###### Chapter Content :

1. Static friction , Kinetic friction , limiting friction 2. Newton’s second law of motion on a rough surface |

** Friction , Types of friction , Static friction , Kinetic friction , limiting friction
**

**When a body slides on a surface, the surface is observed to exert a retarding force on the body, preventing relative motion.**

**When an external horizontal force is applied to a body resting on a rough surface, the body does not immediately start moving : this is due to the face that the force of friction applied by the surface on this body prevents the onset of relative motion.**

**Friction is an important force in several aspects of everyday life.**

**Sometimes, it is important to minimise friction (e.g. in the engine of a car); in other cases it is because of friction that our machines work (from nails to bicycles and cars).**

**The force of friction acts tangential to the surface of contact between two bodies, and acts in such a manner as to prevent relative motion.**

**There are two distinct types of friction :**

**(i) Static friction: Where there is no relative motion between the bodies in contact.**

**(ii) Kinetic friction : Where there is relative motion kinetic friction may be sliding friction or, rolling friction, according as whether the bodies involved slide or, roll on each other.**

__Static friction :__

__Static friction :__**Suppose that a body is kept on a rough surface and an external force Fext is applied on the body in the horizontal direction.**

**The body does not accelerate initially (remains at rest) due to the fictional force F _{f} opposing.**

**Relative motion between the two bodies: The force F _{f} equals the external force .**

**As F _{ext} is increased, F_{f }increases , until it reaches a maximum value F_{lim} , also known as the limiting friction.**

**If F _{ext} is increased thereafter, the force of friction F_{f} suddenly drops and motion commences. Friction is kinetic.**

**The actual force of static friction (F _{f}) is always less then its limiting value F_{lim} before relative motion commences:**

**|F**_{f}| < F_{lim}.

**We must note that F**_{lim}is not the actual force of static friction, it is merely a maximum value. (upper bound)__Kinetic friction :__

__Kinetic friction :__**Once the body is in motion, the frictional force that acts on it is called kinetic (F _{k}).**

**In general, for sliding, F _{k} ≤ F_{lim} (static)**

**♦ Laws of friction ♦**

**♦ Laws of friction ♦**

**Both the Static limiting frictional force F _{lim} and sliding Kinetic frictional force (F_{k}) are proportional to the normal contact reaction (N).**

**The normal reaction N acts on the body perpendicular to the surface.**

**Suppose that the normal contact reaction force on a body from the ground is N.**

**Then,**

**F _{lim} ∝ N**

**F _{lim} = μ_{s} N ……….(i)**

**where μ _{s} is a constant of proportionality called co-efficient of static friction.**

**It depends on the nature of surfaces in contact.**

**For kinetic sliding friction,**

**F _{k} ∝ N**

**F _{k} = μ_{k} N …………(ii)**

**μ _{k} is known as co-efficient of kinetic friction.**

**Physically,**

**F _{k} ≤ F_{lim} , and μ_{k} ≤ μ_{s} ……… (iii)**

**The value of N in equation (i) and (ii) depends on the orientation of surfaces in contact.**

**Angle of friction :**

**Angle of friction :**

**The angle subtended by the resultant of the limiting force of static friction and the normal reaction, with respect to the normal reaction is known as the angle of friction.**

**In figure, the block of mass M is resting under a maximum applied horizontal force F on a rough horizontal surface.**

**The resultant of and is **

** **

**The angle between R ^{→} and N^{→} is λ , called the angle of friction.**

**In fact , F _{f } = μ_{s }N and tanλ = F_{f}/N**

**i.e. tanλ = μ _{s } ………(iv)**

**Angle of Repose**

**Angle of Repose**

**The maximum angle of inclination of an inclined plane w.r.t. horizontal for which an object will remain at rest when placed on it, is known as the angle of repose.**

**Suppose a block of mass M is kept on a rough inclined plane. The angle of inclination of the plane w.r.t. horizontal is gradually increased from 0°.**

**It is found as the angle increases, the tendency of the block slipping increases.**

**Ultimately just at a particular maximum angle of inclination the block is on the verge of slipping as shown in the figure.**

__Calculation of Angle of Repose :__

__Calculation of Angle of Repose :__**Let λ be the angle of friction according to the definition and , the angle of repose. With reference to the chosen X-Y axis in figure, we obtain for equilibrium under static conditions,**

**F _{r} = Mg sinθ …(v)**

**N = Mg cosθ …(vi)**

**and F _{r} = μ_{s }N ..(vii)**

**Therefore from (v), (vi) and (vii)**

**tanθ = μ _{s} …(viii)**

**Again we already know,**

**tanλ = μ _{s}**

**Therefore, λ is numerically equal to θ**

**Illustration : A block starts slipping on an inclined plane. It moves one metre in one second. What is the time taken by block to cover next one metre ?**

**Sol:**

**The forces acting on the block at any moment are**

**(i) mg (ii) N, normal reaction**

**(iii) f, friction force**

**Let a = acceleration of the block (down the plane)**

**For path AB,**

**for path AC,**

**From (i) & (ii), t ^{2} = 2 ;**

** t = √2 sec**

** Time taken to cover the distance BC = (t – 1) sec**

**= (√2 – 1) sec = 0.41 sec.**

**Exercise : A block of mass m = 2 kg is kept on a rough horizontal surface. A horizontal force F = 4.9 N is just able to slide the block. Find the coefficient of static friction. If F = 4 N, then what is the frictional force acting on the block ?**

**Illustration : Two blocks A & B are connected by a light inextensible string passing over a fixed smooth pulley as shown in the figure. The coefficient of friction between the block A & B the horizontal table is μ = 0.2; If the block A is just to slip, find the ratio of the masses of the blocks.**

**Solution : From F.B.D. of A, shown in fig**

**N + T Sinθ = m _{A}g …(1)**

**T Cosθ = fmax = μ N …(2)**

**From (1), N = m _{A}g – T Sinθ …(3)**

**From (2) and (3),**

**T Cosθ = m _{A}g – T Sinθ **

**T (Cosθ + Sinθ ) = m _{A}g …(4)**

**From F.B.D. of B,**

**T = m _{B}g …(5)**

**Taking ratio of (4) and (5)**

**on putting θ = 60° , μ = 0.2**