Centripetal Force

Newton's laws of motion state that an object cannot accelerate unless an unbalanced force acts upon it. In this case, a force which isn't parallel to the direction of motion is required to change the direction of an object's motion, i.e. to get it to turn.

This force is known as the radial force or centripetal force, as the force vector always points towards the centre of the circle of rotation and is hence parallel to the radius. This force can be caused by friction, gravity, tension, an electromagnetic force, or any other such force.

Using $F = m a$ , we can calculate the centripetal force of an object with a known mass:

Formula

$F_{c} = m a_{r}$

Variable Key

$F_{c}$ is the centripetal force, in newtons.
$m$ is the mass of the object, in kilograms.
$a_{r}$ is the radial acceleration, in metres per second squared.

This doesn't match the relationships sheet!

I chose to use $m a_{r}$ instead of $\frac{m v^{2}}{r}$ or $m r ω^{2}$ as I believe it is both simpler and gives a better knowledge of the physics involved, by showing the connection between centripetal force and radial acceleration. If preferred, all instances of $a_{r}$ on this page may be substituted accordingly.

Diagrams

Centripetal force is not a "real force", meaning that centripetal force is not an actual force observed in the real world by itself - it is just the name we give to the force or combination of forces required to keep an object moving in a circular motion. These real forces which make up the centripetal force can, and often do, change - if the magnitude increases then the object falls towards the centre of the circle, and if the magnitude decreases then the object flies outwards at a tangent to the circle.

To demonstrate this, below are a few diagrams showing various systems of circular motion.

Horizontal circular motion

In this example, consider a tethered mass spinning horizontally in a circle around a point - for example, a ball on a string. The centripetal force consists of the force of tension from the tether connecting the object to the centre of the circle. Consider the diagram to be a birds-eye view.

Vertically rotating tethered object

This example is similar to the previous one, however there is an additional force of weight which needs to be taken into account. If you summed up all the forces acting on the object, the component of the resultant vector acting towards the centre of the circle must be the required centripetal force for the circular motion.

Three interesting cases include:

When the mass reaches the top of the circle, both the forces of tension and weight will be acting towards the centre of the circle, therefore $F_{c} = m a_{r} + W$ .
When the mass reaches the bottom of the circle, the force of weight is acting against the force of tension, so $F_{c} = m a_{r} - W$ .
When the mass is exactly halfway up the circle, the force of weight is tangent to the circle and perpendicular to the force of tension, therefore it has no effect on the centripetal force, and $F_{c} = m a_{r}$ .

The forces acting on the mass at all other points in the circle can be calculated using trigonometry by breaking up the forces into component parts parallel to the direction of the centre of the circle.