Bodies in Equilibrium

A bridge, a bookshelf, a ladder against a wall: engineering is mostly about things that must not move. For that, two balances must hold at once.

Two conditions#

$\vec F_{\text{ext}} = M\vec a_{cm}$ says the centre of mass stays at rest only if the external forces add to zero. $\tau = I\alpha$ says the body won’t start turning only if the external torques add to zero. A body at rest in an inertial frame needs both: zero net force in every direction and zero net torque about every line.

When all forces lie in one plane (the x–y plane), this reduces to three equations:

The torques about the x- and y-axes vanish automatically for forces in that plane. And if the net force is zero, the net torque is the same about every perpendicular axis, so you may take torques about whichever point is most convenient.

Standing, tipping and stability#

An object resting on a surface feels its weight at the centre of mass and contact forces spread over its base. The contact forces can only push, so they can only cancel gravity’s torque if the vertical line through the centre of mass lands inside the base (or inside the region bounded by the contact points). That is why you lean the other way when carrying a heavy bag in one hand.

Disturb a body slightly and watch what its centre of mass does:

• Stable equilibrium: the centre of mass rises; gravity brings the body back (a box sitting flat).
• Unstable equilibrium: the centre of mass falls; the body moves further away (a box balanced on an edge, a pencil on its tip).
• Neutral equilibrium: the centre of mass stays at the same height; the body stays where you leave it (a ball on a level floor).

The leaning cyclist#

A cyclist rounding an unbanked turn of radius r at speed v is not in equilibrium in the ground frame (the centre of mass is accelerating). Work instead in a frame rotating with the cyclist at ω = v/r. In that frame we add the centrifugal pseudo-force $Mv^2/r$ at the centre of mass, and then the cyclist is at rest, so the torques about the contact point A must cancel. Gravity tries to topple the cycle inward and the centrifugal force outward. With the cycle at angle θ from the vertical and the centre of mass at distance l from A: