Subject: 

Physics

Term:

FIRST TERM

Week:

WEEK 3

Class:

SS 2

Topic:

Vectors; (a) concept of vectors (b) vector representation (c) addition of vectors (d) resolution of vectors(c) concept of resultant velocity using a vector representation.

 

Previous lesson: 

The pupils have previous knowledge of

Position, Distance and Displacement

that was taught as a topic in the previous lesson

 

Behavioural objectives:

At the end of the lesson, the learners will be able to

 

 

Instructional Materials:

• Wall charts
• Pictures
• Related Online Video
• Flash Cards

 

 

Methods of Teaching:

• Class Discussion
• Group Discussion
• Asking Questions
• Explanation
• Role Modelling
• Role Delegation

 

REFERENCES:

• new school Physics by MW Anyakoha
• New system PHYYSICS for senior secondary schools. Dr. Charles Chew.
• Comprehensive Certificate Physics by Olumuyiwa Awe
• Senior School Physics BY PN Okeke, SF Akande
• STAN Physics.

 

Content:

WEEK 3:

DATE ——————-

TOPIC: VECTORS

CONTENT:

• Concept and examples of scalars
• Concept of vectors
• Distinction & similarities between scalars & vectors
• Examples of vectors
• Representation of vectors
• Addition of vectors
• Resolution of vectors

Sub-Topic1: CONCEPT OF SCALARS

Scalars are physical quantities that have magnitude but no direction. That is, scalar has value and unit but no direction. E.g, 10km. This 10km could be in any direction since there is no actual direction. The ‘10’ is the value- the magnitude. Therefore, just 10km is a scalar quantity. Scalar quantities are always not directional.

Scalar quantities unlike vectors have only magnitude. Example; length, area, volume, temperature, work, energy, power, mechanical advantage, velocity ratio, efficiency, surface tension,

Other examples of scalar quantities include:

• Speed
• Time
• Density
• Mass
• Distance, etc.

10km

Scalars are non directional physical quantities.

CONCEPT OF VECTORS

Vectorsare physical quantities that have both magnitude and direction. This means that vectors quantities have values and are always directional. E.g, 10km due North. Here, the value, which is the magnitude, is ‘10’ while the direction is North.

Examples of vector quantities include: pressure, friction, tension, electric field intensity, magnetic field intensity, moment of forces, torque, upthrust.

Other examples of vector quantities are:

• 25km at
• Displacement
• Force
• Acceleration
• Momentum – Impulse

–	Velocity	25km at
–	Weight, etc.	Vectorsare directional physical quantities.

Evaluation:

1. What are vector quantities?
2. List five examples of each of the two types of the physical quantities.

Sub-Topic2: DISTINCTION BETWEEN SCALARS AND VECTORS.

S/ N	SCALARS	VECTORS
1.	Scalars are non directional physical quantities.	Vectorsare directional physical quantities.
2.	Always directed towards different directions.	Always directed towards a particular direction.
3.	E.g, 100km	100km due east.
4.	E.g, mass	Weight

Their similarities include:

• • They are both physical quantities;
  • They both have values, which are the magnitudes.

Evaluation:

1. State three differences between scalars and vectors.
2. State the similarities between them.

GENERAL EVALUATION:

1.What is the difference between 20km and 20km due South?

1. How would you differentiate a scalar from a vector quantity?
2. List five examples of scalar quantities.
3. Mention five physical quantities you consider as vectors and why.

Types of vectors

• 1. Position vectors; these are vectors whose starting point is fixed to a position
  2. Free vectors; these are vector whose starting point could be anywhere in space.
  3. Unit vector; this is a vector whose magnitude is one. It is often represented as â.
  4. Orthogonal vectors; these are vectors whose lines of action are mutually perpendicular to each other
  5. Collinear vectors; these are vector whose lines of action are parallel to one another.
  6. Coplanar vectors; these are vectors whose lines of action lies on the same plane.
  7. Resultant vector; this is a single vector that has the same effect as a system of vectors.
  8. Null vector: this is a vector whose magnitude is zero.

Representation of vectors

Vectors can be represented by a directed line segment whose length is proportional to the magnitude of the vector and its direction is pointing in the direction of action of the vector.

a

Vectors are represented with bold face letters a, a, orâ.

EVALUATION:

• 1. A measurable quantity that has both magnitude and direction is called —- (a) vector (b) scalar (c) displacement (d) distance
  2. The following are example of vectors except —- (a) moment (b) pressure in gas (c) tension (d) viscosity
  3. A vector whose magnitude is one is called —– (a) collinear vector (b) orthogonal vector (c) unit vector (d) free vector
  4. A set of vector whose line of action lies on the same plane is called —- (a) collinear vectors (b) concurrent vectors (c) coplanar vectors (d) coordinate vectors
  5. Which of the following groups of quantities is NOT all vectors? (a) (a) momentum, velocity, force (b) acceleration, force, momentum (c) momentum, kinetic energy, force (d) magnetic field, acceleration, displacement

PERIOD TWO:Addition of vector

The addition of two or more vector produces a single vector call the resultant vector.

