Introduction

Relevance of the Topic

The Acceleration of Uniformly Varied Circular Motion (UVCM) is fundamental to understand the behavior of bodies in varied circular motions, something very present in our daily lives. From the movement of planets, satellites around the Earth, cars on curves, to pendulums, all are related to concepts involving UVCM. Understanding acceleration in these scenarios is key to decipher the complex dynamics in these physical phenomena.

Contextualization

Located within the broader field of Newtonian Physics - the basis for most of our understandings about the physical world - the unit of Kinematics focuses on aspects of motion and its characteristics, being an essential block for the study of Physics. Varied Circular Motion is a focal point in this unit, as it combines the idea of an object moving in a circular and accelerated path. It is the perfect transition between 'simple' uniform motion and the more complex study of dynamics.

By combining the concepts of constant velocity in the tangential direction and acceleration in the central direction, UVCM adds an extra dimension of complexity and application possibilities in problem-solving. Understanding UVCM acceleration well is, therefore, a prerequisite for studying more advanced topics in Physics, such as Dynamics, and is a valuable tool in practical applications, such as engineering and astronautics.

Theoretical Development

Components

• Circular Motion: We start with a reinforcement of the study of circular motion. Circular motions are characterized by a body that travels a path in the form of a circumference or a part of it. The velocity in circular motion is given by the equation v = ωr, where v is the linear velocity, ω is the angular velocity, and r is the radius of the circle.

• Centripetal Acceleration: This is the acceleration that a body experiences when performing circular motion. It is always pointed towards the center of the circumference. Its formula is a = (v^2)/r, where v is the linear velocity and r is the radius of the circle.

• Tangential Acceleration: In UVCM, the tangential velocity of the moving body can also vary. When the tangential velocity varies with time, tangential acceleration occurs. It is directly related to the variation of the velocity module. It can be calculated by: at = Δv/Δt, where at is the tangential acceleration, Δv is the velocity variation, and Δt is the time interval.

• Resultant Acceleration: The resultant acceleration in UVCM is the vector sum of the centripetal and tangential accelerations.

Key Terms

• UVCM - Uniformly Varied Circular Motion: It is a motion in a circular path in which the velocity module of the object varies uniformly over time.

• Angular Velocity (ω): It is the rate of change of the angle between the direction of a moving object and a fixed reference line in space. It is the relationship between the angular displacement (Δθ) and the elapsed time (Δt): ω = Δθ/Δt.

Examples and Cases

• Case of a Car on a Curve: Suppose a car moving at a constant speed along a curve. As the car makes the curve, it experiences two types of acceleration: a centripetal acceleration that always points towards the center of the curve (the force responsible for keeping it on the curve) and a tangential acceleration, which is the rate of change of velocity along the tangent to the curve.

• Case of a Roller Coaster Cart in a Loop: In a roller coaster loop, the cart experiences a centripetal acceleration pointing towards the center of the loop and a tangential acceleration, as the cart's velocity varies along the loop.

• Case of a Straight Wire with a Weight Tied and Rotating: If a weight is tied to a straight wire and starts rotating, the weight experiences a centripetal acceleration. If the weight starts rotating faster with time, it also experiences a tangential acceleration.

Detailed Summary

Key Points

• Circular Motion: Circular motion is characterized by a body that travels a path in the form of a circumference or a part of it. To describe it, we use concepts such as linear velocity (v), angular velocity (ω), and the radius of the circumference (r).

• Centripetal Acceleration: It is the acceleration that the body experiences when performing circular motion. It is always pointed towards the center of the circumference. Its formula is a = (v^2)/r.

• Tangential Acceleration: It is the acceleration resulting from the variation of the tangential velocity of a body in circular motion. It is directly related to the variation of the velocity module. It can be calculated by: at = Δv/Δt, where at is the tangential acceleration, Δv is the velocity variation, and Δt is the time interval.

• Resultant Acceleration: In UVCM, the resultant acceleration is the vector sum of the centripetal acceleration and the tangential acceleration. This acceleration governs the overall behavior of the body in UVCM.

• Applications of the Topic: The acceleration of UVCM is crucial to understand a wide range of natural and artificial phenomena, including the movement of planets and satellites in space, the movement of cars on curves, and even the mechanics of a roller coaster.

Conclusions

• Acceleration in Circular Motion: In UVCM, acceleration is not only present but is a vital presence. The fact that a body is in circular motion does not imply that it is necessarily in a situation of dynamic equilibrium - acceleration is necessary for this circular motion to be sustained.

• The Importance of Variation: Acceleration in UVCM is not constant, it varies. This point is crucial to understand the behavior of bodies in circular motions, as it is this variation that provides the energy necessary to keep the body on its circular path.

• The Two Faces of Acceleration: A body in UVCM experiences two forms of acceleration - centripetal acceleration, which always points towards the center of the path, and tangential acceleration, which is the manifestation of the body's velocity variation.

Suggested Exercises

1. Question on Varied Circular Velocity: An object is rotating in a circular path with a radius of 2 m. The angular acceleration of this object is 4 rad/s². Calculate the velocity of the object after 3 seconds.

2. Calculation of Resultant Acceleration: In a roller coaster cart, the pilot experiences a G force of 2.5. Assuming the cart is moving in a loop with a radius of 10 m, what is the resultant acceleration that the pilot is experiencing?

3. Analysis of Motion in a Practical Situation: In an amusement park, there is a Ferris wheel with a radius of 30 meters that takes 2 minutes to complete one revolution. If a person is at the highest point of the Ferris wheel, at what distance from the axis of rotation is she and what acceleration is she experiencing? Remember that the centripetal acceleration must be equal to the acceleration due to gravity for the person to remain attached to the seat.