# 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.