**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, gravitationalÂ Â 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)Â 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^{0} to each other, Pythagoras theorem is used.

X |

4N |

O |

Y

Y |

Â 3N |

3N

) Î¸

4NÂ Â Â Â Â Â Â Â Â Â Â Â Â Â X

R^{2} = X^{2} + Y^{2}

R^{2 }= 4^{2} + 3^{2}

R^{2} =16 + 9

R^{2} = 25

RÂ Â = âˆš 25

R = 5N

Tan Î¸ = Y/X

Î¸ = tan^{-1}(Y /X)

Î¸ = tan^{-1}(3/4)

Î¸ = tan^{-1}(0.75)

Î¸ = 36.9^{0}

(d) If the two vectors are inclined at an angle less than 90^{0 }or more than 90^{0}, the resultant is obtained by using Parallelogram law of vector addition.

*Parallelogram law of vector addition states that ifÂ two vectorsÂ are represented in magnitudeÂ and directionÂ by adjacent sides of a parallelogram , the resultantÂ is represented Â in magnitude and directionÂ by the diagonalÂ of theÂ parallelogramÂ 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.

ÆŸ |

F

Y

) Î¸Â Â Â Â Â X

SinÎ¸ = y /F

y = f sin Î¸ (vertical component)

CosÎ¸ = x /F

x = F cos Î¸ (horizontal component)

The direction of F is given by

Tan Î¸ = y/x

Î¸ = 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:

F2Â Â Â Â Â Â Â Â Â Â Â Â Â F1

Î˜_{2}Â Â Â Â Â Â Â Â Â Â Â Â Î¸_{1}

Î˜_{3}Â Â Â Â Â Â Â Î¸_{4}

F3Â Â Â Â Â Â Â Â Â Â Â Â Â Â F4

Figure 2:

YÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â R

) âˆž

X

Add all the resolved horizontal components

Figure 1:

X = F1 cos Î¸1 + (-F2 cos Î¸2 ) + (-F3 cos Î¸3 ) + F4 cos Î¸4

Y= F1 sin Î¸1 + F2 sinÎ¸2 + (-F3 sinÎ¸3) + (-F4 sinÎ¸4)

Figure 2:

R^{2} = X^{2} + Y^{2 }

R = âˆšX^{2}+ Y^{2}

And the direction âˆž is given by

Tan âˆž = y/x

## ONLINE WORK

- Define vector
- What is the difference between scalar and vector
- Find the vertical and horizontal components of 500N force when it is inclined at (i) 60
^{0}(ii) 90^{0}(iii) 150^{0}to the ground level

## ASSIGNMENT

**SECTION A**

- Two forces, whose resultant is 100N, are perpendicular to each other. If one of them makes an angle of 60
^{0}with the resultant, calculate its magnitude: (a) 200.0N (b) 173.2N (c) 86.6 (d) 115.5 - A boy pulling a load of 150N with a string inclined at an angle of 30
^{0}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 - 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
- The resultant of two forces acting on an object is maximum when the angle between them is (a) 180
^{0}(b) 90^{0}(c) 45^{0}(d) 0^{0} - A boy pulls hisÂ toyÂ on a smooth horizontalÂ surfaceÂ withÂ a ropeÂ inclined at 60 to theÂ horizontal .IfÂ the effectiveÂ force pulling the toy along theÂ tension in ropeÂ (a) 2.5 NÂ (b) 4.33N (c) 5.0 N (d) 8.66N (e) 10.0N

**SECTION B**

- A body of
**weight**W newton rests on a smooth plane inclined at an angle Ó© to the horizontal. What is the resolved part of the**weight**in newton along the plane? - A lawn-mower is pushed with a force 50N. If the angle between the handle of the mower and the ground is 30
^{0}, (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?

- Calculate the resultant force in the diagram 12NÂ Â Â Â Â Â Â Â Â Â Â 10N

40^{0}Â Â Â Â Â Â 30^{0}

60^{0}

9NÂ Â Â Â 15N

**Â **

See also