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.

Forces and

its effect

Course :- Diploma Engineering

Sub :- Engineering Mechanics

Unit :- II

Force and its Effects

• A force is any influence that causes an object to

undergo a certain change, either concerning its

movement, direction, or geometrical construction.

• In other words, a force can cause an object with mass

to change its velocity (which includes to begin

moving from a state of rest), i.e., to accelerate, or a

flexible object to deform, or both.

• Force can also be described by intuitive concepts

such as a push or a pull. A force has both magnitude

and direction, making it a vector quantity.

• It is measured in the SI unit of Newton's and

represented by the symbol F.

Force and its Effects

• Related concepts to force include: thrust, which

increases the velocity of an object; drag, which

decreases the velocity of an object; and torque which

produces changes in rotational speed of an object.

• In an extended body, each part usually applies forces

on the adjacent parts; the distribution of such forces

through the body is the so-called mechanical stress.

• Pressure is a simple type of stress. Stress usually

causes deformation of solid materials, or flow in

fluids.

Units and Measurement of Force

• The SI unit used to measure force is the Newton

(symbol N), which is equivalent to kg·m·s−2.

• The earlier CGS unit is the dyne. The relationship

F=m·a can be used with either of these. In

Imperial engineering units, if F is measured in

"pounds force" or "lbf”.

• Characteristics of Force Vector

Representation

• Forces and vectors share three major characteristics:

1. Magnitude

2. Direction

3. Location

Bow’s notation

• In the previous illustrations, the forces have beenidentified as F1,F2,R,etc.

• Another system of identifying forces, called Bow'snotation, is helpful in solving force problems.

• In the space diagram, a boldface capital letter, A, B,C, etc., is placed in the space between two forcesand the force is referred to by the two boldfacecapital letters in the adjoining spaces.

• The force AB in the space diagram is represented bythe vector ab in the force diagram, the letters a and bbeing placed at the beginning and end, respectively,of the vector.

• The letters in the space diagram are usually given inalphabetical order and in a clockwise direction.

Bow’s notation

1

• .

Types of Forces

• The following are the fundamental types of the Forces :

Fundamental Forces

1. Gravity

2. Electromagnetic forces

3. Nuclear forces

Non-fundamental forces

1. Normal force

2. Friction

3. Tension

4. Elastic force

5. Continuum mechanics

6. Fictitious forces

Action and Reaction, Tension, Thrust

and Shear Force

• Reaction

• The third of Newton's laws of motion of classicalmechanics states that forces always occur in pairs.

• This is related to the fact that a force results from theinteraction of two objects. Every force ('action') onone object is accompanied by a 'reaction' on another,of equal magnitude but opposite direction.

• The attribution of which of the two forces is action orreaction is arbitrary.

• Each of the two forces can be considered the action,the other force is its associated reaction.

• Tension

• The forces involved in supporting a ball by a rope.

Tension is the force of the rope on the scaffold, the

force of the rope on the ball, and the balanced forces

acting on and produced by segments of the rope.

• In physics, tension is the pulling force exerted by a

string, cable, chain, or similar solid object on another

object.

• It results from the net electrostatic attraction between

the particles in a solid when it is deformed so that the

particles are further apart from each other than when

at equilibrium, where this force is balanced by

repulsion due to electron shells;

• Tension

• As such, it is the pull exerted by a solid trying to

restore its original, more compressed shape.

• Tension is the opposite of compression.

• Slackening is the reduction of tension. As tension is

the magnitude of a force, it is measured in Newton's

(or sometimes pounds-force) and is always measured

parallel to the string on which it applies.

• There are two basic possibilities for systems of

objects held by strings: Either acceleration is zero

and the system is therefore in equilibrium, or there is

acceleration and therefore a net force is present. Note

that a string is assumed to have negligible mass.

• Thrust

• Thrust is a reaction force described quantitatively

by Newton's second and third laws.

• When a system expels or accelerates mass in one

direction, the accelerated mass will cause a force of

equal magnitude but opposite direction on that

system.

• The force applied on a surface in a direction

perpendicular or normal to the surface is called

thrust.

• In mechanical engineering, force orthogonal to the

main load (such as in parallel helical gears) is

referred to as thrust.

• Shear force

• Shearing forces push in one direction at the top, and

the opposite direction at the bottom, causing shearing

deformation.

