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ENGINEERING MECHANICS




Topic: Introduction of the Concept of Force.

CHANGE IN POSITION:

To know force well, first we have to understand what do we mean by Change. What does it mean when we say the position of the body has been changed? Whenever we find the state of object becomes different than that of the same object before some time say Δt, then we say that there exists a change in the state of the object. Suppose the change occurs in the position of the body. But to find the initial position of a body, we need a co-ordinate system.

THE CAUSE OF CHANGE:

It has been seen that to induced a change or to make a change in the position of an object we must have to change the energy possess by the body. To transfer energy into the object we shall have to apply FORCE on the body. Therefore Force is the agency that makes a change in position of a body.

THE CONCLUSION: GALILEO'S LAW OF INERTIA OR NEWTON'S FIRST LAW OF MOTION.

So, if there is no force on an object the position of the object won't change with respect to time. It means if a body at rest would remain at rest and a body at uniform motion would remain in a steady motion. This law is known as Galileo's Law of Inertia or Newton's first law of motion.

Topic: FORCE SYSTEM

QUESTION: WHAT IS A FORCE SYSTEM? CLASSIFY THEM WITH EXAMPLES.

ANSWER: A force system may be defined as a system where more than one force act on the body. It means that whenever multiple forces act on a body, we term the forces as a force system. We can further classify force system into different sub-categories depending upon the nature of forces and the point of application of the forces.
Different types of force system:

(i) COPLANAR FORCES:

If two or more forces rest on a plane, then they are called coplanar forces. There are many ways in which forces can be manipulated. It is often easier to work with a large, complicated system of forces by reducing it an ever decreasing number of smaller problems. This is called the "resolution" of forces or force systems. This is one way to simplify what may otherwise seem to be an impossible system of forces acting on a body. Certain systems of forces are easier to resolve than others. Coplanar force systems have all the forces acting in in one plane. They may be concurrent, parallel, non-concurrent or non-parallel. All of these systems can be resolved by using graphic statics or algebra.

(ii) CONCURRENT FORCES:

A concurrent coplanar force system is a system of two or more forces whose lines of action ALL intersect at a common point. However, all of the individual vectors might not actually be in contact with the common point. These are the most simple force systems to resolve with any one of many graphical or algebraic options. If the line of actions of two or more forces passes through a certain point simultaneously then they are called concurrent forces. Con-current forces may or may not be coplanar.

(iii) LIKE FORCES:

A parallel coplanar force system consists of two or more forces whose lines of action are ALL parallel. This is commonly the situation when simple beams are analyzed under gravity loads. These can be solved graphically, but are combined most easily using algebraic methods. If the lines of action of two or more forces are parallel to each other, they are called parallel forces and if their directions are same, then they are called LIKE FORCES.

(iv) UNLIKE FORCES:

If the parallel forces are such that their directions are opposite to each other, then they are termed as "UNLIKE FORCE".

(v) NON COPLANAR FORCES:

The last illustration is of a "non-concurrent and non-parallel system". This consists of a number of vectors that do not meet at a single point and none of them are parallel. These systems are essentially a jumble of forces and take considerable care to resolve.

Q.) WHAT IS FORCE ADDITION? WHAT ARE THE OPERATIONS OF FORCE?

A.) Force is a vector quantity. It has magnitude and as well as direction. Like other vectors two forces can be added, or a force can be substituted from another force, or may be a force can be multiplied by scalars as well as another vector. Unlike scalar quantities, two vector can't be added arithmatically, they must be geometrically added. Suppose we have a force 10 kN acting on a particle towards east, and suppose another force of 10 kN is acting towards north. We know that 10 kg mass +10 kg mass = 20 kg mass, but here forces of 10 kN towards east and 10 kN towards north, when added produces a resultant of magnitude =10*sqrt(2)=14.14 kN.

To add two forces acting on a plane we use (i) Triangle's Law and (ii) Paralellogram Law. In case of more than two forces exist, then we use force resolution method to find the resultant.

QUESTION: WHAT IS A RESULTANT OF A FORCE SYSTEM?

ANSWER: We have already discussed about addition of two forces on a plane by either (i) Triangle's Law or (ii) Paralellogram Law. For more than two vectors we use (iii) Polygon Law of Force Addition. (iv) Force Resolution Method.

The resultant of a force system is the Force which produces same effect as the combined forces of the force system would do. So if we replace all the combined forces of the force system would do. So if we replace all components of the force by the resultant force, then there will be no change in effect.

