Loop-the-loop

Advanced

Minimum speed at bottom to complete a loop is √(5gr).

Key relationv_bottom = √(5gr)

To complete a vertical loop of radius r, the minimum speed at the top is √(gr) and so, from energy conservation, the minimum at the bottom is √(5gr).

Worked example

r = 5 m: v_bottom ≥ 15.7 m/s.

Common mistake

At the top the normal force is zero at the critical speed; the object is not weightless in other senses.

In more detail

A loop-the-loop requires a minimum speed at the top to stay on the track. Energy conservation links the speed at the bottom to the speed at the top.

Stunt drivers and roller coaster designers calculate the entry speed for loops.
Testing amusement rides for safety at the highest point.
Stunt planners set the entry speed of a loop so that the car stays on the track all the way round, not just at the top.

How high must the entry be?

The entry must be high enough that the object still has the minimum speed at the top, which usually means about 2.5 times the loop radius.

Does mass matter?

No. The speeds depend on height and radius, not mass.

Does the speed at the bottom have to be larger than at the top?

Yes. Energy conservation means the speed at the bottom is higher than at the top, which is why the loop needs a minimum entry speed.