Products of vectors and torque

MDCAT Physics · Physics · Vectors and equilibrium. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

Vectors can be multiplied in two ways. The scalar (dot) product A · B = AB cos θ gives a number with no direction. It is greatest when the vectors are parallel and zero when they are at right angles. Work is an example.

The vector (cross) product A × B has size AB sin θ and points at right angles to both A and B, as given by the right-hand rule. It is zero for parallel vectors.

Torque is the turning effect of a force: τ = r × F. Its size is the force multiplied by the perpendicular distance from the pivot to the line of the force. A body is in equilibrium when the net force and the net torque on it are both zero.

Rules to remember

scalar product: A · B = AB cos θ = A_xB_x + A_yB_y + A_zB_z; A · A = A²
vector product: size of A × B = AB sin θ, direction perpendicular to the plane of A and B (right-hand rule); B × A = −(A × B)
Unit vectors: i · i = 1, i · j = 0, i × i = 0, i × j = k
tan θ = (size of A × B) ÷ (A · B)
torque = force × perpendicular distance from the pivot (τ = rF sin θ), unit N m
torque of a couple = one force × perpendicular distance between the two forces
Equilibrium: first condition ΣF = 0; second condition Στ = 0 (clockwise torques = anticlockwise torques)

Common mistake

Students put the weight of a uniform beam at its end. The weight acts at the centre, so its distance from a pivot at one end is half the length.

Worked example

A uniform metre rule is pivoted at its 50 cm mark. A 6.0 N weight hangs from the 20 cm mark. At which mark must a 4.0 N weight hang to balance the rule?

The rule's own weight acts at the 50 cm mark, on the pivot, so its torque is zero.
The 6.0 N weight is 50 − 20 = 30 cm from the pivot: anticlockwise torque = 6.0 × 30 = 180 N cm.
For balance, 4.0 × d = 180, so d = 45 cm on the other side of the pivot.
Position = 50 + 45 = 95 cm.

Answer: At the 95 cm mark.

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