Translations

Code	Language		Translator	Run

Credits

Fu-Kwun Hwang and lookang; lookang; tinatan

http://iwant2study.org/lookangejss/02_newtonianmechanics_6circle/ejss_model_circularmotionwee02/circularmotionwee02_Simulation.xhtml

Briefing Document: Uniform Circular Motion Derivation of a = v²/r

Source: Excerpts from "Uniform Circular Motion Derivation of a = v^2/r Model - Open Educational Resources / Open Source Physics @ Singapore"

Date: (Assumed to be 2024 or 2025 based on the site's footer and the mention of a 2024 workshop)

Overview:

This document is part of an online resource from Open Educational Resources / Open Source Physics @ Singapore. It focuses on explaining the derivation of the centripetal acceleration formula (a = v²/r) for an object undergoing uniform circular motion using a simulation and vector analysis. The primary goal is to visually and conceptually demonstrate why an object moving at constant speed in a circle is still accelerating, and why this acceleration points towards the center of the circle.

Key Concepts and Ideas:

1. Acceleration in Circular Motion:

• The fundamental idea is that acceleration isn't just about changing speed but also about changing velocity, which is a vector quantity including both magnitude (speed) and direction. As the document notes, "A particle is under acceleration when it's speed changed over time. When a particle is moving around a circle with constant speed. The velocity is changed constantly. The direction of the velocity vector is changing with time. (Velocity is a vector.)"
• Even if the speed is constant, the continuous change in the direction of the velocity vector means the object is constantly accelerating.

1. Visualizing Velocity Change with Vectors:

• The simulation described in the source uses visual vectors to demonstrate this changing velocity:
• A RED arrow represents the current velocity vector at time t.
• A MAGENTA arrow represents the velocity vector at a previous time step t-dt.
• These two vectors are then drawn from a common origin to highlight the difference.
• The GREEN vector represents the difference between the two velocity vectors, dv (delta-v), calculated as (\vec{v_{f}}(t+dt)-\vec{_{i}v}(t)).

1. Deriving the Centripetal Acceleration Formula:

• The document shows that this change in velocity (the green vector) always points towards the center of the circle. The document states, "The red arrow is (d\vec{v}=\vec{v_{f}}(t+dt)-\vec{_{i}v}(t)) which is always pointing toward the center." This directional component is key for showing centripetal acceleration
• The derivation proceeds from the relationship between the change in angle (dθ), the velocity, the change in time (dt) and the radius R.
• The angle subtended dθ is given as dθ = v * dt / R
• The magnitude of the change in velocity (|dv|) is shown as: |dv| = v * dθ = (v * v * dt) / R = v² * dt / R
• Finally, dividing by dt yields the magnitude of the acceleration a as: a = dv/dt = v²/R
• Thus, the formula a = v²/R is derived and is clearly connected to the vector differences between the velocity vectors at different points in the circular path.

1. Simulation-Based Learning:

• The document emphasizes the use of an interactive simulation to understand these concepts. Users can "click Play to start the simulation and click pause to stop it," and "Click STEP button to make the time step forward and watch the differences." This hands-on approach is meant to improve the learning experience and visual understanding.
• The simulation visualizes the velocity vectors at different time steps and their differences to reinforce the concept of centripetal acceleration.

Important Quotes:

• "A particle is under acceleration when it's speed changed over time. When a particle is moving around a circle with constant speed. The velocity is changed constantly. The direction of the velocity vector is changing with time."
• "The red arrow is (d\vec{v}=\vec{v_{f}}(t+dt)-\vec{_{i}v}(t)) which is always pointing toward the center."
• "(|d\vec{v}|=v d\theta= \frac{ (v)(v)dt}{R}), so (\frac{d\vec{v}}{dt}=\frac{v^2}{R})"

Additional Points:

• Target Audience: The document is designed for "Junior College" students and emphasizes the use of digital tools for education, such as "EasyJavaScriptSimulation".
• Accessibility: The resources are designed to work across various platforms, including "Android/iOS including handphones/Tablets/iPads" and "Windows/MacOSX/Linux including Laptops/Desktops," making it broadly accessible.
• Open Source Nature: The resources are part of an open educational resource and are shared under a Creative Commons Attribution-Share Alike 4.0 Singapore License, supporting free and open sharing of educational materials.
• Wide Array of Resources: The page contains links to many other interactive physics resources, such as simulations and Tracker models, which highlight a commitment to a hands-on, visual approach to physics education.

Conclusion:

This document provides a clear, vector-based, explanation of why an object moving in a circle at constant speed is accelerating towards the center of the circle. It effectively combines a conceptual explanation with a visual simulation and a mathematical derivation of the a = v²/r formula. The interactive nature of the linked resources enhances engagement and allows students to understand the underlying concepts more readily. The document is part of a larger collection of similar materials intended to provide open access to interactive, inquiry-based physics learning resources.

