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Concept of Moment

& Friction

Course :- Diploma EngineeringSub :- Engineering Mechanics

Unit :- III

Concept of Moment• In physics, moment relates to the perpendicular distance from

a point to a line or a surface, and is derived from the

mathematical concept of moments.

• A moment of force being the product of the distance of a force

from an axis times the magnitude of the force, i.e., F × d, where F

is the magnitude of the force and d is the moment of the force.

• See torque for a more complete description of moments of force or

couple for the related concept free moment of force also known as

a force couple.

• The moment of inertia is the "second moment" of mass of a

physical object. This is the object's resistance or inertia to changes

in its angular motion. It is roughly the sum of the squared

distances (i.e., moments) of the object's mass particles about a

particular axis.

Varignon’s theorem

• Statement only

• The theorem states that the moment of a force

about any point is equal to the sum of the

moments of its components about the same

point.

Principle of Moments –Application

of moments to simple mechanism

• The Principle of Moments, also known asVarignon's Theorem, states that the moment ofany force is equal to the algebraic sum of themoments of the components of that force.

• It is a very important principle that is often usedin conjunction with the Principle ofTransmissibility in order to solve systems offorces that are acting upon and/or within astructure.

• This concept will be illustrated by calculating themoment around the bolt caused by the 100 poundforce at points A, B, C, D, and E in theillustration.

Parallel Forces• Statics refers to the bodies in equilibrium.

• Equilibrium deals with the absence of a net force. When the net

equals zero, the forces are in equilibrium provided they are

concurrent (they intersect).

• If they are non-concurrent, the body may rotate even if the

vector sum of the forces equals zero. Hence, there must be

another condition to set forces in equilibrium – that under the

influence of forces, the body must have no tendency toward

translational or rotary motion.

• An example of non-concurrent forces where the vector sum

may be equal to zero but it still causes the body to move is

parallel forces. They act in the same or opposite directions.

Their lines of action are parallel. Forces acting in the same or

opposite directions are parallel.

• Torque (Moment Of Force)

• Torque or moment of force refers to the turningeffect of the force upon a body about a point(fulcrum).

• It is the product of the magnitude of the force andperpendicular distance from the line of action ofthe force to the fulcrum.

• This perpendicular distance is called moment armor lever arm. The greater the distance from theaxis to the point where we apply the force, thegreater the torque.

• Maximum torque occurs when the direction of theapplied force is perpendicular to a line drawnbetween the axis and the point where the force isapplied.

• When the line and the force are in the same

direction, so that the force acts directly toward

or away from the axis of rotation, there is no

torque.

• Mathematical Equation:

Torque = Force X Moment arm

M = F X d

Conditions For A Body To Be In Equilibrium

Under The Influence Of Forces

1. The sum of the forces pulling the body in one

direction must equal the sum of the forces

pulling the body in the opposite direction:

ΣF = 0

2. The sum of moments tending to rotate the

body clockwise about a point must equal the

sum of moments tending to rotate the body

counter clockwise about the same point:

ΣM = 0

Concepts of Couple Properties and

Effect• In mechanics, a couple is a system of forces with a

resultant moment but no resultant force. A better term is

force couple or pure moment.

• Its effect is to create rotation without translation, or more

generally without any acceleration of the centre of mass.

• In rigid body mechanics, force couples are free vectors,

meaning their effects on a body are independent of the

point of application. The resultant moment of a couple is

called a torque.

• Torque has special properties that moment does not have,

in particular the property of being independent of

reference point, as described below.

Simple couple

• Definition- A couple is a pair of forces, equal in

magnitude, oppositely directed, and displaced by

perpendicular distance or moment.

• The simplest kind of couple consists of two equal

and opposite forces whose lines of action do not

coincide. This is called a "simple couple".

• The forces have a turning effect or moment called

a torque about an axis which is normal to the

plane of the forces. The SI unit for the torque of

the couple is Newton meter.

Simple couple

• If the two forces are F and −F, then the

magnitude of the torque is given by the

following formula:

• T = F × d

• where

T is the torque

F is the magnitude of one of the forces

d is the perpendicular distance between the forces,

sometimes called the arm of the couple

Simple couple

• The magnitude of the torque is always equal to

F d, with the direction of the torque given by

the unit vector ê, which is perpendicular to the

plane containing the two forces.

• When d is taken as a vector between the points

of action of the forces, then the couple is the

cross product of d and F.

T = d × F

Applications

• Couples are very important in mechanicalengineering and the physical sciences.

• In a liquid crystal it is the rotation of an optic axiscalled the director that produces the functionality ofthese compounds. As Jerald Erickson explained

• At first glance, it may seem that it is optics orelectronics which is involved, rather thanmechanics. Actually, the changes in opticalbehavior, etc. are associated with changes inorientation. In turn, these are produced by couples.Very roughly, it is similar to bending a wire, byapplying couples.

Moving a force parallel to its line of

action

• In physics, net force is the overall force acting on an

object. In order to perform this calculation the body is

isolated and interactions with the environment or constraints

are introduced as forces and torques forming a free-body

diagram.

• The net force does not have the same effect on the

movement of the object as the original system forces, unless

the point of application of the net force and an associated

torque are determined so that they form the resultant force

and torque.

