right)\right|}{2}
simplify(F_Ay)
mg(cos(πθ90)cos(πθ180)cos(πθ180)+2)2 \frac{\textrm{mg}\,{\left(\cos \left(\frac{\pi \,\theta }{90}\right)\,\left|\cos \left(\frac{\pi \,\theta }{180}\right)\right|-\left|\cos \left(\frac{\pi \,\theta }{180}\right)\right|+2\right)} }{2}

Since FAx=μFAyF_{\textrm{Ax} } ={\mu F}_{\textrm{Ay} } we get

mu = simplify(F_Ax/F_Ay)
sin(πθ90)cos(πθ180)cos(πθ90)cos(πθ180)cos(πθ180)+2 \frac{\sin \left(\frac{\pi \,\theta }{90}\right)\,\left|\cos \left(\frac{\pi \,\theta }{180}\right)\right|}{\cos \left(\frac{\pi \,\theta }{90}\right)\,\left|\cos \left(\frac{\pi \,\theta }{180}\right)\right|-\left|\cos \left(\frac{\pi \,\theta }{180}\right)\right|+2}

Note that θ\theta must be below 90 degrees since it is limited by the length of the rod, or the A's distance to the wall. We can find the maximum value graphically, we set the length to one.

L_theta = subs(norm(OA),L,1);
figure
fplot(L_theta,[0,90]); hold on
fplot(1);
axis([0,90,0,1.1])
title('L(\theta)'); xlabel('\theta'); ylabel('L')
vpasolve(subs(norm(OA)==L,L,1),50)
60.0 60.0

You can try setting the length to any other value. Reflect on why the angle is the independent of the length.

Finally the graph on the friction coefficient as a function on the angle. μs(θ)\mu_s \left(\theta \right) for all angles θ[0,60]\theta \in \left\lbrack 0,{60}^{\circ } \right\rbrack

figure
fplot(mu,[0,60])
title('\mu_s(\theta)'); xlabel('\theta'); ylabel('\mu_s')
theta_max = vpasolve(diff(mu)==0,40)
41.763684779627981055699678449275 41.763684779627981055699678449275
mu_max = vpa(subs(mu,theta,theta_max),4)
0.5538 0.5538

Example 4 - Clutch

Find the relation between the applied axial force PP and the moment MM for the clutch. Assume a static case and that the friction coefficient μs=μ\mu_s =\mu is constant across the surface.

Solution: Consider the right-hand drawing where are small piece dθd\theta of the whole ring at rr is drawn. The contact pressure pp is constant and dP\textrm{dP} is a small contribution of PP i.e., the small contribution of the normal force on the ring at rr. Now we shall take a dive into the painful details of mathematical modelling, which strengthens our skills, we define the properties of an arbitrary ring at the radius rr.

dA=2π  r  dr\textrm{dA}=2\pi \;r\;\textrm{dr} The area of the ring.

dP=p  dA\textrm{dP}=p\;\textrm{dA} Constant pressure.

dF=μ  dP\textrm{dF}=\mu \;\textrm{dP} Fully developed friction, dF  \textrm{dF}\;, acting in the tangential direction of the arrow.

dM=r  dF\textrm{dM}=r\;\textrm{dF} The contribution of the friction force to the moment, rdFr\bot \textrm{dF}

clear
syms da r dr dP p dA dF mu dM dF 
[dP,dM,dF,dA]= solve([dA==2*pi*r*dr ...
                      dP==p*dA ...
                      dF==mu*dP ...
                      dM==r*dF], [dP,dM,dF,dA])
dP=2πdrpr dP = 2\,\pi \,\textrm{dr}\,p\,r
dM=2πdrμpr2 dM = 2\,\pi \,\textrm{dr}\,\mu \,p\,r^2
dF=2πdrμpr dF = 2\,\pi \,\textrm{dr}\,\mu \,p\,r
dA=2πdrr dA = 2\,\pi \,\textrm{dr}\,r

Now we can integrate, i.e., add all small bits together, where we have P,M  &  RP,M\;\&\;R. Solve for the needed relations from the resulting system of equations. We'll get the surface pressure pp as a bonus.

