An amusement park ride consists of a car moving in a...

An amusement park ride consists of a car moving in a vertical circle on the end of a rigid boom of negligible mass. The combined weight of the car and riders is 6.0 kN, and the circle’s radius is 10 m. At the top of the circle, what are the (a) magnitude FB and (b) direction (up or down) of the force on the car from the boom if the car’s speed is v 5.0 m/s? What are (c) FB and (d) the direction if v 12 m/s?

An amusement park ride consists of a car moving in a vertical
circle on the end of a rigid boom of negligible mass. The combined
weight of the car and riders is 6.0 kN, and the circle’s radius is 10 m.
At the top of the circle, what are the (a) magnitude FB and (b) direction (up or down) of the force on the car from the boom if the
car’s speed is v 5.0 m/s? What are (c) FB and (d) the direction if
v 12 m/s?

Key Concepts

Centripetal Acceleration

Centripetal acceleration is the inward acceleration required for an object to follow a circular path. It is defined by the equation a = v²/r, where v is the speed of the object and r is the radius of the circle. This concept is central to understanding how the forces need to be directed towards the center of the circle to maintain circular motion.

Newton's Second Law in Circular Motion

Newton's Second Law states that the net force acting on an object equals the mass of the object multiplied by its acceleration (F = ma). In the context of circular motion, this net force is the centripetal force required to keep the object moving in a circle, and it must be directed toward the center of the circle. This principle helps relate the sum of the forces acting on the object to its speed and the radius of the circle.

Force Analysis in Vertical Circular Motion

When an object is moving in a vertical circle, gravitational force plays a significant role along with other forces such as tension or normal force. At different points in the circle, weight (the force due to gravity) may add to or subtract from the force provided by the support (such as a boom or string) to provide the necessary centripetal force. Analyzing these forces involves breaking them down into their vertical and horizontal components and ensuring their vector sum gives the required centripetal force.

Free-Body Diagram (FBD) Analysis

A free-body diagram is a schematic representation of an object and all the forces acting upon it. In problems involving vertical circular motion, drawing an FBD is essential to identify the magnitudes and directions of forces such as weight and the force from the supporting structure. This visual tool aids in setting up the correct equations to solve for unknown forces and to understand how the net force conforms with the requirements of centripetal acceleration.

Transcript

Please give Ace some feedback