Equilibrium of a rigid body

A point is in equilibrium when the forces on it balance. A rigid body needs two conditions, because a body can stay in one place and still spin:

• Translational equilibrium: the net external force is zero, ΣF = 0.
• Rotational equilibrium: the net external torque is zero, Στ = 0.

A useful fact: once the net force is zero, the net torque comes out the same about every point. So you may take moments about whichever point is easiest, usually one where an unknown force acts, which removes it from the equation.

A couple is two equal and opposite forces whose lines of action are apart: turning a tap, a steering wheel or a key. The net force is zero, so it does not move the body along; it only turns it. Its moment, force × distance between the lines, is the same about every point.

Principle of moments: for a lever in balance, load × load arm = effort × effort arm. Mechanical advantage = load ÷ effort, which is greater than 1 when the effort arm is longer. That is how a crowbar or a bottle opener lets a small push move a big load.

Example: on a see-saw, a 30 kg child sits 2.0 m from the pivot. Where must a 40 kg child sit? 30 × 2.0 = 40 × d, so d = 1.5 m: the heavier child sits nearer the middle.

The centre of gravity is the point about which the total torque of gravity is zero, so the weight can be taken to act there. Hang a body from any point and it settles with the centre of gravity straight below that point. In a uniform field, which is true for everyday bodies near the Earth, it is the same as the centre of mass.

Method for every problem: draw all the external forces, write ΣF = 0 in two directions, then take moments about a smart point.