# Centripetal force

Centripetal force follows directly from our equations for centripetal acceleration that we saw in uniform circular motion:

$Fc=mac=mv2r=4π2mrT2=4π2mrf2.$

Centripetal force causes an object to follow a circular path. That being said, it is not another type of force that you can add in the net force equation. Some specific force like $F→T$ or $F→f$ will be the centripetal force.

If the specific force isn’t supplying enough (for example, ${F}_{\text{T}}<{F}_{\text{c}}$), then the object will not follow the circular path: it will go off on a tangent. If the original centripetal force is now zero (e.g., the string breaks), then it will travel in a straight line. If it was too small but stays that way (e.g., car turning too fast), then it will follow a circle with a wider radius.

The centripetal force does not have to be provided by a single force. For example, when you spin a ball on a string in a vertical circle, there are two forces on the ball at all times: tension and gravity. They will always sum to $F→c$ if the ball moves in a perfect circle. Since gravity doesn’t change, the tension force must change. At the top of the circle, $|F→T|=mac−mg,$ and at the bottom, $|F→T|=mac+mg.$

## Example

A 1105 kg car is entering a level turn with a radius of 20.0 m at 21.5 m/s. If the coefficient of friction is 0.450, what is the maximum safe speed?

There is only one force acting on the car, the force of friction, and it needs to provide a centripetal force for the car to be safe:

$F→net=F→f=ma→c.$

We can substitute for friction and centripetal acceleration:

$μmg=mv2r.$

Now we can solve for speed:

Therefore, the maximum safe speed is 9.39 m/s.