EDUCATION HOLE PRESENTS Engineering Mechanics Unit-I

Force Systems .......................................................................................................................... 3 Definitions ............................................................................................................................................................ 4

Basic assumptions ............................................................................................................................ 5

Scalar quantities .............................................................................................................................. 6

Vector quantities ............................................................................................................................. 6 Free vectors .......................................................................................................................................................... 7

Forced vectors ................................................................................................................................. 7 Parallelogram law................................................................................................................................................. 7 Triangle law .......................................................................................................................................................... 8 Polygon law .......................................................................................................................................................... 8 Calculation of values ............................................................................................................................................ 8 Parallel Forces ...................................................................................................................................................... 8 Fixed vectors ........................................................................................................................................................ 9 Resultant .............................................................................................................................................................. 9

Force System ................................................................................................................................. 10 Concurrent Forces .............................................................................................................................................. 11 Coplanar Forces .................................................................................................................................................. 11 Classification ...................................................................................................................................................... 11

Representation of force ................................................................................................................. 12

Force as a Vector ........................................................................................................................... 12 Example 1 ........................................................................................................................................................... 13 Example 2 ........................................................................................................................................................... 14

Resultant Force ...................................................................................................................... 14

Composition of forces .................................................................................................................... 14

Parallelogram Law ......................................................................................................................... 16 Resolution .......................................................................................................................................................... 16

Principle of Transmissibility of forces ............................................................................................. 18

Moment of a force ......................................................................................................................... 18

Vector representation ............................................................................................................ 19 Vector Rules ....................................................................................................................................................... 20

Moment for coplanar force system ........................................................................................ 20

Varignon’s theorem ............................................................................................................... 21

Couple ........................................................................................................................................... 24

Coplanar Concurrent Force system ........................................................................................ 24

Coplanar Non Concurrent force systems ................................................................................ 27

Equilibrium of coplanar force system ............................................................................................. 28 Conditions of Static Equilibrium of Concurrent Forces ...................................................................................... 28

Free body diagrams ....................................................................................................................... 28 Reactions ............................................................................................................................................................ 29

Equilibrium of a body under three forces ....................................................................................... 30

Lami’s theorem ...................................................................................................................... 31 Proof of Lami's Theorem .................................................................................................................................... 32

Friction .................................................................................................................................. 32 Types of Friction ................................................................................................................................................. 32

Fluid Friction .................................................................................................................................................. 33 Skin friction .................................................................................................................................................... 33 Internal Friction ............................................................................................................................................. 33

Theory of Dry friction ............................................................................................................. 33

Angle of friction ..................................................................................................................... 34

Angle of Repose ............................................................................................................................. 35

Cone of friction .............................................................................................................................. 35

Coulomb’s laws of friction ............................................................................................................. 37

Force Systems

Basic concepts

“Force” is one of those words that gets used both in everyday speech and in physics, but in this case, the technical meaning isn’t all that far removed from the everyday meaning. When you think of forces in an everyday sense, you think of things that you do to try to change the behavior of objects (or people)– pushing them, pulling them, hitting them, threatening to hit them, etc. The basic idea carries over– forces are things that change the motion of objects.

If you have two objects that interact with one another in some way, you describe the size and effect of that interaction in terms of a force. Force, in turn, is related to the motion of the object via Newton’s Laws of Motion, of which there are three, because three is the magic number:

1. An object at rest tends to remain at rest, and an object in motion tends to remain in motion in a straight line at constant speed, unless acted on by a force.

2. The net force on an object is equal to the time rate of change of the momentum, or Fnet = dp/dt (which is equal to mass times acceleration for reasonable-size objects at speeds much lower than the speed of light).

3. If one object exerts a force on a second object, the second object exerts a force on the first that is equal in magnitude to the first force, and in the opposite direction.

These can be summarized as “1) Inertia, 2) F = ma, 3) Action-Reaction,” and anybody who has ever taken physics has seen them. They can be understood a little better by thinking in terms of force as the quantification of interaction. The first law is just the codification of common sense: objects don’t change their motion unless some interaction causes them to do so. If you see a change in the motion of some object, you can deduce that there must’ve been an interaction to cause that change, and indeed that’s how we detected all the forces we know (about which more later).

The second law is just the quantification of the first law: it tells you how big a force you need to get a given change in motion. The units of force are defined in terms of the second law: a one-newton force is the result of an interaction that causes a one-kilogram object to accelerate at one meter per second.

