Wall of death

If the bike makes an angle q with the vertical we have: tanq= v²/rg

Aircraft banking

If the plane banks at an angle of q with the horizontal we have:

Cyclist cornering

If the cyclist leans over at an angle of q with the vertical we have:

R = mg F = mv²/r and tanq= F/R

The coat hangar

Take an ordinary wire coat hanger, bend the frame into a square and bend the hook until it faces the opposite corner of the square. File the end of the hook flat. Now balance a 1p coin on the hook and spin the coat hanger in a vertical circle with your finger in the top of the frame. The centripetal force of the hook on the coin will keep the coin balanced on it although it may be upside down!

Strength of rotating objects - an extension topic

This proof enables you to find the maximum tension of a wire before breaking in terms of its angular velocity.
Let the mass per unit length be r. The m = rL where L is the length of the wire.

Considering the forces acting on a short length of wire:

(T + dT) - T = -rw²dm giving dT = -rw²dr since dm = rdr

Integration with limits gives T = ½(mw²)(L -r²/L)
giving the maximum tension of ½ mw²L when r = 0. That is the wire will break at the point of attachment.

Useful data

Passenger jet – maximum legal banking angle = 27.5^o
Daytona race track = 31^o at 150 mph
Aston Martin test track = 11^o for r = 516 m    v = 70 mph