Circular motion — centripetal force and the formula F = mv²/r

What just happened?

You doubled the speed, and the force needed to keep the body on the circle grew FOURFOLD. The centripetal force grows with the square of the speed.

What is the centripetal force?

It is a force directed toward the CENTER of the circle that constantly "turns" the body, keeping it on the circular path. Without it the body would fly straight.

Formula:

It grows with mass and with the square of the speed, and decreases with radius. The centripetal acceleration:

$a = \dfrac{v^2}{r}$

What does cutting the constraint do?

When the centripetal force disappears (you cut the string), the body does not "fly outward", but travels TANGENTIALLY — along a straight line, following inertia (Newton's first law).

Where does this force come from in practice?

From the tension of a string, the friction of tires on the road in a turn, the gravity keeping a planet in orbit — there is always "something" playing the role of the centripetal force.

Why you need this for the exam:

$F = \tfrac{m v^2}{r}$ and $a = \tfrac{v^2}{r}$ solve problems about turns, satellites and centrifuges. The trick: the centripetal force is not a new force, just a role played by a real force directed toward the center.

Remember: you compute the force on a circle from $F = \tfrac{m v^2}{r}$ (it grows with the square of the speed), and after the constraint is cut the body flies tangentially, not outward.