The third law tells you that interactions always go both ways. If particle A interacts with particle B, particle A is also affected by that interaction. This is why I specify that force is the interaction between two objects– there may be more than two objects in a system (nine eight planets orbiting the Sun, say), but in terms of forces, you think about them two at a time. The interaction between the Earth and the Sun produces a force on the Earth and a force on the Sun. The interaction between the Earth and Jupiter produces a force on Jupiter, and a second force on the Earth, and so on. You determine the motion by adding up the forces due to all possible pairs, and then applying the second law.

So, if forces are interactions, what sorts of interactions are allowed? Modern physics says that there are only four types of interactions possible between fundamental particles (though there’s a sort of cottage industry in looking for a fifth). Every particle interaction you see is due to one of these four fundamental forces: gravity, electromagnetism, the strong nuclear force, and the weak nuclear force.

Definitions

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 newtons and represented by the symbol F.

The original form of Newton's second law states that the net force acting upon an object is equal to the rate at which its momentum changes with time. If the mass of the object is constant, this law implies that the acceleration of an object is directly proportional to the net force acting on the object, is in the direction of the net force, and is inversely proportional to the mass of the object. As a formula, this is expressed as:

where the arrows imply a vector quantity possessing both magnitude and direction.

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.

Basic assumptions

Time is the measure of the succession of events and is a basic quantity in dynamics. Time is not directly involved in the analysis of statics problems.

Space is the geometric region occupied by bodies whose positions are described by linear and angular measurements relative to a coordinate system.

Mass is a measure of the inertia of a body, which is its resistance to a change of velocity. The mass of a body affects the gravitational attraction force between it and other bodies.

Force is the action of one body on another. It may be exerted by actual contact or from a distance, as in the case of gravitational forces and magnetic forces. The action of a force is characterized by its magnitude, the direction of its action and its point of application. Force is represented by vector.

Particle is defined as a body whose size, in a given physical situation, is not significant in the analysis of its response to the forces that act on it. In other words, the body may be modelled as a point of concentrated mass, and the rotational motion of the body can be ignored.

Rigid body may be defined as a body that does not deform under the action of forces. Generally, the size of a rigid body influences its response to forces.

Scalar quantities

Vectors are quantities that require not only a magnitude, but a direction to specify them completely. Let us illustrate by first citing some examples of quantities that are not vectors. The number of gallons of gasoline in the fuel tank of your car is an example of a quantitity that can be specified by a single number---it makes no sense to talk about a "direction" associated with the amount of gasoline in a tank. Such quantities, which can be specified by giving a single number (in appropriate units), are called scalars. Other examples of scalar quantities include the temperature, your weight, or the population of a country; these are scalars because they are completely defined by a single number (with appropriate units).

Vector quantities

Consider a velocity. If we say that a car is going 70 km/hour, we have not completely specified its motion, because we have not specified the direction that it is going. Thus, velocity is an example of a vector quantity. A vector generally requires more than one number to specify it; in this example we could give the magnitude of the velocity (70km/hour), a compass heading to specify the direction (say 30 degrees from North), and an number giving the vertical angle with respect to the Earth's surface (zero degrees except in chase scenes from action movies!). The adjacent figure shows a typical coordinate system for specifying a vector in terms of a length r and two angles, theta and phi.

Free vectors

An example of a scaled vector diagram is shown in the diagram at the right. The vector diagram depicts a displacement vector. Observe that there are several characteristics of this diagram that make it an appropriately drawn vector diagram.

• a scale is clearly listed • a vector arrow (with arrowhead) is drawn in a specified direction. The vector arrow has a

head and a tail. • the magnitude and direction of the vector is clearly labeled. In this case, the diagram

shows the magnitude is 20 m and the direction is (30 degrees West of North)

Forced vectors

Parallelogram law

Two concurrent forces, F1 and F2, are added graphically according to the parallelogram law on their common plane to produce the resultant force P. For addition by this method, the tails of the vectors are placed at a common point, If necessary, the principle of transmissibility can be applied as shown in The forces F1 and F2 are replaced by a single force P without changing the external effects on the body that they acted on. This addition is written mathematically by using the vector expression

P = F1 + F2

From mathematics, it is known that every vector equation can solve for two unknown scalar quantities. Hence, in the solution of this equation, two unknowns can be determined. The two unknowns are normally the magnitude and direction of the resultant force P.

Triangle law

The addition of vectors illustrated by the equation given above can also be carried out graphically by using the triangle rule, The application of this rule requires the shifting of the line of action of one of the forces F1, F2 or P. If the lines of action of F1 and F2 are maintained, then the resultant force P has the correct magnitude and direction but its line of action is not obtained because the line of action does not pass through the point of intersection

Polygon law

In the case of planar forces, the resultant can also be obtained by drawing the vectors of the forces head-to-tail to form the respective force polygon,. Note that a closed polygon is obtained if the resultant force is zero.

Calculation of values

When the three laws are applied, the unknown values are obtained as follows:

1. if a scaled drawing is made, the unknown values are measured directly from the drawing; and

2. if a sketch is made, the sine rule or cosine rule is used to get the unknowns.

Parallel Forces

Parallel forces P and Q are normally added by adding the magnitudes of the forces, |P+Q|=|P|+|Q|. Use the negative sign for opposing direction. The direction of the resultant force is the same as the direction of the original forces, The position of the line of action of the resultant is normally determined by using the principle of moments, The parallelogram law can also be used but not directly. The forces are added using the parallelogram law by first adding two equal , opposite, and collinear forces to the original system. One of the added forces is applied to the point of action of the first original force whilst the other is applied to the point of application of the second original force. The added forces, F and -F, do not give rise to any external effects on the body which the forces act on.

The process of adding of two parallel forces F1 and F2 is carried out according to the order. F is added to F1 to produce the resultant P1. Then -F is added to F2 to produce the resultant P2. The resultant P, the addition of F1 and F2, is obtained by adding P1 and P2. Hence, for two parallel forces, the resultant P is obtained through

P = F1+ F2 = (F1+F) + (F2-F)

The force P obtained has the correct magnitude, direction, and line of action. Note that the magnitude of P is the same as the algebraic sum of the magnitudes of F1 and F2 and its direction is the same as the direction of the forces being summed up. I ought to be stated here that the position of the line of action of the resultant force produced by a parallel-force system is determined by using the principle of moments, as stated above.

The actual solution for the addition of forces is obtained either:

(a) by drawing, using a suitable scale, the polygon of the forces concerned and measuring the required quantities from the polygon; or

(b) by sketching the polygon of forces and then calculating the values required using geometrical calculations.

Fixed vectors

A couple is a pair of equal and opposite force vectors that are some distance apart and that act upon the same body, thus causing a rotation. Imagine that force and force are incident at two locations along a rigid body of total length at positions and , where . (Hint: draw a free body diagram)

Then

In 3D the same rule applies, using that

which means that the moment will be the same around any point in the system.

Resultant

Any system of forces may be reduced to a system of components and a resulting moment.

That is to say, and about the point

and and and then with

Force System Any collection of forces and moments in three-dimensional space is statically equivalent to a single resultant force vector plus a single resultant moment vector. (Either or both of these resultants can be zero.) The x-, y-, and z-components of the resultant force are the sums of the x-, y-, and z-components of the individual forces, respectively.

∑= nFR

∑∑ +∑ +====

n

iiz

n

iiy

n

iix FkFjFi

1,

1,

1, [Three-dimensional]

The resultant moment vector is more complex. It includes the moments of all system forces around the references axes plus the components of all system moments.

∑= nMM

( )∑ ∑+−=

i ixyxx iMizFyFM θcos)(

( )∑ ∑+−=

i iyzxy iMixFzFM θcos)(

( )∑ ∑+−=

i izxyz iMiyFxFM θcos)(

Concurrent Forces

Two or more forces that act at the same point are called concurrent forces. Concurrent forces need not have the same direction. They simply act at the same point. If they do have the same direction, they are collinear forces.

Coplanar Forces

Two or more forces whose directed arrows lie in the same plane are called coplanar forces. Since two concurrent forces always lie in a common plane, they are always coplanar. Three or more concurrent forces are not necessarily coplanar.

Classification

A force is a push or pulls acting upon an object as a result of its interaction with another object. There are a variety of types of forces. Previously in this lesson, a variety of force types were placed into two broad category headings on the basis of whether the force resulted from the contact or non-contact of the two interacting objects.

Contact Forces

Action-at-a-Distance Forces

Frictional Force Gravitational Force

Tension Force Electrical Force

Normal Force Magnetic Force

Air Resistance Force

Applied Force

Spring Force

These types of individual forces will now be discussed in more detail. To read about each force listed above, continue scrolling through this page. Or to read about an individual force, click on its name from the list below.

