Torque and equilibrium

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

Torque is the turning effect of a force about an axis. It depends on the size of the force and on the moment arm, the perpendicular distance from the axis to the line of action of the force. A force whose line of action passes through the axis gives no torque. Anticlockwise torque is taken as positive, clockwise as negative.

A body is in equilibrium when it has no linear and no angular acceleration. It may be at rest or moving with uniform velocity. Two conditions must both hold: the forces must add to zero, and the torques about any axis must add to zero. A pair of equal and opposite forces that do not act along the same line satisfies the first condition but still rotates the body.

Rules to remember

Torque = r × F; magnitude τ = rF sin θ, where θ is the angle between r and F. Unit: N m.
Torque = force × moment arm, where the moment arm is r sin θ.
First condition: ΣF = 0, which means ΣF_x = 0 and ΣF_y = 0.
Second condition: Στ = 0, so total clockwise torque = total anticlockwise torque.
The weight of a uniform body acts at its centre.
Take torques about a point where an unknown force acts, so that force drops out.

Common mistake

Students forget the weight of the beam or rule itself. If the body is uniform and has weight, that weight acts at the centre and gives a torque unless the pivot is at the centre.

Worked example

A uniform metre rule of weight 1.0 N is pivoted at the 30 cm mark. What weight W hung at the 5 cm mark keeps it horizontal?

The rule's weight acts at the 50 cm mark, 0.20 m from the pivot. W acts 0.25 m from the pivot on the other side.
Second condition: W × 0.25 = 1.0 × 0.20.
W = 0.20 ÷ 0.25 = 0.80 N.

Answer: W = 0.80 N (the pivot then pushes up with 1.0 + 0.80 = 1.8 N)

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