Objectives (5 - 7 minutes)

1. Understand the concept of Simple Harmonic Motion (SHM) and how it applies to the study of a mass attached to a spring.

2. Develop the ability to calculate the frequency, period, and amplitude of an SHM, as well as the potential and kinetic energy of the system.

3. Apply the acquired knowledge to solve practical and theoretical problems related to the SHM of a mass attached to a spring.

Secondary Objectives:

• Foster critical and analytical thinking of students in solving physics problems.

• Encourage teamwork, as solving complex problems often requires collaboration from different perspectives and skills.

• Promote the application of theoretical physics concepts to real-world situations, helping students understand the relevance and usefulness of these concepts.

Introduction (10 - 15 minutes)

1. Review of previous concepts: The teacher starts the class by recalling concepts that are fundamental to the understanding of the topic to be addressed, such as the concept of force, energy, and motion. He may ask students questions to verify if they remember these concepts and, if necessary, briefly reinforce them. (3 - 5 minutes)

2. Problem situations: The teacher presents two problem situations to arouse students' interest and contextualize the topic. The first situation could be as follows: 'Imagine we have a pendulum of mass m attached to a spring. If we displace the mass and release it, it will start to oscillate. What determines the speed at which it oscillates and the distance it travels in each oscillation?' The second situation could be: 'How could we calculate the potential and kinetic energy of this system at different points of its oscillation?' (5 - 7 minutes)

3. Contextualization: The teacher explains that the study of Simple Harmonic Motion (SHM) is fundamental in various areas of physics and engineering. For example, it is used to understand the motion of subatomic particles, the behavior of sound waves, and the vibration of structures such as bridges and buildings. In addition, the teacher may mention practical applications of SHM, such as the creation of pendulum clocks and the determination of the mass of particles in particle accelerators. (2 - 3 minutes)

4. Curiosities: To further arouse students' interest, the teacher can share some curiosities about SHM. For example, he may mention that the motion of a pendulum is considered the most classic example of SHM and that the study of SHM was essential for the development of Einstein's theory of relativity. Additionally, the teacher can tell the story of how Galileo's discovery of SHM in the 17th century revolutionized the field of physics. (2 - 3 minutes)

Development (20 - 25 minutes)

1. Theory of Simple Harmonic Motion (SHM) (7 - 10 minutes):

   • The teacher introduces the concept of SHM, explaining that it is a periodic motion that occurs when the restoring force is directly proportional to the displacement and opposite to it.
   • He should then present the mathematical formula for SHM: F = -kx, where F is the restoring force, k is the spring constant, and x is the displacement of the mass relative to the equilibrium position.
   • The teacher should explain that, according to Newton's second law, F = ma, and that, in SHM, the acceleration is proportional and opposite to the displacement, leading to the SHM equation: a = -(k/m)x.
   • He can use a diagram or a physical model to illustrate these concepts.

2. Calculation of Frequency, Period, and Amplitude (5 - 7 minutes):

   • The teacher should explain that the frequency (f) of an SHM is the number of complete cycles the system performs in one second and that the period (T) is the time required for the system to complete one full cycle.
   • He should then present the formulas for calculating f and T: f = 1/T and T = 1/f.
   • The teacher should explain that the amplitude (A) of an SHM is the maximum distance the mass moves relative to the equilibrium position and that the system's energy is proportional to the square of the amplitude.
   • He should present the formula for calculating the amplitude: A = -x_max and the formula for calculating the system's energy: E = (1/2)kA^2.

3. Problem Solving (8 - 10 minutes):

   • The teacher should present some theoretical and practical problems involving the SHM of a mass attached to a spring and guide students in solving these problems.
   • He should start with simpler problems and gradually increase the complexity.
   • The teacher should encourage students to apply the concepts they have learned and the formulas that have been presented to solve the problems.
   • He should also encourage discussion and collaboration among students, in order to promote teamwork and critical thinking.

4. Practical Examples (3 - 5 minutes):

   • The teacher should present some practical examples of applications of the SHM of a mass attached to a spring.
   • For example, he can show how pendulum clocks work based on SHM, or how particle accelerators use SHM to determine the mass of subatomic particles.
   • The teacher should explain how the concepts learned in the class are applied in these examples, in order to reinforce students' understanding and demonstrate the relevance and usefulness of these concepts.

Return (8 - 10 minutes)

1. Connection to the real world (3 - 4 minutes): The teacher should encourage students to relate the content learned to the world around them. He can do this by asking students to think of everyday situations involving SHM. Some examples may include the movement of a clock pendulum, the movement of a spring on a trampoline, or the behavior of sound waves. The teacher should guide students to explain how the concepts and formulas learned in class apply to these situations.

2. Reflection on learning (2 - 3 minutes): The teacher should then ask students to reflect on what they learned in class. He can ask questions such as:

   • What was the most important concept you learned today?
   • What questions have not been answered yet?
   • How can you apply what you learned today in other areas of your study or life?

   The teacher should give students time to think about these questions and, if possible, encourage discussion in the classroom. This reflection will help students consolidate what they have learned and identify any gaps in their understanding.

3. Feedback on the class (2 - 3 minutes): Finally, the teacher should ask for feedback from students about the class. He can ask students to evaluate how well they understood the content, which teaching strategies were most effective, and what could be improved in the next class. Student feedback is valuable to the teacher as it helps adjust and improve teaching practices.

4. Preparation for the next class (1 minute): The teacher should then inform students about the topic of the next class and if there is any reading material or homework assignment they need to complete before the next class. This will help students prepare for the next class and continue learning outside the classroom.

Conclusion (5 - 7 minutes)

1. Recapitulation of content (2 - 3 minutes): The teacher should summarize the main points discussed in class, recalling the concepts of Simple Harmonic Motion (SHM), how to calculate the frequency, period, and amplitude of an SHM, and how to calculate the potential and kinetic energy of the system. He can do this interactively by asking students to recapitulate the concepts and formulas.

2. Connection of theory, practice, and applications (1 - 2 minutes): The teacher should highlight how the class connected theory, practice, and applications. He can reinforce that through theory, students were able to understand the fundamental principles of SHM, and that solving practical problems allowed them to apply these principles. Additionally, he can remind students of the SHM applications that were discussed, showing how theoretical physics is relevant to the real world.

3. Extra materials (1 minute): The teacher should suggest extra materials for students who wish to deepen their understanding of SHM. This may include textbooks, online videos, interactive simulation websites, and additional problems to solve. He should encourage students to explore these resources on their own and to bring any questions they may have to the next class.

4. Importance of the subject (1 minute): Finally, the teacher should emphasize the importance of SHM for everyday life and for science. He may mention again the SHM applications that were discussed, emphasizing how these principles are used in many aspects of our lives, from creating clocks to understanding the behavior of subatomic particles. Additionally, he can remind students that the study of SHM is a fundamental step in understanding more advanced concepts in physics, and that by mastering these concepts, they will be prepared to successfully face future challenges in the discipline.