Question bank: Kinematics: Relationship between Speeds in Circular Motions

The composition of movements is an important technique to understand the behavior of objects in their different types of trajectories. For example, if a car travels in a straight line with a constant speed of 20 m/s and then makes a curve with a radius of 40 meters, what will be the angular velocity of this car?

A Ferris wheel in an amusement park has a radius of 20 meters and completes one full revolution in 1 minute. Suppose a passenger is sitting in one of the chairs at the highest point of the Ferris wheel. (a) First, calculate the angular velocity (in radians/second) of the Ferris wheel. (b) Next, determine the linear velocity of this passenger. (c) Finally, imagine an object is thrown upwards from the highest point of the Ferris wheel with an initial velocity of 5 m/s. What is the total velocity of this object relative to the ground?

Regarding the movement of a car's wheel, what is the difference between uniform motion and uniformly accelerated motion?

An engineer from a roller coaster company is designing a new loop model for an amusement park. The project requires a deep understanding of the forces and accelerations involved in circular movements. Considering a point where a roller coaster car performs a loop of radius R, the engineer wants to calculate the linear velocity necessary for passengers to feel a maximum normal force of 6 times the acceleration due to gravity. Using the concept that at the highest point of the loop, the normal force is given by n = mg - m(v^2)/R, where n is the normal force, m is the mass of the passenger, g is the acceleration due to gravity, and v is the linear velocity of the point. Considering the acceleration due to gravity g = 9.8 m/s^2 and neglecting air resistance, determine the linear velocity necessary for the normal force to be the desired one, and explain how this velocity is related to the angular velocity of the movement.

A Formula 1 car is completing a circular curve on a race track with radius R and constant centripetal acceleration a. At point A of the curve, the car is moving with an angular velocity ωA around the center of the curve. Given that the centripetal acceleration is v^2/R and angular velocity is related to linear velocity by the equation v = ωR, where v is the linear velocity and R is the radius of the curve, determine the mathematical expression relating the linear velocity at point A, vA, with the angular velocity ωA and the radius of the curve R. Considering that the car starts from rest and accelerates uniformly until reaching the angular velocity ωB after traveling half of the curve, where ωB > ωA and the angular acceleration α is kept constant throughout the journey to point B, determine the expression for the angular acceleration α in terms of ωA, ωB, and R. Your answer should take into account the conservation of energy and provide the theoretical justification for the expression found.