P=0PdP=0R2πpr  drP=\int_0^P \textrm{dP}=\int_0^R 2\pi \textrm{pr}\;\textrm{dr} M=0MdM=0R2π  μ  p  r2  drM=\int_0^M \textrm{dM}=\int_0^R 2\pi \;\mu \;p\;r^2 \;\textrm{dr}
syms P R M
ekv = [P == int(dP/dr, r,0,R)
       M == int(dM/dr, r,0,R)]
(P=πR2pM=2πR3μp3) \left(\begin{array}{c} P=\pi \,R^2 \,p\\ M=\frac{2\,\pi \,R^3 \,\mu \,p}{3} \end{array}\right)
[M,p] = solve(ekv,[M,p])
M=2PRμ3 M = \frac{2\,P\,R\,\mu }{3}
p=PR2π p = \frac{P}{R^2 \,\pi }

A quick dimension analysis verifies the model and that pressure relation makes sense as well.

Example 5 - Flexible belts

Ropes and belts which are wrapped around cylinders are a very common application for static friction in mechanics. We shall look closer at the problem.

A rope is wrapped some angle β[0,]\beta \in \left\lbrack 0,\infty \right\rbrack radians around a cylinder. We cut a small piece dθ  d\theta \; at an arbitrary angle θ[0,]\theta \in \left\lbrack 0,\infty \right\rbrack and create a FBD according to the figure. We are interested in the maximum force and examine the limit just before sliding. The force equilibrium is divided into a tangential (t)\left(t\right) and radial (n)\left(n\right) direction

t:  T  cos(dθ2)+μ  dN(T+dT)cos(dθ2)=0t:\;T\;\cos \left(\dfrac{d\theta }{2}\right)+\mu \;\textrm{dN}-\left(T+\textrm{dT}\right)\cos \left(\dfrac{d\theta }{2}\right)=0 n:dN(T+dT)sin(dθ2)T  sin(dθ2)=0n:\textrm{dN}-\left(T+\textrm{dT}\right)\sin \left(\dfrac{d\theta }{2}\right)-T\;\sin \left(\dfrac{d\theta }{2}\right)=0

Since we are dealing with very small angles, we can utilize the small angle approximation, i.e., dθ<<1d\theta <<1.

dθ<<1{cos(dθ)=lim  dθ0      1  sind(dθ)=lim  dθ0  dθ  \overset{d\theta <<1}{\Rightarrow} \left\lbrace \begin{array}{ll} \cos \left(d\theta \right)\overset{\lim \;d\theta \to 0}{=} \;\;\;1 & \;\\ \textrm{sind}\left(d\theta \right)\overset{\lim \;d\theta \to 0}{=} \;d\theta & \; \end{array}\right.

Thus our expression simplifies to

t:  T  +μ  dN(T+dT)=0t:\;T\;+\mu \;\textrm{dN}-\left(T+\textrm{dT}\right)=0 n:dN(T+dT)dθ2T  dθ2=0n:\textrm{dN}-\left(T+\textrm{dT}\right)\dfrac{d\theta }{2}-T\;\dfrac{d\theta }{2}=0

Since both dT\textrm{dT} and dθd\theta are very small, their product is even smaller and can be neglected, i.e., dTdθ<<dT,dθ\textrm{dTd}\theta <<\textrm{dT},d\theta

Thus we end up with

{μ  dN=dT  dN=T  dθ  \left\lbrace \begin{array}{ll} \mu \;\textrm{dN}=\textrm{dT} & \;\\ \textrm{dN}=T\;d\theta & \; \end{array}\right.

