(Force Vectors) - University of · PDF fileThe process of finding out the resultant force of a number of given ... Analytical Graphical (Vector) Method:- ... magnitude and direction

MSC Baraa Hassun Mechanic Engineering2014

Original text book: Engineering Mechanics -Statics, Twelfth Edition, R. C. Hibbeler, 2009.

Babylon University

College of Materials Engineering

Subject: Engineering Mechanics Date:

Time:2 Hrs stage: first

Department :ceramic & polymer . Number of Lecture: 2 ,3

(Force Vectors) 2.1 Scalar and vectors A scalar is any positive or negative physical quantity that can be

completely specified by its magnitude.

A vector is any physical quantity that requires both a magnitude and

direction for its complete description. A vector is shown graphically by

an arrow. The length of the arrow represents the magnitude of the

vector, and a fixed axis defines the direction of its line of action .The

head of the arrow indicates the sense of direction of the vector (Fig 2-1)



System of Forces:



Fig(a)

2.2 Vector operations:-

Multiplication and division of vector by a scalar:

If a vector is multiplied by a positive scalar, its magnitude is increased



by that amount. When multiplied by a negative scalar it will also

change the directional sense of the vector (Fig 2-2).

Amrani

addition Vector

All vector quantities obey the parallelogram law of addition. Fig 2-3 and

Fig 2-4 and Fig 2-5 illustrates addition of vectors 𝐀 and 𝐁 to obtain a

resultant 𝐑 .







The process of finding out the resultant force of a number of given forces is

called composition of forces or compounding of forces.

Graphical

Methods:

Analytical

Graphical (Vector) Method:-

Two forces P & Q are represented by the line segments AB & AD.

Resultant Force:-

R2 = P2 + Q2 - 2PQ cos θ



Ex(1):

Forces of 6N, 3N, and 4N act as shown in the diagram. Find graphically the

magnitude and the direction of the resultant of these forces.

Ans.

By using scale: 1 N=0.5 cm

By measurement, AD=2.35 cm and angle DAB=49º

The resultant is 4.7 N and makes an angle of 49º with the x-axis.



Referring to Fig. 1.8 (b), we can get the resultant AD by constructing triangle

ABD. Line AB is drawn to represent F1 and BD to represent F2. Then AD

should represent the resultant of F1 and F2. Then we have derived triangle

law of forces from fundamental law parallelogram law of forces. The

Triangle Law of Forces may be stated as If two forces acting on a body are

represented one after another by the sides of a triangle, their resultant is

represented by the closing side of the triangle taken from first point to the

last point.

If more than two concurrent forces are acting on a body, two forces at a

time can be combined by triangle law of forces and finally resultant of all

the forces acting on the body may be obtained. A system of 4 concurrent

forces acting on a body are shown in Fig. 1.6. AB represents F1 and BC

represents F2. Hence according to triangle law of forces AC represents the

resultant of F1 and F2, say, R1.

2. DERIVED LAWS:



Fig (1.6)

If CD is drawn to represent F3, then from triangle law of forces AD

represents, the resultant of R1 and F3. In other words AD represents the

resultant of F1, F2 and F3. Let it be called as R2.On the same line logic can

be extended to say that AE represents the resultant of F1, F2, F3 and F4 if

DE represents F4. Thus resultant R is represented by the closing line of the

polygon ABCDE in the direction AE. Thus we have derived polygon of law of

forces and it may bestated as ‘If a number of concurrent forces acting

simultaneously on a body are represented in magnitude and direction by

the sides of a polygon, taken in a order, then the resultant is represented in

magnitude and direction by the closing side of the polygon, taken from first

point to last point.



Examples(2):

Read Example • 2.1 in book.

• Exercise 2.1

P = 75 N Q = 125 N

Determine the resultant using

(1) Parallelogram Law,

(2) Triangle Rule,

(3) Trigonometry.

.









Ex (4):

Two forces act on a body so that the resultant is a force of 50

pounds.The measures of the angles between the resultant and the

forces are 25o and 38o. Find to the nearest pound the magnitude of

the larger applied force?

X



2.3 vector addition of forces: Experimental evidence has shown that a force is a vector quantity since

it has a specified magnitude, direction, and sense and it adds according

to the parallelogram law.



Finding a resultant force: The two component forces 𝐅 𝟏 and 𝐅𝟐 acting on the pin in Fig 2-7 can be

added together to form the resultant force

𝐅𝐑 = 𝐅𝟏 + 𝐅𝟐



Finding the components of a force: Sometimes it is necessary to resolve a force into two components in order to

study its pulling and pushing effect in two specific directions.

For example, in Fig 2.8, F is to be resolved into two components along

two members, defined by u and v (Fig 2.8)

(Force Vectors) - University of · PDF fileThe process of finding out the resultant force of a number of given ... Analytical Graphical (Vector) Method:- ... magnitude and direction

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