Scalars And Vectors SS1 Physics Lesson Note

CONCEPT OF SCALAR AND VECTOR QUANTITIES

Physical quantities are divided into scalar and vector quantities.

A scalar 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 arrowhead.

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 magnitudes 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 directions, the resultant will be 1N.

(c) If the two vectors are inclined at 900 to each other, Pythagoras theorem is used.

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 = 36.90

(d) If the two vectors are inclined at an angle less than 900 or more than 900, the resultant is obtained by using the 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 and later taken as the adjacent sides of the parallelogram.  

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 Resultant Of More Than Two Vectors 

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

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

     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

The direction ∞ is given by  

 Tan ∞ = y/x 

ASSIGNMENT 

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) 600 (ii) 900 (iii) 1500 to the ground level