• Applied Force • Gravitational Force • Normal Force • Frictional Force • Air Resistance Force • Tension Force • Spring Force

Representation of force

In order to perform an inverse dynamics analysis it is necessary to have 9 variables to define the reaction force completely (e.g. Force Vector (Fx, Fy, Fz), Center of Pressure (CPx, CPy, CPz), and Free Moments (Tx, Ty, Tz)). Most force platforms measure only 6 signals of the 9 required. This is not a limitation in gait laboratories because for studies of walking it is possible to make valid assumptions about 3 of the signals, thus providing all of the signals necessary for an Inverse Dynamics analysis of walking on the force platform (The assumptions are: CPz=top surface of the force platform, and Tx=Ty=0).

Note: The assumption that the two free moments are equal to zero are not valid for handles instrumented with 6 degree of freedom load cells, wheel chair rims, or any circumstance in which the person can wrench the force platform, or pull up on it.

Force as a Vector

A force is a vector that can be separated into x and y components which are perpendicular to each other. We separate a force into components so that we can add multiple forces together.

If you draw an accurate enough diagram you can find the x and y components. However this can be tedious and is error prone and is unacceptable on the A-level examination. Instead you need to use trigonometry to separate a vector into x and y components. The force is always the hypotenuse of the triangle. The angle, which is the direction that the force is applied, is usually given in degrees. However sometimes you will also find the angle given in the Cardinal directions (east, south, west, north,etc.).

Value Formulae

Value Formulae

Example 1

Find the x and y components of a 40 Newton force acting on a ball at a 30° angle.

1. Using the mathematical formulae we get: 1. 2.

Example 2

Find the Angle if the resultant force is 24N and the y component of the force is 18.11N.

1. 1. Using the mathematical formulae we get:

2.

Resultant Force When multiple forces are acting on one object you will get a resultant force. The resultant force is how all the forces combined will act on the object. The procedure for finding the resultant force is as follows:

1. Draw the forces and the angles on a Cartesian plane. This is to ensure that you get the right angles.

2. Make all angle in terms of the positive x axis. 3. Separate each force into its x and y components. 4. Add all the x and y components. You can also use the i and j notation, where i is the

horizontal component and j is the vertical component. 5. Find the resultant force. 6. Find the angle. If a force is negative make it positive. Then place the angle in the right

quadrant, because the inverse tangent will place all angles in the first quadrant.

Composition of forces The net force is a vector produced when two or more forces act on a single object. It is calculated by vector addition of the force vectors acting on the object

FIGURE 1.8

This is an analytical method of finding the resultant of multiple forces. In this method, first find the components of each force in two mutually perpendicular directions. Then the components in each direction are algebraically added to get the two components. These two component forces, which are mutually perpendicular, are combined to get the resultant force.

Consider, for instance, forces F1, F2, F3, … acting on a particle O (Fig. 1.8). Their resultant R is defined by the relation

Resolving each force into its rectangular components, we write

From which, it follows that

Parallelogram Law

The parallelogram of forces is a method for solving (or visualizing) the results of applying two forces to an object.

Figure 1: Parallelogram construction for adding vectors

When more than two forces are involved, the geometry is no longer parallelogrammatic, but the same principles apply. 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 Figure 1. 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.

Resolution

A single force F acting on a particle may be replaced by two or more forces which together have the same effect on the particle. These forces are called components of the original force F, and the process of substituting them for F is called resolving the force F into components.

Consider a force F acting on a particle at O is to be resolved along OM and ON making angle α and β with the force, respectively. Components of the force F along OM and ON are P and Q, respectively. Draw AC parallel to OB and BC parallel to OA. Complete the parallelogram OACB as shown in Figure 1.5.

Applying the law of sine to ΔOAC,

From which

If the force F is resolved into two components which are perpendicular to each other, then these are called rectangular components Fx and Fy (Fig. 1.6).

Substituting α + β = 90 and θ = α in the equation

Principle of Transmissibility of forces States that the conditions of equilibrium or conditions of motion of a rigid body will remain unchanged if a force acting at a give point of the rigid body is replaced by a force of the same magnitude and same direction, but acting at a different point, provided that the two forces have the same line of action.

Moment of a force Moment is the name given to the tendency of a force to rotate, turn or twist a rigid body about an actual or assumed pivot point. However rotation is not required for the moment to exist. When a restrained body is acted upon by a moment, there is no rotation.

Moments have primary dimensions of length × force.

Typical units are foot-pounds, inch-pounds, and Newton-meters. Moments are vectors. The moment vector, Mo, for a force about a point O is the cross product of the force, F, and the vector from point O to the point of application of the force, known as the position vector, r. The

scalar product θsinr is known as the moment arm, d.

