Contextualization

Introduction to Simple Harmonic Motion

Simple Harmonic Motion (SHM) is one of the fundamental concepts in Physics, being the mathematical description of phenomena that occur in a periodic and oscillatory way, such as the swing of a pendulum or the movement of a spring. SHM is characterized by having a movement that repeats itself at equal intervals of time, and can be described by sine or cosine functions.

The equation of Simple Harmonic Motion is represented by the following formula: x(t) = A * cos(wt + ø), where x(t) is the position of the object in relation to time, A is the amplitude of the movement, w is the angular frequency of the movement and ø is the initial phase. The study of these equations is extremely important since they are used to describe a wide variety of physical phenomena.

The SHM also has a direct relationship with the mechanical energy of a system. In an idealized oscillatory motion (no friction or other dissipative forces), the mechanical energy is conserved and is periodically transformed between kinetic energy and potential energy.

Importance of SHM

The Simple Harmonic Motion is ubiquitously observed in nature and in several technologies. For example, in quantum physics, the movement of a particle inside a quantum potential well is well approximated by an SHM. In engineering, the principle of oscillations is used in the design of clocks and in the construction of bridges and buildings.

Learning how to describe SHM mathematically and intuitively is crucial to the study of any discipline involving waves because a wave is nothing more than the propagation of a perturbation - and therefore, a type of SHM. Whether in sound waves, light waves, waves on strings, etc., SHM is always present.

Sources of Study

Students can deepen their knowledge about Simple Harmonic Motion and its equations using the following resources:

1. Physics, by Halliday, Resnick and Walker - is one of the most renowned and used Physics books in college courses, available in most libraries.
2. SHM at Khan Academy - this is a detailed and intuitive explanation about SHM, with exercises for fixation.
3. Creating Waves - SHM - Detailed explanation with demonstrations and animations that help visualize the oscillatory motion.

Practical Activity: "Unveiling Simple Harmonic Motion"

Project Goal

The purpose of this project is to allow students to apply their knowledge about simple harmonic motion and the SHM equation in a practical experiment.

Project Description

The project will be carried out in groups of 3 to 5 students. The task is to create a simple harmonic oscillator, observe its movement, collect data and analyze it using the SHM equation.

Necessary Materials

• 1 spring
• 1 object with known weight (to be attached to the spring)
• 1 string (to attach the spring)
• 1 ruler
• Stopwatch
• Cell phone camera (optional)

Practical Activity Detailed Step-by-Step

Step 1: Preparation

1. Hang the spring in a place where it can oscillate freely.
2. Attach the object of known weight to the spring.
3. Ensure that the system is able to execute an oscillatory movement.

Step 2: Running the Experiment

4. Move the object until the spring is extended and release it so that it starts to oscillate.
5. Use a stopwatch to measure the time the object takes to complete 10 full oscillations. Repeat this measurement for five times to obtain an average time.
6. Record all this data.

Step 3: Data Analysis

7. Determine the oscillation period by dividing the average time obtained by 10.
8. Calculate the angular frequency (w) using the formula w = 2π/T (where T is the period).
9. From the data, plot a graph of the oscillatory motion.

Step 4: Conclusions

10. Compare if the object's movement hanging on the spring fits the definition and characteristics of simple harmonic motion.
11. Discuss whether the experimental results agree with the studied theory.
12. Discuss possible sources of error and their effects on the results.

Project Delivery

After the experiment and data analysis are completed, each group must prepare a detailed report, containing the following topics:

1. Introduction: Contextualize the project theme, explaining what SHM is and its importance. Then, present the goal of the experiment carried out.
2. Development: Describe the experiment in detail, its methodology and the results obtained. Discuss the theory involved and compare it with the results of the experiment.
3. Conclusion: Resume the main points of the work, the learning obtained and draw up a synthesis of the results and their conclusions about the project.
4. Bibliography: Indicate the sources consulted for the development of the project.

Remember that this report is not only an exhibition of results, but also an opportunity for you to show your analytical and problem-solving skills, critical thinking and teamwork.