# Motion in a vertical circle

## What is Motion in a Vertical Circle?

• Motion in a vertical circle is a type of two-dimensional motion in which an object moves along a circular path in the vertical plane.
• This motion involves forces acting on the moving object which can fluctuate depending on the object’s position in the circular path.
• In general, the forces involve gravitational force acting downwards and centripetal force that pulls the object towards the centre of the circle.

## Energy in Motion in a Vertical Circle

• When an object moves in a vertical circle, it continually exchanges potential and kinetic energy.
• At the highest point of the circle, the object will have maximum potential energy and a minimum kinetic energy due to its decreased speed.
• At the lowest point, the object’s potential energy is at its lowest and the kinetic energy is highest due to the increased speed.
• Understanding this exchange of kinetic and potential energy could be key to solving problems related to motion in a vertical circle.

## Forces in Motion in a Vertical Circle

• The centripetal force is important for maintaining the object’s circular path.
• The tension force from a string, holding the object in circular motion, plays into the equation of the net force acting on the body.
• If the system is under non-uniform circular motion, there will also be a tangential component of the force, often caused by factors like friction or propulsion.
• Remember the tension is at its greatest at the bottom of the swing and smallest at the top.

## Solving Problems in Motion in a Vertical Circle

• Use Newton’s Second Law to derive the condition for motion in a vertical circle.
• The formula for the centripetal force is given by `Fc = mv^2/r`, where `m` is the mass of the body, `v` its velocity and `r` the radius of the circle.
• Find the speed of the object at various positions around the circle by using conservation of energy principles.
• The tension in the string at any point can be calculated using `T = Fc ± mgcosθ`, where `T` is the tension, `Fc` the centripetal force, `m` the mass of the object, `g` the acceleration due to gravity, and `θ` the angle from the vertical.

## Real-world Applications of Motion in a Vertical Circle

• Insight on this concept is important within physics and mathematics, as well as real-world applications such as roller coaster design, vehicle turning, and the spin of celestial bodies.
• An understanding of the forces involved and the energy exchanges that occur during the motion in a vertical circle are necessary for these applications.