If we push a book on a horizontal table with a certain initial velocity, it eventually comes to rest. This shows that there is some force opposing the motion of the book on the table. This is called the ‘frictional force’. It arises due to interaction between the molecules of the book and of the table.
Actually, whenever the surface of a body slides over the surface of another body, each body exerts a frictional force on the other which is parallel to the surface in contact. The frictional force on each body is opposite to the direction of its motion relative to the other. Thus, when a book slides along the surface of a table from left to right, the frictional force acting on the book is directed to the left while an equal frictional force acts on the table directed to the right. Frictional force may also exist between surfaces even before the relative motion starts.

Static Friction

Let us consider a block at rest on a horizontal table surface. The block is acted upon by two forces: its weight mg acting vertically downward at its centre of gravity, and the reactionary-force N exerted on it by the table which is directed vertically upward and passes through its centre of gravity. Since the block is in equilibrium, N = mg. In the Figure the lines of action of mg and N are shown slightly displaced for clarity.
When we apply a small horizontal force F, say towards right, the block does not move. The force R exerted on the block by the table surface is now so inclined (figure (a)) toward left that R, mg and F may form a ‘closed’ triangle (since the block is still in equilibrium). Figure (b).
The force R can be resolved into two components; parallel and perpendicular to the contact-surfaces. The component parallel to the contact surface is called the ‘force of static friction’ f_, which balances the applied force F (F = f ). The component perpendicular to the contact surfaces is the ‘normal reaction’ N exerted on the block which balances the weight mg of the block (N = mg). Figure (c).
Now, if the applied force F is slightly increased, the block still does not begin to move. This means that the force R is further inclined towards left so that the force of static friction f also increases to become equal to the new value of F. Thus, as the applied force F is increased, the force of static friction f also increases, but after a certain limit f cannot increase any more. At this moment the block is just about to move.
This maximum value of the static frictional force fs is called ‘limiting frictional force’ (it is equal to the smallest force required to start motion). Now, as the applied force is further increased, the block begins to move.

Law of Limiting Friction

The limiting (maximum) static frictional force depends upon the nature of the surfaces in contact. It does not depend upon the size or area of the surfaces. For the given surfaces, the limiting frictional force fs is directly proportional to the normal reaction N.
fs∝N
or fs=μsN (for limiting frictional force)
where the constant of proportionality m_s is called the ‘coefficient of static friction’. The above formula holds only when fs has its maximum (limiting) value. Before this stage, force of friction fs<μsN.
If the direction of the applied force F is reversed, the direction of f also reverses, while the direction of N remains unchanged. In fact f is always opposite to F.

Angle of Friction

In the case of limiting friction, the angle which the resultant R of the limiting frictional force fs and the normal reaction N makes with the normal reaction N is called the angle of friction. If this angle is θs, then
tanθs=fsN
But fsN=μs
tanθs=μs
Thus, the coefficient of static friction is equal to the tangent of the angle of friction.

Kinetic Friction

Once the motion starts, the frictional force acting between the surfaces decreases, so that a smaller force F is required to maintain uniform motion. The force acting between the surfaces in relative motion is called the ‘kinetic frictional force’ fk which is less than the limiting force of static friction fs. We know from daily experiences that a lesser force is required to maintain the motion of a body than the force required to start the body from rest.
Thus, when the block is in uniform motion, the force of kinetic friction is
fk=μkN
where μk is the coefficient of kinetic friction and its value is less than μs.

Friction on an Inclined Plane
In the figure(a), a block is placed on a rough inclined plane making an angle θ with the horizontal. Initially, the block is stationary. The F.B.D. of the block is shown in figure (b).The weight mg can be resolved into two components, mgsinθ parallel to the inclined plane and mgcosθ perpendicular to it. Due to the parallel component mgsinθ, the block has a tendency to slide down the plane, but the static frictional force f exerted on the block by the plane is preventing it from sliding.
Since the block is in equilibrium, the net force on the block parallel to the plane and also perpendicular to the plane is zero. That is
f−mgsinθ=0 and N−mgcosθ=0
On increasing the angle of inclination θ, the component mgsinθ increases, but simultaneously f also increases. When f increases to its limiting (maximum)) value, the block is just about  to slide. Suppose in this position the angle of inclination of the plane is θs.
Then fs=μsN
Substituting this value of f_s in the above expression, we get
μsN=mgsinθs and
N=mgcosθ
Dividing, we get
μs=tanθs
Hence, measuring the angle of inclination at which the block just starts sliding, the coefficient of static friction can be determined.

 Application 1

A block of mass M = 10 kg is placed at rest on a horizontal surface as shown in the figure. The coefficient of friction between the block and the surface is µ = 0.3. It is pulled with a horizontal force  F.
Find the magnitude of the force of friction if
(i) F = 20 N
(ii)F = 40 N

Solution

The maximum value of friction force is
fmax=μN=μMg
or fmax=(0.3)(10)(10)=30N
(i) To keep the block stationary the magnitude of friction force should be f = F = 20 N since F < f_max . Therefore the force of friction is f =20 N
(ii) To keep the block stationary the magnitude of friction force should be f = F = 40 N. Since f cannot be more than f_max = 30 N, therefore, the force of friction is f = f_max = 30 N

Note  that in this case friction force is unable to keep the block stationary and the block accelerators with
a=F−fM=40−3010=1m/s2
Also note that friction force is not always equal to µN. It is the limiting or maximum value of friction. At any stage friction force may attain any value between 0 and µ N.
i.e. 0≤f≤μN

Application 2

For the system shown in the figure, the coefficient of static and kinetic friction between the block m₂ and the horizontal surface is µ_s and µ_k.
If m₂ = 10 kg, µ_s= 0.4 and µ_k = 0.35
(i) Find the maximum value of m₁ so that the block m₂ does not move.
(ii) If m₁ = 5 kg, find the acceleration of the system.

Solution

(i) The force which tries to accelerate the system is m₁g, while the opposing force is f_s. For maximum value of m₁ under equilibrium condition.
(m₁g)_max = f_{s max} =µ_sm₂g
or   m_1max  = µ_sm₂
here µ_s = 0.4, m₂  = 10 kg
Thus, m_1max  = (0.4) (10) = 4 kg
(ii) When m₁ = 5 kg, the system starts accelerating; m₁ accelerates downward and m_2­ accelerates rightward. Let a be the acceleration of the system, then
a=FnetMsystem=m1g−fkm1+m2             or           a=[m1−μkm2m1+m2]g
Putting m₁ = 5kg; m₂ = 10 kg; µ_k = 0.35; g = 10 m/s²
We get, a = [5−(0.35)(10)5+10](10) = 1 m/s²

Application 3

A block is placed at rest on a horizontal surface. The coefficient of friction between the block and the surface is µ. It is pulled with a force F at an angle q with the horizontal as shown in the figure. Find the value of q at which minimum force is required to move the block. Also find the magnitude of this minimum force.

Solution

The free body diagram of the block is shown in the figure. The friction force is maximum because the block is just about to move.
Applying Newton’s Second Law
N+Fsinθ=mg             (1)
Fcosθ=fmax=μN         (2)
or     Fcosθ=μ(mg−Fsinθ)
or     F(cosθ+μsinθ)=μmg   (3)
For minimum value of F,    dFdθ=0
Thus, differentiating equation (3) w.r.t. θ,  by applying chain rule, we get
dFdθ(cosθ+μsinθ)+Fddθ(cosθ+μsinθ)=0
dFdθ(cosθ+μsinθ)+F(−sinθ+μcosθ)=0
Since dFdθ=0, therefore (−sinθ+μcosθ)=0
or tan θ = µ ⇒ θ = tan^–1 µ
Note that minimum force required to move the block is at the angle of friction.
Now from equation (3), Fmin=μmg1+μ2√ with  q = tan^-1(μ)

ROUGH PULLEYS

We know that when a string passes over a smooth pulley, the tension on each side of the string is same. But if the surface of the pulley is rough then the tensions on either side of the string are different. The difference in tension depends on the coefficient of friction (µ) and the angle (θ) through which the string is wrapped over the pulley.

Application 4

Show that the tension on two sides of the string wrapped over a rough pulley are related as
T₂ = T₁ e ^μθ    (T₂ > T₁)
where µ is the coefficient of friction and θ is the angle subtended by the string over the pulley.

Solution

Consider a small element of string, which subtends an angle dθ at the centre.
For the equilibrium of the element
dN=Tsin(dθ2)+(T+dT)sin(dθ2)
or dN=2Tsin(dθ2)=Tdθ      (1)
sin(dθ2)≈(dθ2)
(T+dT)cos(dθ2)−Tcos(dθ2)=dF=μdN  (cos(dθ2) ≈  1)
or   dT=μdN            (2)
From equations (1) and (2)
dT=μTdθ      or      dTT  = dT=μdθ
Integrating both sides, we get
∫T1T2dTT=∫0θμdθ   or    n∣∣T2T1∣∣=μθ
or T₂ = T₁. This shows that T₂ > T₁.

Application 5

Find the minimum and maximum values of M₂ for which the system remains in equilibrium. Take M₁ = 10 kg and µ = 0.2.

Solution

For minimum value of M₂
M1g=M2mingeμπ (θ=π)
or M2min=M1eμπ=M1e−μπ
Putting M₁ = 10 kg; µ = 0.2, after solving
M2min=(10)e−(0.2)(3.14)= 5.33 kg
For maximum value of M₂
M2maxg=M1geμπ
or M2maxg=(10)e(0.2)(3.14) = 18.73 kg