PHYSICS S.S ONE 3RD TERM WEEK 6

 

Topic:              VECTORS  

Sub topic:

Reference materials:

(1) ESSENTIAL PHYSICS, TONALD PUBLISHERS, EWELUKWA (2017)

(2) NEW SCHOOL PHYSICS, AFRICAN FIRST PUBLISHERS, ANYAKOHA M.W (2011)

(3) INTERNET

Instructional materials:

Entry behavior: The students have been taught expansion

Behavioural objective: At the end of the lesson the students should be able to:

I.                    Define scalar and vector quantity

II.                  Differentiate between scalar and vector quantity

III.                Find the addition of vectors

IV.                Resolve vector into quantity

CONTENT

VECTORS

Quantities in Physics can either be classified as scalars or vectors. These quantities are usually handled differently when used in numerical calculation.

Scalar quantities are those which have only magnitude or numerical value but no direction eg length , volume, mass, density, speed, energy, temperature etc

Vector quantity are those which have both magnitude and direction. Eg displacement, velocity, weight, acceleration, force, momentum, electric field etc.

VECTOR REPRESENTATION

A vector is represented in magnitude by a given length of line. It’s represented with an arrowhead at the end of the line drawn. Sometimes it is  indicated by an angle measured anticlockwise from a horizontal reference line.

 

 

The diagram shows a vector of magnitude 8N moving from O to A in the direction of θ to X – axis.

 

 

 

ADDITION OF VECTORS

I.                     Two vectors act in the same direction: the two vectors will be added together.

If two forces vector of magnitude F1 = 5N and F2 = 7N act on a body in the same direction.

solution

 

 

      Rx   = 5N +  7N = 12N

 

I.                     Two vectors act in the opposite direction: the differences of the two vectors will be calculated by subtraction.

If two force vectors of magnitude F1 = 10N and F2 = 6N act in opposite direction as shown below

solution

         Rx =  10N – 6N = 4N in the direction of 10N force.

RESULTANT VECTOR

The resultant vector is that single vector which would have the same effect in magnitude and direction as the original vectors acting together.

There are two methods of adding or compounding vectors to find the resultant. These are :

i.                     The parallelogram method

ii.                   The triangle method

The parallelogram law of vectors states that if two vectors are represented in magnitude and direction by the adjacent sides of parallelogram, the diagonal of parallelogram drawn from the point of intersection of the vectors represents the resultant vector in the magnitude and direction.

Example:

Find the sum of two vector A and B, A = 5m            60^o and B = 6m     130^o

Solution

 

 

………………60^o…………………..

Using cosine rule

C² = A² + B² – 2AB Cos C

  =  5² + 6² -2x5x6 Cos 110

= 25 + 36 – 60 (-0.3420)

= 61 + 20.52

C² = 81.52

C =

C = 9.02m

Angle θ can be calculated by using sine rule:  = 

                        Sin θ  =    =   = 0.6263

                           Θ = sin^-1 0.6263

                          Θ  = 39^o

The magnitude is 9.02 making 39^o with   A.

Note: The resultant vector is that single vector which would have the same effect in magnitude and direction as the original vector acting together.

RESOLUTION OF VECTORS INTO COMPONENTS

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

For example, consider 3 forces acting on a body as shown below 

 

 

 

 

 

We add all the horizontal components and obtain  X = F₂ cos O – F₁ cos O

Note that we have taken the right hand or easterly  direction as positive and the left hand or westerly direction as negative.

We add all the resolved vertical components and obtain

          Fy = F₂ sin O + F₁ sin O – F₃

Note that we have taken the upward direction or northerly as positive and the downward direction or southerly as negative.

We then find the resultant of X  and Y i.e        R = X² + Y² and the direction α is given by  tan α = y/x

Example:

Four forces acts as shown below. Calculate their resultant

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solution:

The forces are resolved into the horizontal and vertical components as shown below

force	Inclination to the horizontal	Horizontal component	Vertical component
10 12 9 15	30o 40o 90o 60o	10 Cos 30 = + 8.66 -12Cos 40= -9.19 9Cos 90 = 0 15Cos60= 7.5                             X = 6.97	10 Sin 30 = + 5 12Sin 40= 7.7 -9Sin90 = -9 -15Sin60= -12.99                                 Y = - 9.27

 

R² = X²  +  Y²

R  = 6.972 + 9.292

R  = 48.58 + 86.30

R = 134.88

R = 11.61N

Tan   =

                Tan    = 1.33

                           = tan^-1 1.33

                          = 53.10 E of S

PRESENTATION

Step I: The teacher defines scalar and vector quantity with relevant example

Step  II: The teacher illustrates how vectors is represented with diagram

Step III: The teacher leads the students in adding two or more vectors together.

Step IV: The teacher explains how resultant of two vectors inclined at angle 0o to each other can be obtained.

Step V: The teacher explains how vector is resolved into components.

EVALUATION

The teacher evaluates the lessons by asking the following questions:

1.       Define scalar and vector quantity

2.       Give two example each of scalar and vector quantities

3.       State the law of parallelogram of vector

ASSIGNMENT

Four forces 5N, 4N, 7N and 6N in the direction north, south , west and east respectively. Find the resultant of the force.