A resultant is a single vector which has the same effect as a system of vectors put together.

Equilibrant is the vector that will bring a system of vector to equilibrium when added to the system. It has the same magnitude as the nt of the system but acting in the opposite direction to the equilibrant.

Consider two vectors a and b, the addition of these vector can be obtained by joining the head of one to the tail of the previous one. The resultant is the vector that joins the beginning to the end. b

c

b

a

Case 1. Parallel vectors acting in the same direction

For two parallel vectors acting in the

V₁ same direction, the angle between the

V₂

vectors is ZERO Resultant R = V₁ + V₂

Example 1: Three men pushed a car out ofa muddy ground by applying the following forces 450N, 600N and 920N. What is the resultant force on the car?

Case 2. Parallel vectors acting in opposite direction

V₁

V

2

Resultant R = V₂ – V₁

Example 2: during a tug of war game, team A pull in the positive x direction with a force of 900N and team B pull in the negative X – direction with a force of 1200N. what is the resultant of the train?

V₁ = 900N V₂ = 1200N = 1200 – 900 = 300N

Case 3. Two perpendicular vectors acting at a point.

R

V

2

V

1

The angle made by the resultant with the direction of V₂ is given as

Example 3: two force 8N and 15N acting along the vertical and the horizontal axis respectively acts on a body of mass 3kg. What is the acceleration of the body?

Solution: _8N

Force = mass x acceleration

F = ma F = R = 17N m = 3kg a =?

17 = 3 X a

Case 4. Two vector acting at a point and at angle to each other.

V

2

V

1

Ø

This case can be solved by using the parallelogram law of vectors.

Parallelogram law of vectors state that:

Parallelogram law of vectors state that when two vectors acting at a point are represented in magnitude and direction by the adjacent sides of a parallelogram, the resultant of the two vectors can also be represented in magnitude and direction by the diagonal of the parallelogram drawn from the common point of the two vectors.

180

–

Ø

R

V

2

V

1

Ø is the angle between the two vectors. The direction the resultant force R made with V₂( can be obtained using the sin rule. This is given as

Example: two forces F₁ and F₂ act on a particle. F₁ has magnitude 5N and in direction 030⁰, and F₂ has a magnitude of 8N and in the direction 090⁰. Find the magnitude and direction of the resultant.

Solution:

N

8

5

N

R

30

0

8

N

5

N

90

0

30

0

ϴ

The angle between the two forces is 090⁰ – 030⁰ = 60⁰

(b) direction of the resultant don’t forget

ϴ

• 120⁰ is obtained from (180⁰ – 60⁰) in the diagram
• we use V₁ because we are looking for the angle between R and V₂.

So the direction of the resultant in three digits ( 90 – 22) = 0680

EXPERIMENT 1:

• Educator should carry out an experiment to verify the parallelogram law of vectors – using the force board

CASE 5. Three vectors acting at a point and in equilibrium.

Consider a metal ball suspended from a ceiling by a string. If is pulled by an horizontal force as shown below, the triangular law of vector may be applied as shown below.

Triangular law of vectors states state that when three vectors F

w

W

T

F

T

acting act a point are in equilibrium, the vectors can be represented in magnitude and direction by the adjacent sides of a triangle by joining the head of one vector to the tail of the previous one.

Example: a 15 kg mass suspended from a ceiling is pulled asides with a horizontal force, F, as shown in the diagram above. Calculate the value of the tension T (g= 10m/s²) Solution:

60

0

150

N

T

F

60

0

T

F

Using the trigonometric ratio;

EXPERIMENT 2:

– An experiment to verify the Lami’s theorem using the force board. N.B. note that the resultant of a system in equilibrium is ZERO.

The triangular law of vectors is also called the Lami’s theorem

EVALUATION

1. The angle between two parallel vectors acting in opposite direction is —- (a) 0⁰ (b) 45⁰ (c) 90⁰ (d) 180⁰
2. The resultant of a system of forces is equilibrium is —- vector. (a) unit (b) free (c) orthogonal (d) null
3. What is the resultant of the forces 6N and 8N acting act an angle 60⁰ to each other?

