FRICTION

When a body tends to move under the action of an external force then some resistance will act opposite to the direction of motion between the contact surfaces.This resistance is known as friction or frictional force.

In the other words , friction is the force that will act between two contact surface opposite to the direction of motion.

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R is the normal reaction [because it acts perpendicullary]

S = resultant reaction

S = (F² + R²)^1/2

φ = The angle made between normal reaction & resultant reaction it is called the angle of friction.

tan φ = F/R = μ = co – efficient of friction.

F = μ R

ANGLE OF FRICTION

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It is the angle made between the normal reaction and the resultant reaction.

It is denoted by `φ’ (φ = Angle of friction)

CO-EFFICIENT OF ERICTION

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It is the ratio of the frictional force to the normal reaction at the contact surface.

μ = frictional force/ normal reaction

μ = F/R = tan φ [F = μR]

here , μ is constant and always less than 1, value of μ depends on the nature of the contact surface.

CONE OF FRICTION

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When a body can move due to the application of an external force , we find the resultant reaction (s) as shown in figure .It we replace the external force (p)  in opposite direction then the resultant reaction also changes it’s position. The locus of the resultant reaction will from an inverted cone like figure .The cone is known as cone friction.

ANGLE OF REPOSE

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If a body kept on a rough inclined plane and on increasing the angle of inclined plane , the body can slides down the inclined plane for a particular value of inclination.

The angle of inclined plane with the horizontal direction is known as angle of repose.

PROVE THAT THE ANGLE OF REPOSE IS EQUAL TO THE ANGLE OF FRICTION I.E. α = φ

LAWS OF FRICTION

LAWS OF STATIC FRICTION (COLUMB’S LAW OF FRICTION)

             1.       The force of friction is always acting opposite to the direction of motion of the body.

             2.       The magnitude of the force of friction is exactly equal to the force which tends to move the body CP = F 

            3.       The force of limiting friction bears a constant ratio with the normal reaction (R) between the contact surface i.e. F/R = constant = μ (co-efficient of froction)

            4.       The force of friction is independent of the area of the contact between the surface.

            5.       The frictional force depends upon the roughness of the surface in contact.

LAWS OF DYNAMIC FRICTION (SLIDING FRICTION)

            1.       The force of friction always acts in a direction opposite to the direction of motion of the body.

            2.       The magnitude of kinetic friction bears a constant ratio with the normal reaction between the two surface.The ratio is called co-efficient of kinetic friction (μ_n) which is slightly less than μ.

            3.       For maximum speeds , force of friction remains constant but the decreases with the increasese in speed.

USEFULL AND HARMFULL EFFECTS IN FRICTION

USEFULL EFFECTS

           1.       It is impossible for every body to walk on  the road without the effects of friction.

           2.       Machineries can do useful work with the aid of friction force.

           3.       When we write something on the exercise book or black board , it is due to friction.

           4.       If we pullout nail from wood the frictional force offer.

           5.       When ladder is placed with its one end on the vertical wall and the other end at floor , the frictional forces prevent the ladder from slipping due to friction.

           6.       When we do some work by our hands , things does not slip from our hand due to frictional effect.

HARMFULL EFFECTS

          1.       Large amount of power has been last due to friction in engines, gears , trains , bearings etc.

          2.       Changes of wirn out of machine tools due to friction.

          3.       When a fluid run over a pipe  there is a chance of producing heat due to friction.

          4.       Heat generated due to fluid friction between air and outer surface of air –craft.

          5.       Due to frictional effects of dry leaves of a forest , changes of great firing.

METHODS OF RESOLUTION

RESOLUTION OF FORCE

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Ḟ  = F < Ө       ( F = Magnitude , Ө = Direction)

from ∆OAC , OA/OC = COS Ө , OA = OC COS Ө = F COS Ө

This is the y – component

PROVE

From ∆OAC , x²=(F COS Ө)²+(F SIN Ө)²

X² = F²(SIN² Ө+ COS² Ө)

X² = F²

X = F (Proved)

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Algebric sum of X – component

(  ͢+) Σf_x= +f₁cos Ө₁- f₂cos Ө₂- f₃cos Ө₃+ f₄cos Ө₄

Algebric sum of Y – component

(+↑)Σf_y = +f₁ sin Ө₁+ f₂ sin Ө₂- f₃ sin Ө₃- f₄ sin Ө₄

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Resultant force R² = Σf_x²+ Σf_y²

Magnitude of  Ṝ = (Σf_x²+ Σf_y²)^1/2

Direction α  =tan^-1(Y-component/X-component)

                  α  =tan^-1(Σf_x/ Σf_y)

                                  TRIANGLE LAW

STATEMENT

If two forces are acting simultaneously on a body may represent by the two adjacent sides of a triangle taken in order , then the third side of the triangle will represent the resultant force taken in opposite order.

EXPLANATION

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AB = F₁ , BC = F₂ , AC = R = resultant force in opposite order.

