Class: Senior Secondary School 2 (SS2 / SSS 2)
Term: Third Term
Week: 7
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Dynamics
Previous Lesson: Dynamics: Meaning, Newton’s Laws and Inclined Plane Motion.
Topic: DYNAMICS
Subject Matter: Motion of connected particles, work, energy and power, impulse and momentum

Specific Objectives

By the end of the lesson, pupils should be able to:

Cognitive Domain:

Define work, energy, power, impulse, and momentum.
State the formulas for work, kinetic energy, potential energy, power, impulse, and momentum.
Explain the application of Newton’s laws of motion to connected particles.

Affective Domain:

Appreciate the importance of dynamics in understanding the motion of objects.
Show interest in solving problems related to forces and motion.

Psychomotor Domain:

Solve problems involving connected particles using Newton’s laws of motion.
Calculate work done, kinetic energy, potential energy, power, impulse, and momentum in various scenarios.
Draw clear force diagrams for systems of connected particles.

Collaborate with peers to discuss and solve problems on dynamics.
Communicate solutions to dynamics problems clearly and logically.

Reference Materials

The following resources were used in planning this lesson:

Senior Secondary Schools Education Curriculum
State Unified Scheme of Work
New Further Mathematics Project 2 for Senior Secondary Schools
https://www.example.com/dynamics-connected-particles
https://www.example.com/work-energy-power-momentum

Instructional Materials

The teacher will teach this lesson with the aid of:

An inclined plane with an object on it
Charts showing formulas for work, energy, power, impulse, and momentum
Whiteboard and markers or chalkboard and chalk

Rationale for the Lesson

This lesson helps pupils understand how forces affect the motion of objects. It provides fundamental concepts that are useful in fields like engineering, sports, and daily life, enabling pupils to analyze and predict motion.

Prerequisite/Previous Knowledge

Pupils are expected to have prior knowledge of basic concepts of force, mass, acceleration, and Newton’s laws of motion, as well as basic algebraic manipulation.

Lesson Content/Board Summary

DYNAMICS

1. Motion of Connected Particles

Connected particles are objects linked together, often by strings, ropes, or by being in direct contact. Their motion is interdependent, and Newton’s laws of motion are applied to each particle individually, considering forces like tension, friction, and gravity.

Newton’s Second Law: F = ma, where F is the net force, m is the mass, and a is the acceleration.

Worked Example 1: Connected Masses over a Smooth Pulley

Two masses, m₁ = 5 kg and m₂ = 3 kg, are connected by a light inextensible string passing over a smooth fixed pulley. Find the acceleration of the system and the tension in the string.

Solution:

Step 1: Draw a diagram and identify forces.
For m₁ (5 kg): Forces are T (upwards) and 5g (downwards). Assume m₁ moves down.
For m₂ (3 kg): Forces are T (upwards) and 3g (downwards). Assume m₂ moves up.

Step 2: Apply Newton’s Second Law (F=ma) to each mass.
For m₁ (moving downwards): 5g – T = 5a — (1)
For m₂ (moving upwards): T – 3g = 3a — (2)

Step 3: Solve the simultaneous equations.
Add (1) and (2): (5g – T) + (T – 3g) = 5a + 3a
2g = 8a
a = 2g / 8 = g / 4

Using g = 9.8 m/s²: a = 9.8 / 4 = 2.45 m/s²

Step 4: Substitute ‘a’ back into one equation to find T.
Using (2): T – 3g = 3a
T = 3g + 3a = 3(9.8) + 3(2.45)
T = 29.4 + 7.35 = 36.75 N

Therefore, the acceleration is 2.45 m/s² and the tension is 36.75 N.

Worked Example 2: Blocks in Contact on a Smooth Horizontal Surface

Two blocks of masses M₁ = 4 kg and M₂ = 2 kg are placed in contact on a smooth horizontal surface. A horizontal force F = 30 N is applied to M₁ towards M₂. Find the acceleration of the system and the force of contact between the blocks.