• A crack or tear may develop in a body from parallel

shearing forces pushing in opposite directions at

different points of the body.

• If the forces were aligned and aimed straight into

each other, they would pinch or compress the body,

rather than tear or crack it.

• Shearing forces are unaligned forces pushing one

part of a body in one direction, and another part

the body in the opposite direction.

• Shear force

• When the forces are aligned into each other, they are

called compression forces.

• An example is a deck of cards being pushed one way

on the top, and the other at the bottom, causing the

cards to slide.

• Another example is when wind blows at the side of a

peaked roof of a home - the side walls experience a

force at their top pushing in the direction of the wind,

and their bottom in the opposite direction, from the

ground or foundation.

• William A. Nash defines shear force thus: "If a plane

is passed through a body, a force acting along this

plane is called shear force or shearing force.

Force System

• Concurrent Force System in Space

• The same method used to solve coplanar

concurrent force systems is used to solve

noncoplanar concurrent systems.

• The plane-table (an early surveying instrument)

weighs 40 pounds and is supported by a tripod,

the legs of which are pushed into the ground. The

force in each leg may be considered to act along

the leg.

• Using the free body and the equations given, solve

for the magnitude of the force in each leg.

Free body diagrams

• Space diagram represents the sketch of the physical problem.

The free body diagram selects the significant particle or

points and draws the force system on that particle or point.

• Steps:

1. Imagine the particle to be isolated or cut free from its

surroundings. Draw or sketch its outlined shape.

2. Indicate on this sketch all the forces that act on the

particle. These include active forces - tend to set the

particle in motion e.g. from cables and weights and

reactive forces caused by constraints or supports that

prevent motion.

3. Label known forces with their magnitudes and directions.

use letters to represent magnitudes and directions of

unknown forces.

• Assume direction of force which may be correctedlater. The crate below has a weight of 50 kg. Draw afree body diagram of the crate, the cord BD and thering at B.

2

Resultant and components concept of equilibrium

• Orthogonal components of forces

• The determination of the resultant of three or

more forces using strictly the Parallelogram Law

in the form of Equations is somewhat tedious and

in the long run almost useless. We need better

tools !

• Determination of resultant of forces

• we now replace the force F1 by its x- and y-component and repeat this step for the two other forces involved.

• The result is that we have replaced the original three forces by six new forces, of which three are aligned with the x-axis and three with the y-axis of our coordinate system.

• The final step is then to add the three force components in the x-direction (no sweat here, that would be just adding/subtracting numbers) to get the x-component of the resultant.

• The y-component of the resultant is obtained in similar fashion.

• Formally we write this as :

• Once we have these components we candetermine the magnitude of the resultant and theangle β between the resultant and the x-axis :

• Last Equation has always two solutions for the angleβ. If for given Rx and R your calculator gives β=80°for example then β=110° is a solution as well.

yyyy

xxxx

FFFR

FFFR

321

321

R

R

RRR

y

yx

)sin(

22

Resultant of forces, a sample case

• When applying these equations it is extremely important toknow about the sign (plus/minus) conventions which goalong with the cos() and sin() function used in theEquations.

• Of course in Statics we don't make up our own rules butfollow strictly the rules of trigonometry. Here is a shortsample case I would recommend you read carefully.

• To some of you it might seem silly to harp on signconventions. However, in practical engineering applicationsnot observing the correct sign amounts often to thedifference between a well designed structure and a failingstructure with possible loss of human life and/or millions ofdollars.

• Forces, being vectors are observed to obey the

laws of vector addition, and so the overall

(resultant) force due to the application of a

number of forces can be found geometrically by

drawing vector arrows for each force.

• For example, see Fig. This construction has the

same result as moving F2 so its tail coincides

with the head of F1, and taking the net force as

the vector joining the tail of F1 to the head of F2.

• This procedure can be repeated to add F3 to the

resultant F1 + F2, and so forth

Equilibrium of Two Forces

• Equilibrium occurs when the resultant force

acting on a point particle is zero (that is, the

vector sum of all forces is zero).

• When dealing with an extended body, it is also

necessary that the net torque in it is 0.

• There are two kinds of equilibrium:

1. static equilibrium

2. dynamic equilibrium.

• Static Equilibrium

• Static equilibrium was understood well before theinvention of classical mechanics.

