Resulation Of Vector,Equilibrium,Couple,Conditions for Equilibrium

RESOLUTION OF VECTOR
	DEFINITION
	The process of splitting a vector into various parts or components is called "RESOLUTION OF VECTOR"
	These parts of a vector may act in different directions and are called "components of vector".
	We can resolve a vector into a number of components .Generally there are three components of vector viz.  Component along X-axis called x-component Component along Y-axis called Y-component Component along Z-axis called Z-component
	Here we will discuss only two components x-component & Y-component which are perpendicular to each other.These components are called rectangular components of vector.
	METHOD OF RESOLVING A VECTOR INTO  RECTANGULAR COMPONENTS
	Consider a vector V acting at a point making an angle q with positive X-axis. Vector V is represented by a line OA.From point A draw a perpendicular AB on X-axis.Suppose OB and BA  represents two vectors.Vector OA is parallel to X-axis and vector BA is parallel to Y-axis.Magnitude of these vectors are Vx and Vy respectively.By the method of head to tail we notice that the sum of these vectors is equal to vector V.Thus Vx and Vy are the rectangular components of vector V.  Vx = Horizontal component of V. Vy = Vertical component of V.
	MAGNITUDE OF  HORIZONTAL COMPONENT
	Consider right angled triangle DOAB
	MAGNITUDE OF  VERTICAL COMPONENT
	Consider right angled triangle DOAB

Moments

The moment (or torque) of a force about a turning point is the force multiplied by the perpendicular distance to the force from the turning point.

Moments are measured in newton metres (Nm).

Moment = F d

§  F = the force in newtons (N)

§  d = perpendicular distance in metres (m)

Example; A 10N force acts at a perpendicular distance of 0.50m from the turning point. What is the moment of the force?

Moment = Fd
= 10 x 0.50
= 5.0 Nm

The principle of moments.

” When an object is in equilibrium the sum of the anticlockwise moments about a turning point must be equal to the sum of the clockwise moments.”

sum of anticlockwise moments = sum clockwise moments

Example;

Couple

If two opposite moments act to cause an object to rotate, such as when your two hand are at the 'quarter-past-three' position on a car steering wheel, it is called a couple. The moment of a couple is called the torque. It is quite often said of engines and applys to the ability of the engine to turn the wheels, or wrongly by Jeremy Clarkson from 'Top Gear' as in, "This engine has a lot of torques."

Figure 2. A couple.

The torque is a psuedo-vector quantity. That is, it has magnitude and it has a direction that is projected perpendicular to the plane of the vectors F andr. The direction of the vector from the plane follows a right-handed corkscrew rule or equivalently the right-hand screw rule.

Torque is a psuedo-vector quanitity with the direction given by the right-hand screw rule.

The magnitude of the torque of a couple can be found by finding the net torque from clockwise and anti clockwise moments acting on an arbitrary point of rotation.

T =F(x + d)

T =F x
Equating, F x +F d = F x.
Net Torque. T = F d

Conditions for Equilibrium

An object at equilibrium has no net influences to cause it to move, either in translation (linear motion) or rotation. The basic conditions for equilibrium are:

The conditions for equilibrium are basic to the design of any load-bearing structure such as a bridge or a building since such structures must be able to maintain equilibrium under load. They are also important for the study of machines, since one must first establish equilibrium and then apply extra force or torque to produce the desired movement of the machine. The conditions of equilibrium are used to analyze the "simple machines" which are the building blocks for more complex machines.