Equilibrium of forces

AS Level Physics · Forces, density and pressure. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.

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The idea

An object is in equilibrium when two conditions are both true: there is no resultant force on it and no resultant torque about any point. Zero resultant force alone is not enough. A couple has zero resultant force but still makes an object rotate.

The principle of moments deals with the second condition. To use it, choose a pivot. A good choice is the point where an unknown force acts, because that force then has no moment and drops out of the equation.

For three coplanar forces in equilibrium, the force vectors drawn head to tail form a closed vector triangle. You can also resolve: the components must balance in any two perpendicular directions.

Rules to remember

Common mistake

Leaving out the weight of the beam, or measuring distances from the end of the beam instead of from the chosen pivot.

Worked example

A uniform rod of weight 40 N and length 1.2 m is hinged to a wall at one end. It is held horizontal by a string fixed to its other end. The string makes an angle of 30° with the rod. What is the tension in the string? (sin 30° = 0.500)

  1. Take moments about the hinge, so the force at the hinge has no moment.
  2. Clockwise moment of the weight = 40 × 0.60 = 24 N m (the weight acts at the midpoint).
  3. Anticlockwise moment of the tension = T × sin 30° × 1.2 = 0.60T.
  4. 0.60T = 24, so T = 24 ÷ 0.60 = 40 N.

Answer: 40 N

Practice questions

Try each one, then open the answer.

1. Three coplanar forces act on a point object in equilibrium. What can be said about the forces?

  1. A
    they are all equal
  2. B
    their vectors form a closed triangle when drawn head to tail
  3. C
    they act along the same line
  4. D
    two of them must be at right angles
Show answer

Answer: B. If the resultant is zero the vector polygon closes. For three forces this is the triangle of forces.

2. A uniform plank of weight 120 N and length 3.0 m rests on a pivot 1.0 m from one end. What vertical force applied at that nearer end keeps the plank horizontal?

  1. A
    40 N
  2. B
    60 N
  3. C
    120 N
  4. D
    180 N
Show answer

Answer: B. The weight acts at the centre, 1.5 m from the end, which is 0.50 m beyond the pivot. Moments: F × 1.0 = 120 × 0.50, so F = 60 N downwards.

3. For a rigid body to be in equilibrium

  1. A
    the resultant force must be zero
  2. B
    the resultant moment must be zero
  3. C
    both the resultant force and the resultant moment must be zero
  4. D
    it must be at rest
Show answer

Answer: C. Zero resultant force prevents linear acceleration; zero resultant moment prevents rotation. A body moving at constant velocity is also in equilibrium.

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