The Resultant of a force system is a vector addition of all the components of the force system. The magnitude as well as direction of a resultant can be measured through analytical method. Almost any system of known forces can be resolved into a single force called a resultant force or simply a Resultant. The resultant is a representative force which has the same effect on the body as the group of forces it replaces. (A couple is an exception to this) It, as one single force, can represent any number of forces and is very useful when resolving multiple groups of forces. One can progressively resolve pairs or small groups of forces into resultants. Then another resultant of the resultants can be found and so on until all of the forces have been combined into one force. This is one way to save time with the tedious "bookkeeping" involved with a large number of individual forces. Resultants can be determined both graphically and algebraically.

The Parallelogram Method and the Triangle Method are used to find the resultant of a force system. It is important to note that for any given system of forces, there is only one resultant.

QUESTION: WHAT IS A COMPONENT OF A FORCE? WHAT IS FORCE RESOLUTION?

ANSWER: It is often convenient to decompose a single force into two distinct forces. These forces, when acting together, have the same external effect on a body as the original force. They are known as components. Finding the components of a force can be viewed as the converse of finding a resultant. There are an infinite number of components to any single force. And, the correct choice of the pair to represent a force depends upon the most convenient geometry. For simplicity, the most convenient is often the coordinate axis of a structure.

A force can be represented as a pair of components that correspond with the X and Y axis. These are known as the rectangular components of a force. Rectangular components can be thought of as the two sides of a right angle which are at ninety degrees to each other. The resultant of these components is the hypotenuse of the triangle. The rectangular components for any force can be found with trigonometrical relationships:
Component of a force F along X-axis is Fx = Fcosθ and component along Y-axis is Fy = Fsinθ.

EXTRA NOTE: When forces are being represented as vectors, it is important to should show a clear distinction between a resultant and its components. The resultant could be shown with color or as a dashed line and the components as solid lines, or vice versa. NEVER represent the resultant in the same graphic way as its components.

The Steps to find a Resultant of a force system: (for con-current forces)

STEP 1:

RESOLVE ALL THE COMPONENT FORCES ALONG X-AXIS AND Y-AXIS. If a force F acts on an object at an angle ß with the positive X-axis, then its component along X-axis is
F cosß, and that along Y-axis is F sinß.

STEP 2:

Add all the X-components or Horizontal components and it is denoted by ΣFx. Add all the Y-components and denote it as ΣFy.

STEP 3:

MAGNITUDE OF THE RESULTANT R will be equal to the square root of the sum of square of ΣFx and ΣFy.

STEP 4:

DIRECTION OF THE RESULTANT α
α equals to the tan inverse of (ΣFy/ΣFx).

EQUILIBRANT:

Any concurrent set of forces, not in equilibrium, can be put into a state of equilibrium by a single force. This force is called the Equilibrant. It is equal in magnitude, opposite in sense and co-linear with the resultant. When this force is added to the force system, the sum of all of the forces is equal to zero. A non-concurrent or a parallel force system can actually be in equilibrium with respect to all of the forces, but not be in equilibrium with respect to moments.

EXTRA NOTE: Graphic Statics and graphical methods of force resolution were developed before the turn of the century by Karl Culmann. They were the only methods of structural analysis for many years. These methods can help to develop an intuitive understanding of the action of the forces. Today, the Algebraic Method is considered to be more applicable to structural design. Despite this, graphical methods are a very easy way to get a quick answer for a structural design problem and can aid in the determination of structural form.

QUESTION: WHAT IS STATIC EQUILIBRIUM? What are the conditions of static equilibrium for (i) con-current force system (ìi) coplanar non concurrent force system?

Ans: A body is said to be in equilibrium when there is no change in position as well as no rotation exist on the body. So to be in equilibrium process, there must not be any kind of motions ie there must not be any kind of translational motion as well as rotational motion.

We also know that to have a linear translational motion we need a net force acting on the object towards the direction of motion, again to induce an any kind of rotational motion, a net moment must exists acting on the body. Further it can be said that any kind of complex motion can be resolved into a translational motion coupled with a rotating motion.

Therefore a body subjected to a force system would be at rest if and only if the net force as well as the net moment on the body be zero. Therefore the general condition of any system to be in static equilibrium we have to satisfy two conditions

(i) net force on the body must be zero ie, ...
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