• Objects which are at rest have zero net force actingon them. The simplest case of static equilibriumoccurs when two forces are equal in magnitude butopposite in direction.

• For example, an object on a level surface is pulled(attracted) downward toward the center of the Earthby the force of gravity. At the same time, surfaceforces resist the downward force with equal upwardforce (called the normal force).

• The situation is one of zero net force and noacceleration.

• Pushing against an object on a frictional surface

can result in a situation where the object does not

move because the applied force is opposed by

static friction, generated between the object and

the table surface.

• For a situation with no movement, the static

friction force exactly balances the applied force

resulting in no acceleration.

• The static friction increases or decreases in

response to the applied force up to an upper limit

determined by the characteristics of the contact

between the surface and the object.

• Dynamic equilibrium

• Galileo Galilei was the first to point out the inherentcontradictions contained in Aristotle's description offorces.

• Dynamic equilibrium was first described by Galileowho noticed that certain assumptions of Aristotelianphysics were contradicted by observations and logic.

• Galileo realized that simple velocity additiondemands that the concept of an "absolute rest frame"did not exist.

• Galileo concluded that motion in a constant velocitywas completely equivalent to rest. This was contraryto Aristotle's notion of a "natural state" of rest thatobjects with mass naturally approached.

• A simple case of dynamic equilibrium occurs inconstant velocity motion across a surface withkinetic friction.

• In such a situation, a force is applied in thedirection of motion while the kinetic friction forceexactly opposes the applied force. This results inzero net force, but since the object started with anon-zero velocity, it continues to move with anon-zero velocity.

• Aristotle misinterpreted this motion as beingcaused by the applied force. However, whenkinetic friction is taken into consideration it isclear that there is no net force causing constantvelocity motion.

Superposition and Transmissibility of

Forces• Principle of superposition of forces

• Net effect of forces applied in any sequence on abody is given by the algebraic sum of effect ofindividual forces on the body.

3

Principle of transmissibility of forces

• The point of application of a force on a rigid body

can be changed along the same line of action

maintaining the same magnitude and direction

without affecting the effect of the force on the

body.

• Limitation of principle of transmissibility:

Principle of transmissibility can be used only

for rigid bodies and cannot be used for

deformable bodies.

Newton’s third law

• Newton's Third Law is a result of applying symmetry to

situations where forces can be attributed to the presence of

different objects.

• The third law means that all forces are interactions between

different bodies and thus that there is no such thing as a

unidirectional force or a force that acts on only one body.

• Whenever a first body exerts a force F on a second body,

the second body exerts a force −F on the first body. F

and −F are equal in magnitude and opposite in

direction.

• This law is sometimes referred to as the action-reaction

law, with F called the "action" and −F the "reaction". The

action and the reaction are simultaneous:

• If object 1 and object 2 are considered to be in thesame system, then the net force on the system due tothe interactions between objects 1 and 2 is zero since

• This means that in a closed system of particles, thereare no internal forces that are unbalanced.

• That is, the action-reaction force shared between anytwo objects in a closed system will not cause thecenter of mass of the system to accelerate.

2112

FF

02112

FF

0

F

Triangle of Forces

• When there are three forces acting on a body andthey are in equilibrium, we use the triangle law tosolve such problems:

• If three forces acting at a point are in equilibrium,they can be represented in magnitude and directionby the sides of a triangle taken in order.

• When the triangle law is applied to three forces inequilibrium, the resulting triangle will be a closedfigure, ie all the vectors will be head-to tail. Such avector diagram implies that the resultant force iszero.

Two Force Systems

• A two-force member is a rigid body with no force

couples, acted upon by a system of forces composed of,

or reducible to, two forces at different locations.

• The most common example of the a two force member is

a structural brace where each end is pinned to other

members as shown at the left.

4

Two Force Systems

• In the diagram, notice that member BD is pinned at only

two locations and thus only two forces will be acting on

the member (not considering components, just the total

force at the pinned joint).

• Two-force members are special since the two forces

must be co-linear and equal.

• This can be proven by taking a two force member with

forces at arbitrary angles as shown at the left. If moments

are summed at point B then force FD cannot not have

any horizontal component. This requires FD to be

vertical. Then the forces are summed in both directions,

it shows FB must also be vertical. Furthermore, the two

forces must be equal.