                                                         POLYGON LAW

STATEMENT

If a number of forces (more than two) are acting simultaneously on abody or a particle may be represented by the side of a polygon taken in order , then the closing side of polygon will represent the resultant force taken opposite order.

EXPLANATION

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Assuming F₁ ,F₂ , F₃ ,F₄ are the forces acting simultaneously on a body as soon in figure.

Then , according to polygon law R be the resultant force.

MOMENT AND COUPLE

MOMENT OF FORCE

Moment of force may be define as the turning tendency of a body with respect to some point or axis.

It is the force multiple by the perpendicular distance from the point or axis.

MOMENT = FORCE X PERPWNDICULAR DISTANCE

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Moment about point `a’ = f x d

TYPES OF MOMENT

Moment are two types (1.) clock wise moment. (2.) anti clock wise moment

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COUPLE

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If two equal parallel and opposite direction force are acting on a body

Then the turning effect of the body is called couple

It is one of the force multiplied of the distance between them called moment of couple.

Moment of couple = F x D

TYPES OF COUPLE

            (1.)  clock wise couple. (2.) anti clock wise couple.

VARIGNON’S THEOREM OR LAW OF MOMENT OR PRINCIPLE OF MOMENT

STATEMENT

If a number of coplanner force are acting simultaneously on a body or particle ‘the algebraic sum of the moments of all the forces about any point is equal to the moment of their resultant force about the same point.

EXPLANATION

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Moment of all the forces about `A’ ΣM_A = ± f₁x₁ ± f₂x₂ ± f₃x₃…………..

Moment of their resultant about `a’ = R x X

According to varignon’s theorem Rx = f₁x₁ ± f₂x₂ ± f₃x₃…………..

PROVE

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Moment of the force about `A’ M_A = F x X

BC = |F| = F

Area of the ∆ABC = ½ x BC x x = ½ x F x X = ½ x M_A

M_A  = 2 x ∆ABC

From □ABCD

∆ABC = ∆ACD = ∆ABD

Angle , ∆BCD = ∆BCE = ∆ABD

∆EBD = ∆ABE + ∆ABD = ∆ABE + ∆BCE

2 X ∆EBD = 2 X ∆ABE + 2 X ∆BCE

Moment of R about E = moment Qabout E + moment of P about E (proved)

CONDITION OF EQUILLIBRIUM

 

PRINCIPLE

A body or particle is said to be in equilibrium if the net resultant force acting on a body becomes zero i.e. the net resultant effect of the body is in stable condition.

EXPLANATION

 

R² = Σf_x²+ Σf_y²,  R = 0 , (for equilibrium condition)

Σf_x²+ Σf_y² = 0, Σf_x = 0, Σf_y = 0 Σm_A = 0

CONDITION

R² = Σf_x²+ Σf_y² (according to principle of equilibrium of force)

R = 0, Σf_x²+ Σf_y² = 0, Σf_x = 0, Σf_y = 0 Σm_A = 0

A body or particle is said to be in equilibrium if the following conditions are satisfied

1. Algebric sum of all the forces are acting in the X – direction is equal to zero [(  ͢+) Σf_x = 0] 
2. Algebric sum of all the forces are acting in the Y – direction is equal to zero [ (+↑)Σf_y = 0]
3. Algebric sum of all the moments of all the forces about any point or axis is equal to zero [ΣM_A = 0]

FREE BODY DIAGRAM(F.B.D)

for the shake of analysis we can draw the figure  seperatly in free space swoing all the active forces and the reactive forces replacing all the contact surface called the free body diagram / space diagram /F.B.D 

BOW’S NOTATION

DEFINATION

For graphical representation of a force same conventional methodology is adopted to denote the force , called bow’s notation.

In bow’s notation , two English capital latter is palsed on each side of the line of action of force

In figure force F is represented by (A) (B) bow’s notation.

EXAMPLE

let F₁, F₂ & F₃ are the forces are acting on a body and R be the resultant of all the forces as soon in figure.

(A)(B)(C)(D) are the notation are used to represent forces.

(A)(B) → F₁ & ab = F₁

(B)(C) → F₂ & bc = F₂

(C)(D) → F₃ & cd = F₃

(D)(A) → F₄ & da = F₄

[A represents the area between F₁ & R]

LAMI’S THEOREM

STATEMENT 

“If three coplanar forces are meeting at a point in rigid body or particle be the equilibrium then each force is directly proportional to the “sine” of the angle between the other two forces”

 EXPLANATION

Let , P, Q, R are three forces acting on a particle as shown in figure.According to the Lami’s theorem

P α sine α ; P = K sine α

Q α sine β ; Q = K sine β     [ K = constant]

R α sine λ ; R = K sine λ

Constant = K = P/ sine α  = Q/sine β = R/ sine λ

PROOF

From , ∆ABC

Now from triangle law we can say                                                                   

AB/sine(180ᶱ - λ) = BC/sine(180ᶱ - α) = AC/sine(180ᶱ - β)

P/ sine α  = Q/sine β = R/ sine λ (proof)