# Lesson Notes By Weeks and Term - Senior Secondary School 2

SCALAR AND VECTOR QUANTITIES

SUBJECT: PHYSICS

CLASS:  SS 2

DATE:

TERM: 1st TERM

REFERENCE TEXT

• New School Physics by M.W Anyakoha
• SSCE WAEC Past Questions
• UTME Past Questions

WEEK TWO

TOPIC: SCALAR AND VECTOR QUANTITIES

CONTENT

• Concept of scalar and vector quantities.
• Vector representation, addition of vectors.
• Resolution of vectors and resultant.

CONCEPT OF SCALAR AND VECTOR QUANTITIES

Physical quantities are divided into two types

i      Scalar quantity

ii     Vector quantity

A scalar is one which has only magnitude (size ) but no direction e.g. distance, speed , temperature , volume , work , energy, power , mass ,electric potential  , gravitational potential  electric  charge .

A vector quantity has both magnitude (size) and direction e.g. force, weight, velocity, acceleration, momentum displacement, magnetic flux, electric fields and gravitational   fields.

Scalar quantities are added according to the ordinary rules of arithmetic.  For example , a mark of  50 added to a mark of 40 produces a mark of 90 –no directional  property .But a force of 50N  combined with a force of 40N  may  produce 90N  if they  are acting  in same direction. But they are acting in opposite direction it would produce a different result. These vectors are combined or added by a special law the parallelogram law of addition of vectors.

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  line indicates the direction in which the  vector  quantity  act  and  it is shown by an arrow head . E.g a distance of 5km west represented by 5cm length of line where 1km = 1cm

N

5cm                    w

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).

3N        +        4N        =            7N

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

4N                _        3N            1N

-

(c) If the two vectors  are inclined at an angle less than  900 or more than 900 , the resultant cannot be obtained by  Pythagoras  theorem  but by vector addition,. Parallelogram law of vector, trigonometric or scale drawings can be used to calculate the magnitude and direction of the resultant             4N

Φ            3N        φ < 900

VECTORS AT RIGHT ANGLES

1. Parallelogram law of vectors 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

Y

3N                                      3N    R

4N            X                            4N

R2 = X2 + Y2    =  42 + 32     =  16 + 9   = 25

R   = √ 25       = 5N

Tan θ = Y/X

θ = tan-1  (Y /X)     = tan-1 (3/4)

θ  = tan-1 (0.75)

θ  = 36.90

1. If  the two vectors are inclined  at an angle  less  than 900 , the scale  drawing or trigonometric  method  can used . In using scale drawing (graphical ) methods, a convenient  scale is chosen ( if the  magnitude of the forces given is large ) and then draw the lengths corresponding to the magnitude of the forces . A Protractor is used to draw the angle in between the forces. The parallelogram is completed and the resultant and its fraction obtained

R                                   R

RESOLUTION OF VECTORS

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

F               Y

X

sin θ  = Y /F

Y = F sin θ (vertical component)

cos θ = 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

F2              F1                  Y

Θ2    θ1

Θ3    θ4

X

Add all the resolved horizontal components

X = F1 cos θ1 + (-F2 cosθ2 ) + (-F3 cos θ3 ) + F4 cos θ4

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

R = √X2+ Y2

And the direction ∞ is given by

Tan ∞ = y/x

EVALUATION

1           Calculate the resultant of five  coplanar  forces of values10N, 12N , 16N , 20N , 15N on an object as shown  below

20N                             12N

40 O      500

30O10N

16N      15N

F(N)  inclination      Hor.comp.      Vert. comp.

10          0                10cos θ=10.00        10 sin θ= 0

12        50                12 cos 50 =7.71     12 sin50= 9.19

20        40               -20 cos 40 =-15.32    20sin40= 12.85

16        90                16 cos 90 = 0          -16 sin 90= -16.00

15       60                 15cos60 =7.50          -15 sin60 =-12.99

9.89                            -6.95

R = √(19.892 + (6.952

R = 12.09

Tan ∞ = 6.95/9.89

∞ = -35.10    54.9

90 – 35.1

=54.9

The direction of the resultant is S 54.90

GENERAL EVALAUTION

1. A body of mass 3.0Kg is acted upon by a force of 24N, if the frictional force on the body is 13N.Calculate the acceleration of the body.
2. For the body in question 1 above, what distance would it move if the force was applied for a period of 7s?

WEEKEND ASSIGNMENT

1. Which of the following is not a vector quantity (a) speed  (b) velocity  (c) force

(d) acceleration  (e) Electric field

1. Which of the following  is not a  scalar quantity (a) density (b) weight (c) speed (d) mass

(e) temperature

1. Two  forces , whose resultant is  100N  are perpendicular to each other.If one of the

makes an angle  of  60 with the resultant , calculate its  magnitude

(sin60 = 0.8660 ,cos 60 = 0.500)    (a) 200N (b) 173.2N (c) 115 .5N (d)  86.6 N

1. 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

1. A boy is pulling a load of 150N  with a string   inclined at an angle of 30 to the horizontal.

If the tension in the lift the load off the ground is  ( sin 30 = ½  , cos 30 = √3/2 and tan30 = 1/√3 ) (a) 255N (b) 202.5N  (c) 105  √3/2 N (d) 75N (e) 52.5N

THEORY

1. Two forces of magnitude 12N and 9N act at right angle to each other f ind the resulrant?                             12N
2. Four forces act as shown below.     9N              10N

400      600

300    15N

Calculate their resultant