Triangle of Forces

• If three forces acting at a point are in equilibrium,

they can be represented in magnitude and

direction by the sides of a triangle taken in order.

• When the triangle law is applied to three forces in

equilibrium, the resulting triangle will be a closed

figure, ie all the vectors will be head-to tail. Such

a vector diagram implies that the resultant force is

zero.

• Extension of parallelogram law and triangle law to many forces acting at one point polygon law of forces

• If two forces acting at a point are represented, in magnitude and direction, by the sides of a parallelogram drawn from the point, their resultant force is represented, both in magnitude and in direction, by the diagonal of the parallelogram drawn through that point.

• let the two forces F1 and F2, acting at the point O be represented, in magnitude and direction, by the directed line OA and OB inclined at an angle θ with each other. Then if the parallelogram OACB be completed, the resultant force, R, will be represented by the diagonal OC.

• Method of resolution into orthogonalcomponents for finding the resultant

• The method to find a resultant, is generally slow andcan be complicated. Taking components of forces canbe used to find the resultant force more quickly.

• In two dimensions, a force can be resolved into twomutually perpendicular components whose vectorsum is equal to the given force.

• The components are often taken to be parallel to thex- and y-axes. In two dimensions we use theperpendicular unit vectors i and j (and in threedimensions they are i, j and k).

• Let F be a force, of magnitude F with components Xand Y in the directions of the x- and y-axes,respectively.

Graphical Methods

• The graphical method of solving mechanicalproblems involving forces is often used because itis quick and accurate. The force is showngraphically. To describe completely the force, thefollowing particulars must be given:

1. Its magnitude

2. Its point of application

3. Its direction

4. Its sense, i.e., whether it is pushing or pulling

Graphical Methods

• A line is drawn to a given length to represent themagnitude of the force. The direction of this lineis parallel to the direction of the force.

• The sense of the force is indicated by an arrow onthe line indicating whether it is acting toward oraway from the point of application.

• The graphical representation of the force is calleda vector. Thus a pull of 6 tons (T) acting at a pointA at 45° to the horizontal would be representedby the vector AB. Using the scale .25 in. = I T,the length of the vector would be 1.50 in.

Graphical Methods

• A body is said to be in equilibrium if the forcesacting at a point balance one another. If two equaland opposite forces act at a point in a straightline, the body is in equilibrium. Examples are tiebars, which are bars under pull or tension, andstruts or columns, which are bars under push orcompression

TWO FORCES ACTING AT A POINT

• Two or more forces acting at a point may be replaced by one force that will produce the same effect. This force is called the resultant of the forces. If two opposite forces of 8 and 5 T act at a point 0 in a straight line, a resultant force of 3 T acting in the same direction as the 8 T force could replace the two original forces.

Lami’s Theorem

• In statics, Lami's theorem is an equation relating themagnitudes of three coplanar, concurrent and non-collinear forces, which keeps an object in staticequilibrium, with the angles directly opposite to thecorresponding forces.

• A,B,C where A, B and C are the magnitudes of threecoplanar, concurrent and non-collinear forces, whichkeep the object in static equilibrium, and α, β and γare the angles directly opposite to the forces A, Band C respectively.

• Lami's theorem is applied in static analysis ofmechanical and structural systems. The theorem isnamed after Bernard Lamy.

Lami’s theorem

5

Proof of Lami's Theorem

• Suppose there are three coplanar, concurrent and non-collinear forces, which keeps the object in static equilibrium. By the triangle law, we can re-construct the diagram as follow:

6

• By the law of sines,

sinsinsin

)sin()sin()sin(

CBA

CBA

IMAGE REFERENCES

http://1.bp.blogspot.com/-4iRIsHRFBNU/UnDisg5p_8I/AAAAAAAAAHM/KhkFjPYAuJ4/s1600/UntitledB.N+1.png

Sr. No. Source/Links

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http://hyperphysics.phy-astr.gsu.edu/hbase/imgmec/acmfb.gif

http://upload.wikimedia.org/wikipedia/en/0/02/LamiProof.png

http://www.meritnation.com/img/shared/discuss_editlive/2191102/2012_04_26_21_06_08/image7486998204120897254.jpg

http://cnx.org/resources/26b654a6b36167297f734f48032b32a8/Figure_10_01_04a.jpg

http://upload.wikimedia.org/wikipedia/en/d/da/Lami.png

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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