FrM O ×=

FdFrMM OO === θsin ]180[ ≤θ

Right-hand rule: Place the position and force vectors tail to tail. Close your right hand and position over the pivot point. Rotate the position vector into the force vector and position your

hand such that your fingers curl in the same direction as the position vector rotates. Your extended thumb will coincide with the direction of the moment. The direction cosines of a force can be used to determine the components of the moment about the coordinate axes.

xx MM θcos=

yy MM θcos=

zz MM θcos=

Alternately, the following three equations can be used to determine the components of the moment from the component of a force applied at point (x, y, z) referenced to an origin at (0,0,0).

yzx zFyFM −= 31.13

zxy xFzFM −= 31.14

xyz yFxFM −= 31.15

The resultant moment magnitude can be reconstituted from its components.

222

zyx MMMM ++= 31.16

Vector representation Typically for vectors the two numbers are a magnitude and a direction. For a displacement vector from the origin, this is the length (r in the example below) and the angle θ measured counter clockwise from the x-axis. This is equivalent to giving the polar coordinates of the point at the head of the vector. We’re probably all more familiar with giving the xy or Cartesian coordinates of a point and we can define a vector similarly. We give the x- and y-components. These are the lengths of a vector along the x-axis and one along the y-axis, the vector sum of which is the original vector. As pictured, the sum of these two right-angled vectors is the diagonal across the rectangular box suggested by the two components. Finding the magnitudes of these two vectors is called breaking a vector down into components.

For a displacement vector you can look at the vector representation as a set of directions to go from the origin to a particular point. The Cartesian description tells you to go a certain distance due east (or west), then turn and go another distance due north (or south). Polar coordinates tell you to go a certain distance in a particular direction. With both sets of direction you end up at the same location.

For a vector at an angle θ measured in the counterclockwise direction from the positive x-axis that has x- and y-components x and y, the transformations between the two descriptions is

Vector Rules

If you are adding a bunch of vectors, the analytical technique is to

1. Break all the vectors down into components.

2. Add all the x-components as straight numbers.

3. Add all the y-components similarly.

4. Convert back to the magnitude and direction description to state the final single vector.

Two vectors are equal if and only if they have the same magnitude and direction or, equivalently, if both the x-components are equal and the y-components are equal. Note that equal vectors do not have to occupy the same space. This means that as long as we keep the same orientation, we can put a vector anywhere we want it

Moment for coplanar force system Parallel forces can be in the same or in opposite directions. The sign of the direction can be chosen arbitrarily, meaning, taking one direction as positive makes the opposite direction

negative. The complete definition of the resultant is according to its magnitude, direction, and line of action.

Varignon’s theorem The Principle of Moments, also known as Varignon's Theorem, states that the moment of any force is equal to the algebraic sum of the moments of the components of that force. It is a very important principle that is often used in conjunction with the Principle of Transmissibility in order to solve systems of forces that are acting upon and/or within a structure. This concept will be illustrated by calculating the moment around the bolt caused by the 100 pound force at points A, B, C, D, and E in the illustration.

First consider the 100 pound force Since the line of action of the force is not perpendicular to the wrench at A, the force is broken down into its orthagonal components by inspection. The line of action of the the 100 pound force can be inspected to determine if there are any convenient geometries to aid in the decomposition of the 100 pound force. The 4 inch horizontal and the 5 inch diagonal measurement near point A should be recognized as belonging to a 3-4-5 triangle. Therefore, Fx = -4/5(100 pounds) or -80 pounds and Fy = -3/5(100 pounds) or -60 pounds. Consider Point A The line of action of Fx at A passes through the handle of the wrench to the bolt (which is also the center of moments). This means that the magnitude of the moment arm is zero and therefore the moment due to FAx is zero. FAy at A has a moment arm of twenty inches and will tend to cause a positive moment.

FAy d = (60 pounds)(20in) = 1200 pound-inches or 100 pound- feet

The total moment caused by the 100 pound force F at point A is 1200 pound-inches.

Consider Point B At this point the 100 pound force is perpendicular to the wrench. Thus, the total moment due to the force can easily be found without breaking it into components.

FB d = (100 pounds)(12in) = 1200 pound-inches

The total moment caused by the 100 pound force F at point B is again 1200 pound-inches.

Consider Point C The force must once again be decomposed into components. This time the vertical component passes through the center of moments. The horizontal component FCx causes the entire moment.

