Objectives (5 - 7 minutes)

1. Understand the concept of Simple Harmonic Motion (SHM) and its application in nature.

2. Learn to derive the equation of simple harmonic motion from the forces involved and Newton's laws.

3. Apply the SHM equation to determine the physical quantities involved in a system subject to this type of motion, such as amplitude, frequency, period, kinetic and potential energy.

Secondary Objectives:

• Develop problem-solving skills in the field of physics, using mathematics as a fundamental tool.
• Stimulate critical thinking and abstraction skills, relating the theory studied to observable phenomena in everyday life.
• Promote interaction and collaboration among students through group activities and class discussions.

Introduction (10 - 15 minutes)

1. Recalling Previous Content: The teacher should start the lesson by recalling the concepts of motion, force, and energy that were previously studied. It is important for students to have a good understanding of these concepts to comprehend Simple Harmonic Motion (SHM) and the equation of motion. (3 - 5 minutes)

2. Problem Situation 1: The teacher should propose the following problem situation: Imagine a spring attached to a wall with a mass hanging on the other end. If the mass is displaced from the equilibrium position and released, it will oscillate back and forth. How can we mathematically describe this motion? (2 - 3 minutes)

3. Contextualization: The teacher should highlight the importance of SHM in nature and practical applications, such as describing the motion of pendulums, springs, sound waves, planetary motion, among others. Examples from everyday life can be mentioned, such as the swinging of a clock pendulum, the swaying of a hanging earring, among others. (2 - 3 minutes)

4. Curiosities: The teacher can share some curiosities to spark students' interest. For example, it can be mentioned that SHM is one of the most common motions in nature and, therefore, one of the first topics studied in physics. Additionally, it can be mentioned that SHM is used in various technological applications, such as in the construction of precise clocks and in earthquake measurements. (2 - 3 minutes)

5. Problem Situations 2 and 3: The teacher should present two more problem situations:

   • An object is attached to a spring and pulled down and then released. How can we calculate its maximum velocity and maximum acceleration?
   • If the object is subjected to a force opposite to its direction of motion, how can we calculate the time needed for it to come to a complete stop? (2 - 3 minutes)

Development (20 - 25 minutes)

1. Theory of Simple Harmonic Motion (SHM) (10 - 12 minutes)

   • Definition and Characteristics: The teacher should start by explaining that SHM is a periodic motion, where the resultant force is proportional to the displacement of the object from its equilibrium position and always points towards that position. It should be emphasized that the acceleration is maximum at the maximum distance position from the equilibrium position and null at the equilibrium position.
   • Equation of Motion: The teacher should then present the equation of SHM: x(t) = A * cos(ωt + φ), where x is the position of the object as a function of time, A is the amplitude of the motion, ω is the angular frequency (2πf, where f is the frequency), t is the time, and φ is the initial phase.
   • Deriving the Equation of Motion: The teacher should demonstrate how the equation of SHM can be derived from Newton's laws and the forces involved. It should be shown that the restoring force (F = -kx, where k is the spring constant) and the second law of Newton (F = ma) lead to the differential equation x'' + (k/m)x = 0, whose solution is the equation of SHM.
   • Units of Measurement: The teacher should explain the units of measurement involved in the equation of SHM: x in meters, A in meters, ω in radians per second, t in seconds, and φ in radians.

2. Solving the Equation of SHM (5 - 7 minutes)

   • Determining the Amplitude: The teacher should teach how to determine the amplitude of the motion from the graph of position versus time or from the equation of motion.
   • Determining the Frequency and Period: The teacher should explain how to determine the frequency and period of the motion from the equation of motion.
   • Determining the Initial Phase: The teacher should demonstrate how to determine the initial phase of the motion from the graph of position versus time.
   • Determining the System's Energy: The teacher should teach how to determine the kinetic and potential energy of the system from the equation of motion.

3. Solving the Problem Situations (5 - 6 minutes)

   • Situation 1: The teacher should solve the first problem situation presented in the Introduction, calculating the object's maximum velocity and maximum acceleration.
   • Situation 2: The teacher should solve the second problem situation, calculating the time needed for the object to come to a complete stop.

4. Guided Practice (5 - 7 minutes)

   • The teacher should propose some exercises for students to solve in class, with the teacher's guidance. The exercises should involve the application of the concepts learned in the lesson.
   • Students should be encouraged to discuss the exercise solutions in groups, promoting interaction among them.

Feedback (10 - 12 minutes)

1. Group Discussion (3 - 4 minutes)

   • The teacher should facilitate a group discussion, where each group will have the opportunity to share their resolutions for the problem situations proposed in the Introduction.
   • Each group should explain how they arrived at their resolution, which formulas and steps they used, and how they interpreted the results.
   • The teacher should encourage other groups to ask questions and make comments, promoting a collaborative learning environment.

2. Connection with Theory (3 - 4 minutes)

   • The teacher should then make the connection between the group resolutions and the theory presented.
   • It should be highlighted how the concepts of SHM and the equation of motion were applied to solve the problem situations.
   • The teacher should reinforce the most important concepts, clarify any doubts, and correct any possible misconceptions.

3. Individual Reflection (2 - 3 minutes)

   • The teacher should suggest that students engage in individual reflection on what they learned in the lesson.
   • Questions like: What was the most important concept you learned today? What questions have not been answered yet? should be asked.
   • Students should have a minute to think about these questions and then have the opportunity to share their answers with the class.
   • The teacher should encourage students to express their opinions and doubts, and should answer all questions to the best of their ability, reinforcing concepts that have not been understood yet.

4. Feedback and Closure (2 - 3 minutes)

   • Finally, the teacher should provide overall feedback on the lesson, praising strengths and suggesting improvements for future lessons.
   • The lesson should be concluded by reinforcing the importance of SHM and the equation of motion, and motivating students to continue studying the topic at home.
   • The teacher should also announce the theme of the next lesson and any homework assignments, if applicable.

Conclusion (5 - 7 minutes)

1. Summary of Contents (2 - 3 minutes)

   • The teacher should start the Conclusion by summarizing the main points covered during the lesson. It should review the definition of Simple Harmonic Motion (SHM), its characteristics, the derivation of the equation of motion, and how the SHM equation can be used to determine the physical quantities of the system in SHM.
   • It should also review the problem situations presented and how they were solved using the SHM theory.
   • It is important for the teacher to make this summary clear and concise so that students can consolidate what they learned during the lesson.

2. Theory-Practice Connection (1 - 2 minutes)

   • The teacher should emphasize how the lesson connected the theory of SHM with practice, through the resolution of the problem situations.
   • It should be reinforced that the theoretical understanding of SHM allowed students to solve practical problems, and that practice, in turn, helped reinforce theoretical understanding.

3. Additional Materials (1 - 2 minutes)

   • The teacher should suggest additional materials for students who wish to deepen their knowledge of SHM.
   • Physics books, websites, YouTube videos, among other resources that explain SHM clearly and didactically can be recommended.
   • Students should be encouraged to explore these materials at home to complement what they learned in the classroom.

4. Practical Application (1 minute)

   • Finally, the teacher should reinforce the importance of SHM and the equation of motion in everyday life and in different areas of science and technology.
   • Examples of SHM applications, such as the motion of pendulums in clocks, sound wave motion, among others, and how the SHM equation is used to describe and predict these motions can be mentioned again.
   • The lesson should end by encouraging students to observe SHM around them and reflect on how physics is present in our daily lives.