 

Uniform Circular Motion Study Guide

Quiz

Instructions: Answer the following questions in 2-3 sentences each.

1. In the context of circular motion, what does it mean for a particle to have constant speed?
2. Why is a particle undergoing uniform circular motion considered to be accelerating, even if its speed remains constant?
3. What is the significance of the magenta arrow in the simulation described in the text?
4. What is the significance of the red arrow in the simulation described in the text?
5. How does the simulation visually represent the change in velocity during uniform circular motion?
6. Explain the relationship between (d\theta), (v), (dt), and (R) as presented in the text.
7. What is the direction of the vector (d\vec{v}) during uniform circular motion, and why?
8. How is the magnitude of (d\vec{v}) determined from (v) and (d\theta)?
9. What mathematical expression is derived for acceleration in uniform circular motion?
10. What is the final form of the formula for centripetal acceleration derived in this text?

Answer Key

1. Constant speed in circular motion means the particle’s rate of movement remains the same; it covers the same distance along the circular path per unit of time. However, the direction of the particle’s velocity is continuously changing.
2. Even with a constant speed, a particle in uniform circular motion is accelerating because acceleration is defined as a change in velocity, and velocity is a vector with both magnitude and direction. Since the direction of the velocity vector is always changing, there is acceleration.
3. The magenta arrow in the simulation shows the instantaneous velocity vector of the particle at a particular time, denoted as (v_{i}(t)), representing the initial velocity at the time it was recorded.
4. The red arrow represents the change in velocity, (d\vec{v}), which is the difference between the final and initial velocity vectors ((\vec{v_{f}}(t+dt)-\vec{v_{i}}(t))). It always points toward the center of the circle.
5. The simulation uses a green vector to visually display the difference between the magenta (initial velocity) and the red (final velocity) vectors when they are placed with a common origin.
6. The relationship is (d\theta = v * dt/R), where (d\theta) is the angular displacement, (v) is the speed, (dt) is the time interval, and (R) is the radius of the circular path.
7. The vector (d\vec{v}) points towards the center of the circle because it represents the change in the direction of the velocity vector. This means the force and acceleration are directed toward the center of the circular path.
8. The magnitude of (d\vec{v}) is given by (|d\vec{v}|= v d\theta), where (v) is the speed and (d\theta) is the angular displacement. This can also be expressed as (|d\vec{v}|= \frac{v^2dt}{R}) when using (d\theta = v*dt/R).
9. The mathematical expression derived for the acceleration in uniform circular motion is found by dividing the magnitude of the change in velocity (|d\vec{v}|) by the time interval (dt), which leads to (\frac{d\vec{v}}{dt}=\frac{v^2}{R}).
10. The final form of the formula for centripetal acceleration is (a = \frac{v^2}{R}). This shows acceleration pointing toward the center of the circle with a magnitude equal to speed squared divided by the radius.

Essay Questions

Instructions: Respond to the following essay questions in a detailed and organized manner.

1. Explain in detail how the simulation, using visual aids like arrows, effectively demonstrates the concept of centripetal acceleration in uniform circular motion. Describe specifically how different vectors represent instantaneous velocity, the change in velocity, and the direction of acceleration.
2. Discuss the importance of understanding vector quantities when analyzing circular motion, and explain how the concepts of velocity and acceleration differ in linear versus circular motion contexts. Provide at least two specific examples.
3. Derive the formula for centripetal acceleration, (a = v^2/R), using the information and relationships presented in the text. Be sure to explain each step in your derivation clearly.
4. Analyze the implications of centripetal acceleration on the motion of a particle. Discuss how this force is constantly influencing direction, and explain why this type of motion requires a continuous centripetal force.
5. Compare and contrast the characteristics of uniform circular motion with non-uniform circular motion, using real-world examples to illustrate the differences and the impact on the derived equations.

Glossary of Key Terms

• Uniform Circular Motion: Motion of an object in a circle at a constant speed. Although speed is constant, velocity is changing because the direction is continuously changing.
• Velocity (Vector): A vector quantity that describes the rate at which an object changes its position, including both its speed and direction. It is often represented by an arrow, where the arrow's length represents speed, and its orientation represents direction.
• Acceleration (Vector): A vector quantity describing the rate at which an object’s velocity changes. This can be a change in speed or direction, or both.
• Instantaneous Velocity: The velocity of an object at a specific moment in time. In circular motion, this velocity vector is always tangent to the circular path.
• Centripetal Acceleration: Acceleration directed towards the center of the circular path necessary to keep an object moving in a circle; given by the formula (a = v^2/R) where v is speed, and R is radius.
• (d\vec{v}): Represents the change in velocity, which is the difference between the final and initial velocity vectors ((\vec{v_{f}}(t+dt)-\vec{v_{i}}(t))) over a given time interval dt.
• (d\theta): Angular displacement which is the angle through which the object has moved along the circular path during time interval dt.
• R: The radius of the circle in circular motion.
• v: The speed of the particle in the circular motion, with a magnitude equal to the rate of the particle moving around the circular path.
• dt: The time interval used in calculations, often a small time step.