(a) 9.0N (b) 10.1 N (c) 11.1N (d) 13.5N

1. Two vectors a and b act on a body. What will be the angel between a and b for the resultant to be maximum. (a) 0⁰ (b) 45⁰ (c) 90⁰ (d) 180⁰

PERIOD THREE: Resolution of vectors

Any position vector can be resolved into two components which are perpendicular to each other. Consider the vector P acting at angleϴ to the horizontal as shown below,

P

x

P

y

P

For a system which consist of several vectors, each vector in the system can be revolved into two components as shown above. V1

V₂

V

3

Consider a system of vectors as shown below,

If are the angles made the vectors respectively, then the component of the resultant along the horizontal is given as:

And the vertical component of the resultant vector is given as:

N.B the angles the vectors V₁, V₂, V₃, and V₄ makes with the positive x direction The magnitude resultant R is given as:

The direction of the resultant with respect to the positive x direction is given as

Example: a boy pull a nail from the wall with a string tied to the nail. The string is inclined to the wall at angle 600. If the tension in the string is 4N. What is the effective force used in pulling the nail?

Solution;

rope

nail

T

60

0

T_y = T cos60

60

0

T

x

=

T

sin

60

The tension has two components T_x and T_y. note that the value of T_x and T_y were obtained using the trigonometric ratio.

The component of T to extract the nail is T_x.

EXAMPLE: four forces act at appoint as shown below. Calculate the magnitude and direction of the resultant force. ₁₂N

40

0

15

N

9

N

30

0

10

N

Solution:

Vector F	Fi	Angles ϴ with +ve x direction	Fx = Fcos ϴ		Fy=Fsin ϴ
F1	10N	300	10cos30 =	8.66	10sin30= 5.00
F2	12N	(180 – 40) = 1400	12cos140=	– 9.192	12sin140= 7.713
F3	9N	(180 + 90) = 2700	9cos 270=	-0.000	9sin270= -9.00
F4	15N	(360 – 60) = 3000	15cos300=	7.50	15sin300= -12.99

OR

Vertical components

Direction of the resultant force

ϴ is negative. Tan ϴ is negative in the 2^nd and the 4^th quadrant. Looking at the geometry of the forces, R will be in the 4th quadrant.

This is the angle made by the resultant with the positive x –axis.

EVALUATION.

1. 1. The component of a force along the vertical and the horizontal axis is given as 24N and 7N respectively. What is the magnitude and direction of the resultant force?
   2. The resultant of two forces 12N and 5N is 13N. what is the angle between the two forces? (a) 0⁰ (b) 45⁰ (c) 90⁰ (d) 180⁰
   3. Two uniform velocities are represented by V₁ and V₂. If the angle between them is .

   Where 0⁰the magnitude of their resultant is —-(a) ^1/2 (b)

   (c) (d)

   1. Below is the diagram of an experiment to determine the resultant of a system using a force board, calculate the angle between the 25N and the 35N.

   50

   N

   25

   N

   1. The wind velocity is 30ms^-1, 30⁰ north of West. Find the component in the north and

   West direction

   1. A force of 15N acts in the positive x-direction. In what direction to the positive x-direction will a force of 20N be applied to give a resultant whose component along the x-direction is zero?

   GENERAL EVALUATION:

   1. A boy drag a heavy crate along the horizontal ground with a string inclined to the horizontal at 50⁰.if the tension in the string is 1500N, calculate the effective force pulling the crate along the ground.
   2. A body is in equilibrium under the action three forces. One of the force is 6.0N acting due East and one is 3.0N in a direction 60⁰ North of East. What is the magnitude and direction of the third force?
   3. Two forces acting at a point makes angles of 25⁰ and 65⁰ respectively with their resultant which is of the magnitude 15N. find the magnitudes of the two component forces.
   4. Differentiate between scalar and vector
   5. The resultant of two forces acting on an object is maximum if the angle between them is (a) 180⁰ (b) 90⁰ (c) 45⁰ (d) 0⁰

   WEEKEND ASSIGNMENT 1:

   1. 200inches due east is an example of —
      1. Displacement
      2. Speed
      3. Acceleration
      4. Distance
   2. 20km due east is an example of —
      1. Scalar quantity
      2. Vectorquantity
      3. Fundamental quantity
      4. Basic quantity
   3. 500 miles is an example of,..
      1. Displacement
      2. Distance
      3. Force
      4. Speed
   4. The following are examples of scalar quantities. Except —
      1. Time
      2. Density
      3. Velocity
      4. Mass
   5. One of the following is a vector quantity.
      1. Force
      2. Distance
      3. Mass
      4. Speed

   Essay

   Tabulate the below physical quantities into scalars and vectors.

   Force, acceleration, speed, velocity, time, mass, weight, distance, momentum, displacement.

   READING ASSIGNMENT

   Read up the topic: ‘’Work, Energy and Power’’ in the following text books.

   i. Senior Secondary School Physics by P.N.Okeke et al. ii. New School Physics for Senior Secondary Schools by Anyakoha, M.W.

   WEEKEND ASSSIGNEMT 2:

   • 1. Differentiate between a resultant of force and equilibrant
     2. Describe an experiment to determine the resultant of a system of force using the force board.

   READING ASSIGNMENT: Students should read pages 112-118 of New School Physics by MA Anyakoha and answer question 13 and 14 on page 121.