FCx d = (80 pounds)(15inches) = 1200 pound-inches

Consider Point D. The force must once again be decomposed into components. Both components will contribute to the total moment.

FDx d = (80 pounds)(21inches) = 1680 pound-inches FDy d = (60 pounds)(8in) = -480 pound-inches

Note that the y component in this case would create a counterclockwise or negative rotation. The total moment at D due to the 100 pound force is determined by adding the two component moments. Not surprisingly, this yields 1200 pound-inches.

Consider Point E Varignon's Theorem applies even though point E is removed from the physical object. Following the same procedure as at point D;

FEx d = (80 pounds)(3in) = -240 pound-inches FEy d = (60 pounds)(24in) = 1440 pound-inches

However, this time Fx tends to cause a negative moment. Once again the total moment is 1200 pound-inches.

Couple

Couple is a system of forces whose magnitude of the resultant is zero and yet has a moment sum. Geometrically, couple is composed of two equal forces that are parallel to each other and acting in opposite direction. The magnitude of the couple is given by

Where F are the two forces and is the moment arm, or the perpendicular distance between the forces.

Couple is independent of the moment center, thus, the effect is unchanged in the following conditions.

• The couple is rotated through any angle in its plane. • The couple is shifted to any other position in its plane. • The couple is shifted to a parallel plane.

In a case where a system is composed entirely of couples in the same plane or parallel planes, the resultant is a couple whose magnitude is the algebraic sum of the original couples

Coplanar Concurrent Force system Resultant of a force system is a force or a couple that will have the same effect to the body, both in translation and rotation, if all the forces are removed and replaced by the resultant.The equation involving the resultant of force system are the following

1. The x-component of the resultant is equal to the summation of forces in the x-direction.

2. The y-component of the resultant is equal to the summation of forces in the y-direction.

3. The z-component of the resultant is equal to the summation of forces in the z-direction.

Resultant of Coplanar Concurrent Force System The line of action of each forces in coplanar concurrent force system are on the same plane. All of these forces meet at a common point, thus concurrent. In x-y plane, the resultant can be found by the following formulas:

Resultant of Spatial Concurrent Force System Spatial concurrent forces (forces in 3-dimensional space) meet at a common point but do not lie in a single plane. The resultant can be found as follows:

Direction Cosines

Vector Notation of the Resultant

Where

Coplanar Non Concurrent force systems The resultant of non-concurrent force system is defined according to magnitude, inclination, and position. The magnitude of the resultant can be found as follows

The inclination from the horizontal is defined by

The position of the resultant can be determined according to the principle of moments.

Where, Fx = component of forces in the x-direction Fy = component of forces in the y-direction Rx = component of thew resultant in x-direction Ry = component of thew resultant in y-direction

R = magnitude of the resultant θx = angle made by a force from the x-axis MO = moment of forces about any point O d = moment arm MR = moment at a point due to resultant force ix = x-intercept of the resultant R iy = y-intercept of the resultant R

Equilibrium of coplanar force system

In static, a body is said to be in equilibrium when the force system acting upon it has a zero resultant.

Conditions of Static Equilibrium of Concurrent Forces

The sum of all forces in the x-direction or horizontal is zero.

or

The sum of all forces in the y-direction or vertical is zero.

or

Important Points for Equilibrium Forces

• Two forces are in equilibrium if they are equal and oppositely directed. • Three coplanar forces in equilibrium are concurrent. • Three or more concurrent forces in equilibrium form a close polygon when connected in

head-to-tail manner.

Free body diagrams A free body diagram, sometimes called a force diagram, is a pictorial device, often a rough working sketch, used by engineers and physicists to analyze the forces and moments acting on a body. The body itself may consist of multiple components, an automobile for example, or just a part of a component, a short section of a beam for example, anything in fact that may be considered to act as a single body, if only for a moment. A whole series of such diagrams may be necessary to analyze forces in a complex problem. The free body in a free body diagram is not free of constraints, it is just that the constraints have been replaced by arrows representing the forces and moments they generate

Reactions The first step in solving most statics problems, after drawing the free-body diagram, is to determine the reaction forces (i.e., the reactions) supporting the body.

Procedure for determining the forces:

The procedure for finding determinate reactions in two-dimensional problems is straightforward. Determinate structures will have either a roller support and pinned support or two roller supports.

Step 1: Establish a convenient set of coordinate axes. (To simplify the analysis, one of the coordinate directions should coincide with the direction of the forces and reactions.)