Solution:

Step 1: Consider the blocks as a single system to find acceleration.
Total mass (M) = M₁ + M₂ = 4 kg + 2 kg = 6 kg
Net force (F) = 30 N
Using F = Ma: 30 = 6a
a = 30 / 6 = 5 m/s²

Step 2: Isolate one block to find the contact force.
Let R be the force of contact exerted by M₁ on M₂ (and vice-versa).
Consider M₂ (2 kg block): The only horizontal force acting on M₂ is R, from M₁.
Using F = ma for M₂: R = M₂a
R = 2 kg × 5 m/s² = 10 N

Alternatively, consider M₁ (4 kg block):
Forces on M₁ are F (30 N) in the direction of motion, and R (contact force from M₂) opposing the motion.
Using F = ma for M₁: 30 – R = M₁a
30 – R = 4 kg × 5 m/s²
30 – R = 20
R = 30 – 20 = 10 N

Therefore, the acceleration of the system is 5 m/s² and the force of contact is 10 N.

2. Work, Energy, and Power

Work Done (W)

Work is done when a force causes a displacement of an object in the direction of the force. It is a scalar quantity.

Formula: W = Fd cosθ
Where: F = force (N), d = displacement (m), θ = angle between force and displacement.
Unit: Joules (J).

Example: A force of 50 N pulls an object 10 m horizontally. Work done = 50 N × 10 m = 500 J.

Energy (E)

Energy is the capacity to do work. It exists in various forms, including kinetic and potential energy.

Unit: Joules (J).

The following are types of mechanical energy:

Kinetic Energy (KE): Energy possessed by an object due to its motion.
Formula: KE = ½mv²
Where: m = mass (kg), v = velocity (m/s).

Example: A 2 kg object moving at 3 m/s has KE = ½ × 2 × (3)² = 9 J.
Potential Energy (PE): Energy possessed by an object due to its position or state. For gravitational potential energy:
Formula: PE = mgh
Where: m = mass (kg), g = acceleration due to gravity (m/s²), h = height (m).

Example: A 5 kg object at a height of 2 m (g=9.8 m/s²) has PE = 5 × 9.8 × 2 = 98 J.

Power (P)

Power is the rate at which work is done or energy is transferred. It is a scalar quantity.

Formula: P = W/t = Fv
Where: W = work done (J), t = time (s), F = force (N), v = velocity (m/s).
Unit: Watts (W).

Example: If 500 J of work is done in 5 seconds, Power = 500 J / 5 s = 100 W.

3. Impulse and Momentum

Momentum (p)

Momentum is a measure of the quantity of motion an object possesses. It is a vector quantity.

Formula: p = mv
Where: m = mass (kg), v = velocity (m/s).
Unit: kilogram-meter per second (kg m/s) or Newton-second (Ns).

Principle of Conservation of Momentum: In an isolated system, the total momentum before a collision or interaction is equal to the total momentum after the collision or interaction.

Formula: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
Where: u = initial velocity, v = final velocity.

Example: A 10 kg mass moving at 5 m/s has momentum = 10 kg × 5 m/s = 50 kg m/s.

Impulse (I)

Impulse is the change in momentum of an object. It is a vector quantity.

Formula: I = Ft = Δp = mv – mu
Where: F = force (N), t = time interval (s), Δp = change in momentum, m = mass (kg), u = initial velocity (m/s), v = final velocity (m/s).
Unit: Newton-second (Ns) or kilogram-meter per second (kg m/s).

Example: A 2 kg ball changes its velocity from 10 m/s to 4 m/s. Impulse = 2(4) – 2(10) = 8 – 20 = -12 Ns. (The negative sign indicates impulse in the opposite direction of initial motion).

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher greets the pupils, reviews the previous lesson on Newton’s laws of motion, and introduces the topic “Dynamics” by asking pupils what they understand by motion and forces.
Pupils’ Activity: Pupils respond to the greetings, recall previous knowledge, and share their ideas about motion and forces.
Learning Point: Pupils connect previous knowledge to the new topic and are prepared for the lesson.

Step 2: Motion of Connected Particles (Part 1)

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher explains the concept of connected particles and guides pupils through setting up equations for masses connected by a string over a smooth pulley, solving Worked Example 1 on the board.
Pupils’ Activity: Pupils listen attentively, ask questions, and follow the steps in solving the problem.
Learning Point: Pupils understand how to apply Newton’s laws to connected particles over a pulley.

