Lesson Plan | Traditional Methodology | Simple Harmonic Motion: Definition

Objectives

Duration: (10 - 15 minutes)

The purpose of this stage is to establish the objectives of the lesson, providing a clear view of what students should learn. This will help guide the focus of the lesson, ensuring that students understand the definition and characteristics of Simple Harmonic Motion, as well as the ability to identify and verify the presence of this type of motion in different situations.

Main Objectives

1. Understand that Simple Harmonic Motion (SHM) is characterized by an acceleration that is directly proportional and opposite to the displacement.

2. Identify the necessary conditions for an object to be in SHM.

3. Apply the theoretical concepts of SHM to determine whether an object is in SHM or not.

Introduction

Duration: (10 - 15 minutes)

The purpose of this stage is to spark students' interest in the topic of the lesson, providing an initial context that connects theoretical content to everyday life and the world around them. This initial engagement is crucial to ensure that students are motivated to learn more about Simple Harmonic Motion and its applications.

Context

Start the lesson with a brief review of the concepts of motion and force, reminding students how force can influence the motion of an object. Explain that today the lesson will address a specific type of motion known as Simple Harmonic Motion (SHM), which is widely observed in nature and in man-made systems. Use examples such as the motion of a clock pendulum or the oscillations of a spring to illustrate SHM.

Curiosities

Did you know that Simple Harmonic Motion is the basis for the functioning of many musical instruments, such as guitars and violins? When a string is played, it vibrates in a pattern that can be described by SHM, producing sounds that are pleasant to the human ear. Furthermore, the principles of SHM are used in many technological devices, such as accelerometers in smartphones.

Development

Duration: (40 - 50 minutes)

The purpose of this stage is to deepen students' understanding of Simple Harmonic Motion (SHM) through a detailed explanation of theoretical concepts, practical examples, and problem-solving. This will allow students to consolidate the knowledge acquired and apply the concepts in practical situations, developing analytical and critical skills.

Covered Topics

1. Definition of Simple Harmonic Motion (SHM): Explain that SHM is a type of oscillatory motion where the restoring force is directly proportional to the displacement and acts in the opposite direction. Use the equation F = -kx to illustrate this concept. 2. Displacement, Velocity, and Acceleration in SHM: Detail how displacement (x), velocity (v), and acceleration (a) vary with time in an SHM. Use graphs to show the relationship between these quantities and time. 3. Energy in SHM: Explain the conservation of energy in an SHM system, addressing kinetic and potential energies. Use the total energy equation E = 1/2 kA² to illustrate how energy is distributed in the motion. 4. Practical Examples of SHM: Provide practical everyday examples that involve SHM, such as the simple pendulum, the mass-spring system, and oscillations of an LC circuit. Explain each example in detail, including specific motion equations.

Classroom Questions

1. 1. Considering an ideal mass-spring system, if the mass is 2 kg and the spring constant is 50 N/m, what is the angular frequency of the system? 2. 2. A simple pendulum has a length of 1 meter. What is the oscillation period of this pendulum in an environment where the gravitational acceleration is 9.8 m/s²? 3. 3. An object in SHM has an amplitude of 0.5 meters and a spring constant of 100 N/m. What is the total energy of the system?

Questions Discussion

Duration: (20 - 25 minutes)

The purpose of this stage is to consolidate students' learning through discussion and detailed analysis of the resolved questions. This moment allows students to review key concepts, clarify doubts, and reinforce their understanding of the principles of Simple Harmonic Motion. Student engagement is encouraged through reflective questions that promote critical thinking and practical application of the studied concepts.

Discussion

• Discussion of the Presented Questions:

• 1. Angular Frequency of a Mass-Spring System:
• • Data: mass (m) = 2 kg, spring constant (k) = 50 N/m.
• • Formula: ω = √(k/m)
• • Calculation: ω = √(50/2) = √25 = 5 rad/s.
• • Explanation: The angular frequency (ω) is the rate at which the system oscillates in radians per second. In this case, a mass of 2 kg and a spring constant of 50 N/m results in an angular frequency of 5 rad/s.
• 2. Oscillation Period of a Simple Pendulum:
• • Data: length of the pendulum (L) = 1 m, gravitational acceleration (g) = 9.8 m/s².
• • Formula: T = 2π√(L/g)
• • Calculation: T = 2π√(1/9.8) ≈ 2π√(0.102) ≈ 2π(0.32) ≈ 2 s.
• • Explanation: The period (T) is the time that the pendulum takes to complete one full oscillation. With a length of 1 meter and a gravitational acceleration of 9.8 m/s², the oscillation period is approximately 2 seconds.
• 3. Total Energy of a System in SHM:
• • Data: amplitude (A) = 0.5 m, spring constant (k) = 100 N/m.
• • Formula: E = 1/2 kA²
• • Calculation: E = 1/2 * 100 * (0.5)² = 1/2 * 100 * 0.25 = 12.5 J.
• • Explanation: The total energy (E) is the sum of the kinetic and potential energies in an SHM system. With an amplitude of 0.5 meters and a spring constant of 100 N/m, the total energy of the system is 12.5 joules.

Student Engagement

1. Questions and Reflections to Engage Students: 2. 1. How does angular frequency change if the mass of the mass-spring system is increased? Explain based on the formula. 3. 2. If the length of the pendulum were doubled, how would this affect the oscillation period? Justify your answer. 4. 3. In a mass-spring system, if the amplitude of the motion were reduced by half, what would be the new total energy of the system? Show your calculations. 5. 4. What are some practical applications of Simple Harmonic Motion that you can observe in your daily life beyond the examples discussed in class? 6. 5. How does the concept of energy conservation apply to other types of oscillatory motions, such as waves on a guitar string or vibrations in a tuning fork?

Conclusion

Duration: (10 - 15 minutes)

The purpose of this stage is to review and consolidate the main concepts covered during the lesson, ensuring that students have a clear and complete understanding of Simple Harmonic Motion. This review moment helps reinforce learning and clarify any remaining doubts, ensuring that students are well-prepared to apply concepts in future academic and practical situations.

Summary

• Simple Harmonic Motion (SHM) is an oscillatory motion where the restoring force is directly proportional to the displacement and acts in the opposite direction.
• The basic equation that describes SHM is F = -kx.
• Displacement, velocity, and acceleration in SHM vary sinusoidally with time.
• The total energy in an SHM is conserved and can be described by the formula E = 1/2 kA².
• Practical examples of SHM include the simple pendulum, the mass-spring system, and oscillations in an LC circuit.

The lesson connected theory with practice by presenting real and everyday examples of SHM, such as the movement of pendulums and springs, and by solving practical problems that helped students visualize how theoretical concepts are applied in real-world situations.

The study of Simple Harmonic Motion is essential for understanding natural and technological phenomena. For example, SHM is fundamental to music, as it describes the vibrations of instrument strings. Additionally, it is used in technological devices such as smartphones, where motion sensors, based on SHM principles, are widely employed.