Balance Forces and Moments, and Choose the Pivot That Removes Unknowns
Torque is the rotational counterpart of force, and a rigid body is in equilibrium only if both the net force and the net torque vanish. Choosing the point about which to take moments is what turns most of these problems into a single equation.
Motion and force questions become simpler when direction, starting condition, time and the acting forces are marked first.
Torque as a vector
For a force F = 2i - 3k at position 0.5j - 2k measured from a reference point (2, 0, -3), the relative position is r = (-2, 0.5, 1), and τ = r × F = (-1.5, -4, -1) N m. Always subtract the reference point from the position before taking the cross product.
Equilibrium and moments
Take moments about a point that two unknown forces pass through. A ladder of weight 500 N whose upper end is 6 m up a smooth wall, with its foot 4 m out, has its centre of gravity one third of the way up from the foot, 4/3 m from the foot horizontally. Moments about the foot give R x 6 = 500 x (4/3), so the wall reaction is about 111 N. A ladder that is not in balance can be made stable by increasing its angle with the ground, since a steeper ladder needs less friction at the floor.
Supports and centre of mass
For a horizontal block hung by two wires with its centre of mass closer to wire B, the torques of the two wires about the centre of mass are equal in magnitude for equilibrium.
Structures such as brackets for street lights are more robust when the support gives a larger counter-torque for the same tension, so compare the lever arms of the supports about the pivot.
OlympIQ exam lens
- ✓τ = r × F with r from the reference point to the point of action; τ is perpendicular to both r and F
- ✓Equilibrium: ΣF = 0 and Στ = 0 about any point
- ✓Choose the pivot so that unknown forces have zero moment arm
- ✓For a ladder, take moments about the foot; steeper ladders need less friction
- ✓The support nearer the centre of mass carries the larger load
Do not take r from the origin when the question gives a different reference point. The torque is about that point, so the position vector must be measured from it.
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