Eliminating dN\textrm{dN} and moving thing around we can formulate a simple ordinary differential equation

dT2dθ=μT2\dfrac{ {\textrm{dT} }_2 }{d\theta }=\mu T_2

with the boundary condition T2(0)=T1T_2 \left(0\right)=T_1

For which we can easily get the solution using Matlabs symbolic differential equation solver dsolve:

clear
syms T1 T2(theta) theta mu
DE = diff(T2(theta), theta) == mu*T2(theta)
θ  T2(θ)=μT2(θ) \frac{\partial }{\partial \theta }\;T_2 \left(\theta \right)=\mu \,T_2 \left(\theta \right)
BV = T2(0)==T1
T2(0)=T1 T_2 \left(0\right)=T_1
T2(theta) = dsolve(DE,BV)
T2=T1eμθ \boxed{ T_2 = T_1 \,{\mathrm{e} }^{\mu \theta } }

As an alternative solution, we can collect TT and θ\theta on each side: dTT=μdθ\frac{\textrm{dT} }{T}=\mu d\theta, and integrate:

T1T11TdT=0θμdθ\int_{T_1 }^{T_1 } \dfrac{1}{T}\textrm{dT}=\int_0^{\theta } \mu d\theta ln  T2T1=μθ\ln \;\dfrac{T_2 }{T_1 }=\mu \theta \Rightarrow T2=T1eμθ \boxed{ T_2 =T_1 e^{\mu \theta } }

Ok, so with, e.g., μs=0.25\mu_s =0\ldotp 25 (hamp against steel according to some table) and wrapped five times we get

mu = 0.25;
T2 = T1*exp(5*2*pi*mu);
vpa(T2,6)
ans =
2575.97T1 2575.97\,T_1

The force T2T_2 needs to be over 2500 times larger to balance T1T_1! Now we understand why this simple mechanism works and has worked for ages!

Screws

A trapezoidal screw is used for transforming a rotational movement to a linear movement. On the other hand, a common screw profile is used for maximizing axial loads. Regardless of the type of profile, the analysis is the same.

Typical measures for dimensioning are mean diameter dmd_m and pitch ll.

The pitch can be defined as the height increase of one rotation. Unwrapping a screw leads to the situation in the figure below.

Thus, we can express the pitch-angle θ\theta as a functin of the pitch

θ=tan1(lπdm)\theta =\tan^{-1} \left(\dfrac{l}{\pi d_m }\right)

Compared to simple bolts and screws, a trapezoidal screw has higher pitch to more efficiently transfer rotation to linear movement with fewer losses. If even more precision is needed (trapezoidal screws tend to have some backlash) then ball-screws can be used, for even less frictional losses. The tradeoff however, except the (much) higher cost of manufacturing is that ball screws can easily un-wind when an axial load is applied, thus a motor torque is needed to hold a load.

Typical pitch valued for trapezoidal screws are given tables available at various suppliers, e.g., mekanex.

Mechanical analysis

Typical questions for an analysis are: What moment MM is needed to transfer an axial force WW according to the figure above ovan given that the geometrical dimensions are dm,ld_m ,l and the coefficient of friction is μs\mu_s.

We will start the analysis by looking a small cut of the screw according to the figure below.

This situation describes a screw working against a load.

Sometimes the angle of friction, α\alpha is used, in this case the resultant, R=F+NR=F+N is used in the FBD.

We will not use RR in the following equilibrium equations.

clear
syms M r F theta N W mu

ekv = [M/r - F*cos(theta) - N*sin(theta) == 0
       N*cos(theta) - W - F*sin(theta) == 0
       F == mu*N]
(MrFcos(θ)Nsin(θ)=0Ncos(θ)WFsin(θ)=0F=Nμ) \left(\begin{array}{c} \frac{M}{r}-F\,\cos \left(\theta \right)-N\,\sin \left(\theta \right)=0\\ N\,\cos \left(\theta \right)-W-F\,\sin \left(\theta \right)=0\\ F=N\,\mu \end{array}\right)
[M, N, F] = solve(ekv, [M, N, F] )
M=r(Wsin(θ)+Wμcos(θ))cos(θ)μsin(θ) M = \frac{r\,{\left(W\,\sin \left(\theta \right)+W\,\mu \,\cos \left(\theta \right)\right)} }{\cos \left(\theta \right)-\mu \,\sin \left(\theta \right)}
N=Wcos(θ)μsin(θ) N = \frac{W}{\cos \left(\theta \right)-\mu \,\sin \left(\theta \right)}
F=Wμcos(θ)μsin(θ) F = \frac{W\,\mu }{\cos \left(\theta \right)-\mu \,\sin \left(\theta \right)}