Motorcyclist leaning into a curve

Problem

Why does a motorcyclist lean into a curve? Explain in terms of forces and moments about the contact point.

The weight acts at the centre of mass (up high), the centripetal friction force at the contact point (down low): two offset forces.

Solution

The forces. In a curve the motorcycle needs a centripetal force $\vv F_c = m v^2/r$ directed towards the centre of the curve. This force is supplied by the friction between tyre and asphalt, applied at the point of contact with the ground. The weight $m\vv g$, on the other hand, acts at the centre of mass, which is up high. The normal force $\vv N$ acts at the contact point.

The moment problem. Consider the moments about the contact point. The centripetal friction is applied exactly there, so it generates no moment. The weight, applied at the high centre of mass, and the centripetal requirement (equivalently the apparent “centrifugal push” felt by the rider, applied at the centre of mass) are offset relative to the contact point: they form a couple that tends to tip the motorcycle outward if it stayed upright.

Why lean. By leaning the motorcycle inward, the centre of mass moves towards the centre of the curve, coming to sit above (or nearly above) the line joining the contact point to the resultant of the forces. In this way the tipping moment of the weight and the one due to the centripetal force cancel each other out: the resultant of weight and ground reaction passes through the contact point and the motorcycle is in rotational equilibrium. The correct lean angle satisfies $\tan\varphi = \frac{v^2}{g\, r}$ that is, the tighter the curve and the higher the speed, the greater the lean required.

$\ev{\text{You lean until the centre of mass moves above the contact point: the moments cancel and there is no tipping.}}$

Linked items

Topics: Dinamica · Attrito Concepts: Forza centripeta · Forza normale · Attrito statico Skills: Diagramma di corpo libero