Step 3: Motion of Connected Particles (Part 2)

Time: 7 minutes
Teaching Skill: Problem Solving/Guidance
Teacher’s Activity: The teacher guides pupils through solving problems involving blocks in contact on a horizontal surface, using Worked Example 2.
Pupils’ Activity: Pupils actively participate in solving the example problem, contributing steps and calculations.
Learning Point: Pupils learn to solve problems involving contact forces between connected blocks.

Step 4: Work Done

Time: 5 minutes
Teaching Skill: Explanation/Definition
Teacher’s Activity: The teacher defines work done, states its formula (W=Fd cosθ) and unit, and provides a simple example.
Pupils’ Activity: Pupils copy the definition, formula, and example into their notes.
Learning Point: Pupils understand the concept and calculation of work done.

Step 5: Energy

Time: 5 minutes
Teaching Skill: Explanation/Elaboration
Teacher’s Activity: The teacher defines energy, explains kinetic and potential energy, states their formulas (KE=½mv², PE=mgh), and provides examples for each.
Pupils’ Activity: Pupils listen, take notes on the definitions, formulas, and examples of different forms of energy.
Learning Point: Pupils understand the different types of mechanical energy and their calculations.

Step 6: Power

Time: 4 minutes
Teaching Skill: Explanation/Application
Teacher’s Activity: The teacher defines power, states its formulas (P=W/t, P=Fv) and unit, and gives a practical example.
Pupils’ Activity: Pupils write down the definition, formulas, and example for power.
Learning Point: Pupils learn how to define and calculate power.

Step 7: Impulse and Momentum

Time: 6 minutes
Teaching Skill: Explanation/Comparison
Teacher’s Activity: The teacher defines momentum and impulse, states their formulas (p=mv, I=Ft=Δp), and provides examples for each concept.
Pupils’ Activity: Pupils note down the definitions, formulas, and examples, paying attention to the relationship between impulse and momentum.
Learning Point: Pupils understand the concepts of impulse and momentum and their calculations.

Step 8: Evaluation/Review

Time: 5 minutes
Teaching Skill: Questioning/Assessment
Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

Define work done and state its SI unit.
State the formula for kinetic energy and explain the terms involved.
A 6 kg mass and a 4 kg mass are connected by a light inextensible string over a smooth pulley. Calculate the acceleration of the system. (Take g = 10 m/s²)
What is impulse? How is it related to momentum?

Pupils’ Activity: Pupils answer orally and in writing.
Learning Point: Pupils demonstrate understanding of the lesson.

Step 9: Conclusion

Time: 3 minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key concepts of dynamics covered, emphasizing the formulas and problem-solving techniques. The teacher then assigns homework involving more problems on connected particles, work, energy, power, impulse, and momentum.
Pupils’ Activity: Pupils listen to the summary and copy down the assigned homework.
Learning Point: Pupils consolidate their understanding and are given tasks to practice the concepts.

Lesson Keywords

Dynamics – The branch of mechanics concerned with the study of forces and their effect on motion.
Connected particles – Objects linked together whose motions are interdependent.
Tension – The pulling force transmitted axially by a string, rope, chain, or similar object.
Work – The energy transferred to or from an object by applying force along a displacement.
Energy – The capacity of a physical system to perform work.
Power – The rate at which work is done or energy is transferred.
Impulse – The change in momentum of an object.
Momentum – The product of an object’s mass and velocity.
Newton’s Laws – Fundamental principles describing the relationship between a body and the forces acting upon it, and its motion in response to those forces.

Differentiation

For struggling learners, the teacher will provide simplified examples and offer additional one-on-one guidance during problem-solving sessions. Advanced learners will be given more complex problems involving inclined planes or friction to challenge their understanding and problem-solving skills.

Note for teachers using this lesson plan

Teachers should ensure that pupils have a strong grasp of vector notation and resolving forces. Emphasize drawing clear free-body diagrams for connected particles and consistent use of units. Encourage pupils to show all steps in their calculations to avoid errors and understand the problem-solving process.