# SCALAR AND VECTOR QUANTITIES

## CONCEPT OF SCALAR AND VECTOR QUANTITIES

Physical quantities are divided into scalar and vector quantities.

A scalar is one which has only magnitude (size) e.g. distance, speed, temperature, volume, work, energy, power, mass etc.

A vector quantity has both magnitude and direction e.g. force, weight, magnetic flux, electric fields, gravitational   fields etc.

## VECTOR REPRESENTATION

A vector quantity can be graphically represented by a line drawn so that the length of the line denotes the magnitude of the quantity. The direction of the vector is shown by the arrow head.

## ADDITION AND SUBTRACTION OF VECTORS

Two or more vectors acting on a body in a specified direction can be combined to produce a single vector having the same effect. The single vector is called the resultant.

For example:

(a)  Two forces Y and X with magnitude of 3N and 4N respectively acting along the same direction will produce a resultant of 7N (algebraic sum of the two vectors).

(b) If Y and X act in opposite direction, the resultant will be 1N.

(c) If the two vectors are inclined at 90 to each other, Pythagoras theorem is used.

 X
 4N
 O

Y

 Y
 3N

3N

) θ

4N               X

R2 = X2 + Y2

R2 = 42 + 32

R2 =16 + 9

R2 = 25

R   = √ 25

R = 5N

Tan θ = Y/X

θ = tan-1(Y /X)

θ = tan-1(3/4)

θ = tan-1(0.75)

θ = 36.9

(d) If the two vectors are inclined at an angle less than 900 or more than 90, the resultant is obtained by using Parallelogram law of vector addition.

Parallelogram law of vector addition states that if  two vectors  are represented in magnitude  and direction  by adjacent sides of a parallelogram , the resultant  is represented  in magnitude and direction  by the diagonal  of the  parallelogram  drawn from the common point

## RESOLUTION OF VECTORS

A single vector can be resolved into two vectors called components. A vector F represented as the diagonal of the parallelogram can be resolved into its component later taken as the adjacent sides of the parallelogram.

 Ɵ

F

Y

) θ      X

Sinθ = y /F

y = f sin θ (vertical component)

Cosθ = x /F

x = F cos θ (horizontal component)

The direction of F is given by

Tan θ = y/x

θ = tan-1 (y/x)

## THE RESULATNT OF MORE THAN TWO VECTORS

To find the resultant of more than two vectors, we resolve each vector in two perpendicular direction s add all the horizontal components X, and all the vertical components, Y.

For example, consider four forces acting on a body as shown below

Figure 1:

F2              F1

Θ2              θ1

Θ3        θ4

F3               F4

Figure 2:

Y                          R

) ∞

X

Add all the resolved horizontal components

Figure 1:

X = F1 cos θ1 + (-F2 cos θ2 ) + (-F3 cos θ3 ) + F4 cos θ4

Y= F1 sin θ1 + F2 sinθ2 + (-F3 sinθ3) + (-F4 sinθ4)

Figure 2:

R2 = X2 + Y2

R = √X2+ Y2

And the direction ∞ is given by

Tan ∞ = y/x

## ONLINE WORK

1. Define vector
2. What is the difference between scalar and vector
3. Find the vertical and horizontal components of 500N force when it is inclined at (i) 60 (ii) 90 (iii) 150 to the ground level

## ASSIGNMENT

SECTION A

1. Two forces, whose resultant is 100N, are perpendicular to each other. If one of them makes an angle of 60 with the resultant, calculate its magnitude: (a) 200.0N (b) 173.2N (c) 86.6 (d) 115.5
2. A boy pulling a load of 150N with a string inclined at an angle of 30 to the horizontal. If the tension in the string is 105N, the force tending to lift the load off the ground is: (a) 52.5N (b) 202.5N (c) 75N (d) 255N
3. A lorry travels 10km northwards, 4km eastwards, 6km southwards and 4km westwards to arrive at a point T. What is the total displacement? (a) 6km east (b) 4km north (c) 6km north (d) 4km east
4. The resultant of two forces acting on an object is maximum when the angle between them is (a) 180 (b) 90 (c) 45 (d) 0
5. A boy pulls his  toy  on a smooth horizontal  surface  with  a rope  inclined at 60 to the  horizontal .If  the effective  force pulling the toy along the  tension in rope  (a) 2.5 N  (b) 4.33N (c) 5.0 N (d) 8.66N (e) 10.0N

SECTION B

1. A body of weight W newton rests on a smooth plane inclined at an angle ө to the horizontal. What is the resolved part of the weight in newton along the plane?
2. A lawn-mower is pushed with a force 50N. If the angle between the handle of the mower and the ground is 30, (a) calculate the magnitude of the force that is pressing the lawn-mover directly into the ground (b) calculate the effective force that moves the mower forward (c) why does the lawn mower move forward and not downward into the ground?

1. Calculate the resultant force in the diagram 12N            10N

40        30

60

9N     15N

POSITION

ELECTROMAGNETIC FIELD

ELECTROMAGNETIC FIELD

MAGNETIC FIELD

ELECTRIC MEASUREMENT

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