chapter 2 particle

7/28/2019 Chapter 2 Particle

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Chapter 2Statics of Particles

2.0 Force on a particle

All the forces acting on a particle will be assumed to beapplied at the same point, that is the forces are assumedconcurrent.

There can be many forces acting on a particle.

The resultant of a system of forces on a particle is the

single force which has the same effectas the system offorces. The resultant of two forces can be found usingtheparallelogram law.


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2.1 Scalars and vectors

ScalarA mathematical quantity possessing magnitude only. Eg: area, volume, mass, length

VectorsA mathematical quantity possessing magnitude anddirection. Eg: forces, velocity, displacement, moment

Magnitude is designated as or simply A

Representation of vector Bold Word Processors Book uses this.

Arrow Long Hand, Word Processors

Underline R Long Hand, Typewriter, Word Processors

Represented graphically as an arrow- Length of arrow = Magnitude of Vector

-Angle between the reference axis and arrows line of action =

Direction of Vector

-Arrowhead = Sense of Vector

or F or F or FF

A


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Example:

Magnitude of Vector = 4 units

Direction of Vector = 20

measured counterclockwise from thehorizontal axis

Sense of Vector = Upward and to the right

The point O is called tail of the vector and the point P is called

the tip orhead


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For 2 vectors to be equal they must have the same:

1). Magnitude A A

2). Direction

They do not need to have the same point of application.

A negative vector of a given vector has same

magnitude but opposite direction.

A -A

A andA are equal and opposite A + (-A) = 0


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2.1.1 Vector operations

Product of a scalar and a vector A + A + A = 4A (the number 4 is a scalar)

This is a vector in the same direction as A but 4 times as long.

(+n)A = vector same direction as A, n times as long

(-n)A = vector opposite direction as A, n times as long


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Vector addition

Addition of two vectorsA and B gives aresultant vectorR bytheparallelogram law.

Result R can be foundby triangle

construction (i.e. head-to-tail fashion: byconnecting the head Ato the tail ofB. Theresultant R extendsfrom the tail ofA to the

head ofB. Addition of vectors is

communicative:

R = A + B = B + A


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Special case: Vectors A and

B are collinear(both havethe same line of action)

Vector subtraction

Vector subtraction isdefined as the addition of

the corresponding vevector.

R = A B = A + ( - B )


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2.1.2 Resolution of vector

A single vectorR can be represented by 2 or more vectors.These vectors are components of the original vector.

Finding these is called resolving the vector into its

components by theparallelogram law

The two components A and B are drawn such that they

extend from the tail orR to points of intersection


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When two or more forces are added, successive applications of theparallelogram lawis carried out to find the resultant

Eg: Forces F1, F2 and F3 acts at a point O- First, find resultant of

F1 + F2- Resultant,

FR = ( F1 + F2 ) + F3

2.1.3 Procedure for AnalysisMake a sketch using theparallelogram law

Label all the known and unknown force magnitudes and angles

Redraw half portion of the parallelogram- Magnitude of the resultant force can be determined by the law ofcosines

- Direction if the resultant force can be determined by the law ofsines


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- Magnitude of the two components can be determined by

the law of sines


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2.2 Rectangular components of a force

In many problems, it is desirable toresolve force F into two perpendicularcomponents in the x and y directions.

Fx and Fy are called rectangular vectorcomponents.

Sense of direction along positive x and yaxes

Sense of direction along positive x and

negative y axes

In two-dimensions, the cartesian unitvectors i andj are used to designate thedirections of x and y axes.

Fx = Fx i and Fy = Fy j

i.e. F = Fx i + Fyj

Fx

and Fy

are scalar components ofF

yx FFF

yx FFF '''


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Unit vectors i and j have

dimensionless magnitude of unity( = 1 ) F = Fxi + Fy(-j) orF = Fxi Fyj

Consider three coplanar forces Cartesian vector notation

F1 = F1xi + F1yjF2 = - F2xi + F2yj

F3 = F3xi F3yj


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Vector resultant is thereforeFR = F1 + F2 + F3

= F1xi + F1yj - F2xi + F2yj + F3xi F3yj

= (F1x- F2x+ F3x)i + (F1y+ F2y F3y)j

= (FRx)i + (FRy)j

If scalar notation are usedFRx= (F1x- F2x+ F3x)

FRy= (F1y+ F2y F3y)

In all cases,FRx= FxFRy= Fy


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Magnitude of FR can befound by Pythagoras`Theorem

Direction angle (orientation of the force)

can be found bytrigonometry

RyRxRFFF 22

Rx

Ry

F

F1

tan


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2.3 Equilibrium of particle

When the resultant of all forces acting on a particle is zero, the particle isin equilibrium.

Newtons First Law: If the resultant force on a particle is zero, the particlewill remain at rest or will continue at constant speed in a straight line.

Particle acted upon by

two forces:

- equal magnitude

- same line of action

- opposite sense

Particle acted upon by three or more forces:- graphical solution yields a closed polygon

- algebraic solution

00

0

yx FF

FR


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We must account for all the forces acting on a particle. The best way to do

this is to draw an FBD. An FBD is a sketch of the particle isolated (or free) from its surroundings

shown with all the forces that act on the particle.

Steps for Drawing a FBD 1).Decide which body to analyze

2).Separate this body from everything else and sketch the contour

3).Draw all applied forces 4).Include any necessary dimensions and coordinate axis

Steps 3 and 4- These are external forces

Applied forces - think of these as forces that try to get the particle to move.

Reaction forces - forces that try to prevent motion.

Note: When drawing the forces, if you don't know the direction, assume a direction and let

the sign of the answer tell you if the direction is correct or not.

Rules of FBD 1). Magnitude and direction of all forces should be clearly indicated.

2). Indicate the direction of the force on the body.


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Space Diagram: A sketch showing

the physical conditions of the

problem.

Free-Body Diagram: A sketch showing

only the forces on the selected particle.

chapter 2 particle

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