Worksheets 

Other Resources

Sample Learning Goals

[text]

For Teachers

A particle is under acceleration when it's speed changed over time. When a particle is moving around a circle with constant speed.
The velocity is changed constantly. The direction of the velocity vector is changing with time. (Velocity is a vector.)

You can click Play to start the simulation and click pause to stop it.
More information will be displayed when the simulation is paused.
When the simulation is paused at time t: The RED arrow shows the velocity vector at that time.
Another vector in Magenta shows its velocity at previous time step t-dt.
(The starting point for the above two vector is different)

We also draw those two vectors again from the same starting point (which is the center of the circle).
You can find out the difference between two vectors (GREEN vector).

Click STEP button to make the time step forward and watch the differences.

Magenta colour arrow is the instance velocity \(v_{i}(t)\), where subscript i is for initial
arrow with color magenta after some time dt, is \(v_{f}(t+dt)\) where subscript f is for final
The red arrow is \(d\vec{v}=\vec{v_{f}}(t+dt)-\vec{_{i}v}(t)\) which is always pointing toward the center.
with  \(d\theta=v*dt/R\) ,
so It's length \(|d\vec{v}|=v d\theta= \frac{ (v)(v)dt}{R}\),
so \(\frac{d\vec{v}}{dt}=\frac{v^2}{R}\)

Research

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Video

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 Version:

Other Resources

https://ggbm.at/rjza5pwy by tan send kwang

FAQ on Uniform Circular Motion

1. What is uniform circular motion?
2. Uniform circular motion is the movement of an object along a circular path with a constant speed. While the speed remains the same, the velocity, which is a vector quantity, is constantly changing due to the changing direction of motion. This continuous change in direction results in acceleration towards the center of the circle.
3. Why is there acceleration in uniform circular motion even if the speed is constant?
4. Acceleration is defined as the rate of change of velocity. Velocity is a vector, meaning it has both magnitude (speed) and direction. In uniform circular motion, the speed is constant, but the direction of the velocity is always changing. This change in direction means that the velocity is changing, and therefore there is an acceleration.
5. What is the direction of acceleration in uniform circular motion?
6. The acceleration in uniform circular motion, often called centripetal acceleration, is always directed towards the center of the circular path. This is why the change in velocity vector always points towards the center of the circle. This acceleration is what constantly changes the direction of the object's velocity, forcing it to move in a circle.
7. How is the centripetal acceleration (a) calculated?
8. The magnitude of centripetal acceleration (a) can be calculated using the formula a = v²/r, where 'v' is the speed of the object and 'r' is the radius of the circular path. This formula can be derived by analyzing the change in velocity vectors over a small time interval.
9. How can the change in velocity be visualized in uniform circular motion?
10. The change in velocity can be visualized by drawing the velocity vectors at two slightly different times. If these vectors are placed tail-to-tail, the vector connecting the tips of these two velocity vectors represents the change in velocity. This vector will always point towards the center of the circular path.
11. How does the angular displacement relate to the speed and radius in uniform circular motion?
12. The angular displacement (dθ), which is the angle covered by the object along the circular path, can be expressed as dθ = v*dt/R. Here, 'v' is the speed of the object, 'dt' is a small time interval, and 'R' is the radius of the circle. This relationship highlights how the object's speed, the radius, and the angle it covers are interconnected in circular motion.
13. What is the role of simulations in understanding uniform circular motion?
14. Simulations are extremely useful for understanding uniform circular motion as they allow us to visualize the changes in velocity vectors and the resulting acceleration. By using a simulation, we can pause at different points in time to analyze the vectors, step through time increments, and get a more concrete feel for the dynamics of the motion. These interactive tools can make abstract concepts like vector change more understandable.
15. What other resources are available to learn more about uniform circular motion and related physics concepts? There is a wide variety of resources including simulations, models, and interactive exercises available such as those found in the provided text. The source includes many examples of simulations on the Open Educational Resources / Open Source Physics @ Singapore website. These include not only circular motion models but also models and simulations for topics such as collisions, energy, and various physics concepts.