Step 2: Draw the free-body diagram.

Step 3: Resolve the reaction at the pinned support (if any) into components normal and parallel to the coordinate axes

Step 4: Establish a positive direction of rotation (e.g., clockwise) for purposes of taking the moments.

Step 5: Write the equilibrium equation for moments about the pinned connection.

Step 6: write the equilibrium equation for the forces in the vertical direction. Usually, this Equation will have two unknown vertical reactions.

Step 7: Substitute the known vertical reaction from step5 into the equilibrium equation from step6. This will determine the second vertical reaction.

Step 8: Write the equilibrium equation for the forces at the horizontal direction.

Step 9: If necessary, combine the vertical and horizontal force components at the pinned connection into a resultant reaction.

Equilibrium of a body under three forces

A rigid body is said to be in equilibrium when the external forces (active and reactive too) acting on it forms a system equivalent to zero.

For a body in space we have

= , = × =

These two vector equations are equivalent to the following six scalar equations written in rectangular components of each force and each moment:

Fix = 0, Fiy = 0, Fiz = 0

Mix = 0, Miy = 0, Miz = 0

The equations may be used to determine unknown forces applied to the rigid body in space or unknown reactions exerted by its support.

These equations may be solved for just six unknowns. If they involve more than six unknowns the body is said to be statically indeterminate. If they involve fewer than six unknowns, the body is said to be partially constrained.

The statement above is not valid absolutely. The solvability of the six equations depends on the properties of the system matrix.

Generally speaking the problem of the equilibrium of a body is always transformed to the problem of the equilibrium of the system of forces that act on the body. To identify all such forces the free-body diagram is essential.

From what has been said it follows that the equilibrium of a particular force system is always simpler than the general case. For example the equilibrium of a body in a plane we may solve using three scalar equations only.

Lami’s theorem Lami's theorem is an equation relating the magnitudes of three coplanar, concurrent and non-collinear forces, which keeps an object in static equilibrium, with the angles directly opposite to the corresponding forces. A,B,C

where A, B and C are the magnitudes of three coplanar, concurrent and non-collinear forces, which keep the object in static equilibrium, and

α, β and γ are the angles directly opposite to the forces A, B and C respectively.

Lami's theorem is applied in static analysis of mechanical and structural systems. The theorem is named after Bernard Lamy.

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:

By the law of sines,

Friction Friction is the contact resistance exerted by one body when the second body moves or tends to move past the first body. Friction is a retarding force that always acts opposite to the motion or to the tendency to move.

Types of Friction Dry Friction

Dry friction, also called Coulomb friction, occurs when unlubricated surfaces of two solids are in contact and slide or tend to slide from each other. If lubricant separates these two surfaces, the friction created is called lubricated friction. This section will deal only with dry friction.

Fluid Friction Fluid friction occurs when layers of two viscous fluids moves at different velocities. The relative velocity between layers causes frictional forces between fluid elements, thus, no fluid friction occurs when there is no relative velocity.

Skin friction Skin friction also called friction drag is a component of the force resisting the motion of a solid body through a fluid.

Internal Friction Internal friction is associated with shear deformation of the solid materials subjected to cyclical loading. As deformation undergo during loading, internal friction may accompany this deformation.

Theory of Dry friction Elements of Dry Friction

= Total reaction perpendicular to the contact surface = Friction force = Coefficient of friction = Resultant of f and N = angle of friction

Consider the block shown to the right that weighs . It is placed upon a plane that inclined at an angle with the horizontal.

• If the maximum available friction force is less than thus, the block will slide down the plane.

• If the friction force will just equate to thus, the block is in impending motion down the plane.

• If the maximum available frictional resistance is greater than thus, the block is stationary.

We can therefore conclude that the maximum angle that a plane may be inclined without causing the body to slide down is equal to the angle of friction .

Angle of friction

Consider a block placed on a rough floor. Now, the reaction force is because it is equal and

opposite to the weight . Now apply a horizontal force so that the block just begins to slide, i.e., the frictional force is equal to the limiting friction. When this condition is satisfied, the angle which the resultant (between the normal and external force) makes with the vertical is called the angle of friction.

The tangent of the angle of friction is equal to the coefficient of static friction.

Angle of Repose

The downhill movement of soil and loose unconsolidated sediments is due to the force of gravity and is resisted by friction. The forces of gravity and friction are in balance at the angle of repose which is the maximum slope angle that unconsolidated materials can maintain. At angles steeper than the angle of repose friction is not sufficient to counter gravity and mass wasting occurs. At angles less than the angle of repose gravity cannot overcome friction and sediments may accumulate to form steeper slopes.