• It is always possible to determine the torque associated with

a point of application of a net force so that it maintains the

movement of the object under the original system of forces.

Moving a force parallel to its line of

action

• With its associated torque, the net force becomesthe resultant force and has the same effect on therotational motion of the object as all actual forcestaken together.

• It is possible for a system of forces to define atorque-free resultant force.

• In this case, the net force when applied at theproper line of action has the same effect on thebody as all of the forces at their points ofapplication.

• It is not always possible to find a torque-freeresultant force.

General Cases of Coplanar Force

System

• Nonconcurrent Force Systems

• You already have some understanding of the

conditions which determine whether a body

subject to nonconcurrent forces is in

equilibrium. Look at the following cases and

tell in which ones

1. Sum of Forces = 0

2. The system is likely to be in equilibrium

Friction

• Concept of friction• Friction is the force resisting the relative motion of

solid surfaces, fluid layers, and material elementssliding against each other. There are several types offriction:

• Dry friction resists relative lateral motion of two solidsurfaces in contact. Dry friction is subdivided into staticfriction ("stiction") between non-moving surfaces, andkinetic friction between moving surfaces.

• Fluid friction describes the friction between layerswithin a viscous fluid that are moving relative to eachother.

Concept of Friction

• Lubricated friction is a case of fluid friction

where a fluid separates two solid surfaces.

• Skin friction is a component of drag, the force

resisting the motion of a fluid across the

surface of a body.

• Internal friction is the force resisting motion

between the elements making up a solid

material while it undergoes deformation.

Concept of Friction

• Energy of friction

• According to the law of conservation of energy,no energy is destroyed due to friction, though itmay be lost to the system of concern.

• Energy is transformed from other forms into heat.A sliding hockey puck comes to rest becausefriction converts its kinetic energy into heat.

• Since heat quickly dissipates, many earlyphilosophers, including Aristotle, wronglyconcluded that moving objects lose energywithout a driving force.

Concept of Friction

• Work of friction

• In the reference frame of the interface between two surfaces,

static friction does no work, because there is never displacement

between the surfaces.

• In the same reference frame, kinetic friction is always in the

direction opposite the motion, and does negative work. However,

friction can do positive work in certain frames of reference.

• Thus, the kinetic friction between the box and rug accelerates the

box in the same direction that the box moves, doing positive

work. The work done by friction can translate into deformation,

wear, and heat that can affect the contact surface properties (even

the coefficient of friction between the surfaces). This can be

beneficial as in polishing

Applications• Transportation

• Automobile brakes inherently rely on friction

• Rail adhesion refers to the grip wheels of a train have on the

rails

• Measurement

• A tribometer is an instrument that measures friction on a

surface.

• A profilograph is a device used to measure pavement surface

roughness.

• Household usage

• • Friction is used to heat and ignite matchsticks (friction

between the head of a matchstick

• and the rubbing surface of the match box).

Laws Of Friction

• The Three Laws of Friction

• The frictional force being independent of the

area of contact

• The frictional force being proportional to the

load

• The frictional force being independent of the

speed of movement

• The amount of Friction that can be exerted between

two surfaces is limited and if the forces acting on the

body are made sufficiently great, motion will occur.

Hence, we define limiting friction as the friction

which is exerted when equilibrium is on the point of

being broken by one body sliding on another. The

magnitude of limiting friction is given by the

following three laws.

• Law 1

• When two bodies are in contact the direction of the

forces of Friction on one of them at it's point of

contact, is opposite to the direction in which the point

of contact tends to move relative to the other.

• Law 2

• If the bodies are in equilibrium, the force ofFriction is just sufficient to prevent motion andmay therefore be determined by applying theconditions of equilibrium of all the forcesacting on the body.

• Law 3

• The ratio of the limiting friction to the Normal reaction between two surfaces depends on the substances of which the surfaces are composed, and not on the magnitude of the Normal reaction.

Limiting Friction and Coefficient of

Friction

• The maximum value of static friction, when motion is

impending, is sometimes referred to as limiting friction,

although this term is not used universally. we find that the

maximum value of static friction and the force of kinetic

friction are each proportional to the normal force; that is,

fsmax = μs n

and

fk = μk n

• These μ's are called the coefficients of friction. μs is the

coefficient of static friction and μk is the coefficient of

kinetic friction. Since fs,max > fk, this means μs > μk. If it

is clear from context, it is common to say simply the

"coefficient of friction" and to label it merely as μ.

Sliding Friction• Sliding friction is the kind of friction that is caused by two surfaces

that slide against each other. This kind of friction is alternatively

known as kinetic. Sliding friction is intended to stop an object from

moving.

• Understanding Sliding Friction

• The amount of sliding friction created by objects is expressed as a

coefficient which takes into consideration the various factors that

can affect the level of friction. These various factors that can impact

sliding friction include the following:

• The surface deformation of objects

• The roughness/smoothness of the surface of the objects

• The original speed of either object

• The size of object

• The amount of pressure on either object

• The adhesion of the surface

CONTENT REFERENCES

A TEXT BOOK OF ENGINEERING MECHANICS , R.S.KHURMI , S.CHAND & COMPANY PVT. LTD.A TEXT BOOK OF ENGINEERING MECHANICS , Dr. R.K.BANSAL , LAXMI PUBLICATION

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