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 900 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 = 36.90
(d) If the two vectors are inclined at an angle less than 900 or more than 900, 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
- 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) 600 (ii) 900 (iii) 1500 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 600 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 300 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) 1800 (b) 900 (c) 450 (d) 00
- 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
- 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?
- A lawn-mower is pushed with a force 50N. If the angle between the handle of the mower and the ground is 300, (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 12N 10N
400 300
600
9N 15N
See also