Contextualization
Circular or rotary movements are omnipresent in our daily life - analog clocks, car wheels, planets orbiting the sun - are all examples of circular movements. Understanding the characteristics of these movements, such as velocity, is crucial in physics and engineering. The relationship between velocities in circular motions is a fundamental concept in physics and is widely used in mechanical engineering, atmospheric sciences, astrophysics, and many other disciplines that involve the motion of bodies.
There are two types of velocities in a circular motion: linear velocity and angular velocity. Linear velocity, sometimes referred to as tangential velocity, is the velocity you would have if you were to move along a straight line tangent to the circle at a given point. Angular velocity is the rate of change of the angle with respect to time, and it takes into account the circular path that the point is traveling.
The relationship between these two velocities is given by the formula v = ωr, where v is the tangential velocity, ω is the angular velocity and r is the radius of the circle. This formula shows that the tangential velocity is directly proportional to the angular velocity and the radius. Thus, when the radius is constant, increasing the angular velocity will result in an increase in the tangential velocity, and vice versa.
This might all sound a bit abstract, but let's bring this down to the real world. For instance, when you ride a bicycle, your bike's velocity (tangential velocity) can be determined by how fast you pedal (the angular velocity of the wheels) combined with the size of the wheels (the radius). That is, velocities in circular motion are not only relevant to theoretical physics, but have tangible and direct applications in our everyday lives.
Practical Activity: Race of Velocities - Linear Vs Angular
Project Objective
The objective of this project is to materialize the theoretical concept of the relationship between velocities in circular motions and its interaction, through a playful and engaging hands-on activity. The students will perform a "race" involving circles of different sizes in order to verify the formula v = ωr in action.
Project Description
In this activity, the students will be divided into groups of 3 to 5 participants. Each group will build two "racetracks" out of cardboard or plastic (it can be an empty PET bottle cut in half): a big one and a small one. The "racetrack" will be a circular strip, like a small running track.
The groups will then hold a race competition between a spinning top, which will represent the "angular velocity" and a marble or similar sphere, which will represent the "linear velocity". Both will travel the track/racetrack in the same amount of time, and the students will observe the relationship between the linear velocity of the marble and the angular velocity of the spinning top, in relation to the size of the racetrack (big and small).
Materials
- Cardboard or PET bottles (to build the racetracks)
- Spinning tops (representing the angular velocity)
- Marbles or similar spheres (representing the linear velocity)
- Stopwatches (to measure the time)
- Ruler or measuring tape (to measure the radius of the tracks)
- Pens and paper (to take notes of observations and results)
Step by Step
- Divide the class into groups of 3 to 5 students.
- Each group must build two racetracks, a big one and a small one.
- Measure and write down the radius of each racetrack.
- Start the "race" with the marble and the spinning top in each racetrack, at the same time.
- Use the stopwatch to measure the time it took for the marble and the spinning top to complete one lap in each racetrack.
- Write down the times.
- Calculate the linear velocity of the marble and the angular velocity of the spinning top, using the formula v = ωr.
- Compare the results obtained in both racetracks and discuss the results in the group.
The activity will take around 5 to 10 hours to be fully executed, considering the time to build the racetracks, to perform the "races", the measurements, the calculations, and the group discussions.
Final Outcome
After carrying out the hands-on activity, the students must elaborate a report containing the following topics:
Introduction: Contextualization of the theme and the project, justification of the relevance of the theme and objectives of the work carried out.
Development: Details of the step by step of the activity, with the description of the construction of the racetracks, the performance of the races, the measurements and the calculations. In this topic, it is expected that the students discuss important details such as the concept of angular and linear velocity, how the calculation of these velocities was performed and what observations they were able to make throughout the activity. They must also present and discuss the results obtained by comparing the velocities in the big racetrack and the small racetrack.
Conclusion: The students must revisit the main points of the work, highlight the learnings obtained in practice about the relationship between velocities in circular motions, and draw conclusions about how the size of the racetrack (radius) influences the linear and angular velocities.
Bibliography: Indication of the sources consulted for the elaboration of the project.
