There are 2 variations of a vertical circular motion diagram; both with different set of equations.

Diagram 1 shows a ball, attached to a string, being spun vertically. The forces experienced by the ball are labelled at 4 locations. Since force is a vector quantity, multiple forces can be added to determine the net force. In our case, the net force is the centripetal force which causes the ball to accelerate towards the centre of the vertical circle. Equations which relate the net force (i.e. centripetal force), force of gravity, and tension force, are shown for when the ball is at the top of the vertical circle and at the bottom of the vertical circle. It is important to note that when using the equations, don’t forget to allocate which direction is positive. For example, typically vertically down is considered negative and vertically up is considered positive. So a force acting down, the force of gravity for example, will be a negative value, whilst a force acting up, tension force at the bottom of the vertical circle, will be a positive value. In this ball on a string example, tension is the greatest at the bottom of the circle and the lowest at the top. This is because the mass remains the same so the force of gravity doesn’t change and it is travelling in uniform circular motion so the speed and are constant, so no variable in the centripetal force equation changes, causing the centripetal force to not change. Hence at the bottom of the circle, tension (which acts up), has to cancel out the force of gravity and also provide the centripetal force. You can also think of it by using the equation when rearranged for tension. It’s an addition when at the bottom of the circle and a subtraction when at the top.

Diagram 2 shows a rollercoaster on a track. The same ideology is used as above but the only change is the direction of the normal force as shown in the diagram. This has an effect on the signs in the equation. When solving vertical circular motion problems, it is important to analyse the scenario so you can use the appropriate equations.

Note: In the equations given in the diagram, the signs of the forces have already been allocated in the equation, so you would only need to use the magnitude of the force. For example, you would substitute in +9.8 for Fg instead of -9.8 as there is already a negative sign in the equation.

Travelling Upside Down Without Falling

Imagine a rollercoaster traveling through a loop-the-loop as seen in the diagram. What is the minimum speed required so the riders do not fall out of the rollercoaster when travelling upside down. This minimum speed is known as the critical speed and the equation to calculate the critical speed is:

The derivation of the equation is as follows. The minimum speed occurs when the normal force is equal to 0. So,

Hence our net force equation turns into this:

Which simplifies down to:

This equation can be used in numerous examples. Imagine a ball being spun vertically. At what speed is the tension in the string equal to 0? Imagine a bucket filled with water being spun vertically. At what minimum speed will the bucket need to travel so the water doesn’t spill when the bucket is at the top? We can use the above equation to solve these problems.