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 $5.0 \mathrm{kN},$ and the circle's radius is $10 \mathrm{~m} .$ At the top of the circle, what are the (a) magnitude $F_{B}$ and $(\mathrm{b})$ direction (up or down) of the force on the car from the boom if the car's speed is $v=5.0 \mathrm{~m} / \mathrm{s} ?$ What are (c) $F_{B}$ and (d) the direction if $v=12 \mathrm{~m} / \mathrm{s} ?$

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 $5.0 \mathrm{kN},$ and the circle's radius is $10 \mathrm{~m} .$ At the top of the circle, what are the (a) magnitude $F_{B}$ and $(\mathrm{b})$ direction (up or down) of the force on the car from the boom if the car's speed is $v=5.0 \mathrm{~m} / \mathrm{s} ?$ What are
(c) $F_{B}$ and
(d) the direction if $v=12 \mathrm{~m} / \mathrm{s} ?$

Key Concepts

Centripetal Force

Centripetal force is the net force required to keep an object moving in a circular path, and it always points towards the center of the circle. It is calculated by the formula mass multiplied by the square of the velocity divided by the circle's radius. This concept is fundamental in analyzing any scenario involving circular motion, as it explains why an object continuously changes direction even if its speed remains constant.

Gravitational Force

Gravitational force is the force that attracts any object with mass towards the center of the Earth. In vertical circular motion, this force is always directed downward and plays a significant role in altering the net force acting on the object, especially at points in the path such as the top or bottom of the circle. Its interaction with other forces determines the net acceleration experienced by the object.

Free Body Diagram Analysis

A free body diagram is a graphical representation used to visualize all the forces acting on a body. In problems involving objects moving in a vertical circle, drawing a free body diagram helps in clearly identifying the directions and magnitudes of forces such as gravitational force, tension or normal force from the support, and the required centripetal force. This tool is crucial for setting up the proper force balance equations based on Newton's laws.

Newton’s Second Law for Circular Motion

Newton’s second law states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration. In circular motion, the acceleration is centripetal, meaning it is directed toward the center of the circle. Using this law, one can set up the equation where the sum of the forces (like gravitational force and the force from the boom) equals the mass times the centripetal acceleration (v²/r). This equation is central to solving for unknown forces in circular motion problems.

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