Water plays an important role in mass wasting. Dry sediments, other than clays, have no cohesion. Damp sediments are cohesive because water coats the sedimentary grains and holds them together with its surface tension (surface tension is the result of the dipolar nature of water). The angle of repose increases (think of a sand castle). In saturated sediments all the pore spaces are filled with water. The weight of water in the interconnected pores exerts pressure. Pore water pressure acts to counter the weight of grain-on-grain contact. The friction between the grains is thereby decreased and so is the angle of repose, possibly resulting in mass wasting.

The two basic classes of mass wasting are flows and slides. In flows, the material behaves as a fluid. Soil creep, earthflows, and mudflows are examples. In slides, the material behaves as a rigid solid that detaches along a basal surface. Slumps and landslides are examples.

Steady soaking rains, undercutting slopes for road building and house sites, and removal of vegetation by fires, etc may induce mass wasting.

Cone of friction

A cone clutch serves the same purpose as a disk or plate clutch. However, instead of mating two spinning disks, the cone clutch uses two conical surfaces to transmit torque by friction.

The cone clutch transfers a higher torque than plate or disk clutches of the same size due to the wedging action and increased surface area. Cone clutches are generally now only used in low peripheral speed applications although they were once common in automobiles and other

combustion engine transmissions. This section describes their relationship and goes on to explain the Cone of friction:

Let R be the normal reaction at a point of contact O and F the frictional force acting in a direction perpendicular to R. Then the total force at O is given by

(2) acting in a direction making an angle

(3) with the normal reaction. If friction is limiting, and the action at O makes an angle of with the normal reaction. This angle is denoted by . Thus

(4) and the magnitude of the limiting friction can be found if either or is known. If the direction in which the body tends to move is varied the the force of limiting friction will always lie in the plane through O perpendicular to the normal reaction and the direction of the total action at O will always lie 0n a cone with it's vertex at O and axis along the line of the

normal reaction. The semi vertical angle will be

This cone is called the cone of friction. If friction is not limiting, the angle made by the total action at O will be less than . Hence whether the friction be limiting or not, the direction of the total action at O must be inside or on the cone of friction. It follows that if P is the resultant of the other forces acting on the body which is in equilibrium, the direction of P must lie inside or on the cone o friction, since P must balance the total action at O

Coulomb’s laws of friction

Coulomb friction is a simplified quantification of the friction force that exists between two dry surfaces in contact with each other. All friction calculations are approximations, and this measurement, which was developed in 1785 by Charles-Augustin de Coulomb as a refinement of Leonardo da Vinci's classical model, is dependent only on the fundamental principles of motion. It assumes that the contact surfaces are fairly uniform and that the coefficient of friction that must be overcome for motion to begin is well-established for the materials in contact. It also accounts for the normal force involving gravitational pull, whether in direct horizontal movement to the normal force or at a vectored incline.

Mechanical engineering calculations often use Coulomb friction formulas due to their simplicity, and they can be adapted to accommodate static friction of bodies not in motion or kinetic friction of bodies sliding against each other. This model assumes that the materials are rigid solids, without lubricants or other liquids or gasses between them. Though Coulomb friction law works well with these materials, where semi-soft compounds such as rubber are involved or polished metal surfaces, the calculations are less accurate.

The Coulomb friction coefficient is a static force that is slightly higher than motive force when two materials are at rest while in contact with each other. This coefficient of friction is well-known for many simple, pure materials and is given as a unit-less number. For dry surfaces, the coefficient of friction for wood against concrete is 0.62, for polystyrene against steel of 0.3 to 0.35, and for steel against Teflon® of 0.04. These numbers are used to calculate the force required to overcome static friction, known as the friction force, by multiplying the coefficient of friction times the normal force. The normal force is the mass of the materials times gravitational pull, with vector calculations added in if the two surfaces are moving up or down an incline against gravitation pull, or towards it.

Coulomb friction damping is the effect of friction always opposing the direction of motion. It is expressed as the release of heat energy between the surfaces, which reduces the net kinetic energy of movement. Coulomb friction torque involves rotational forces when two materials are not moving in linear fashion while in contact, and is another example of where basic formulas are incorporated into more complex calculations of the actual friction taking place. These calculations take the Coulomb formulas and expand upon them to include a variety of friction

environments, including viscous fluid friction, internal friction in materials where deformation takes place, and more.