Equilibrium and Newton's third law

The idea

A particle is in equilibrium when the forces on it add to zero. Newton's third law: if $A$ pushes on $B$ , then $B$ pushes on $A$ with an equal and opposite force. These two forces act on different bodies.

Add the forces as vectors: $\mathbf{F}_1 + \mathbf{F}_2 + \mathbf{F}_3 = \mathbf{0}$ .
Solve for the unknown force component by component.
For a person in a lift, the floor's push on them equals their push on the floor.

Watch out A third-law pair never balances on one body: the two forces act on different objects.

See it worked

Worked example

Forces $(3\mathbf{i} + 4\mathbf{j})\text{ N}$ and $(\mathbf{i} - 6\mathbf{j})\text{ N}$ act on a particle with a third force $\mathbf{F}$ . It is in equilibrium. Find $\mathbf{F}$ .

Sum of the known forces: $4\mathbf{i} - 2\mathbf{j}$ .
Equilibrium: $\mathbf{F} + 4\mathbf{i} - 2\mathbf{j} = \mathbf{0}$ .
$\mathbf{F} = -4\mathbf{i} + 2\mathbf{j}$ .

Answer $(-4\mathbf{i} + 2\mathbf{j})\text{ N}$

Your turn

Question 1Non-calculator3 marks

Forces $(2\mathbf{i} - 3\mathbf{j})\text{ N}$ , $(-5\mathbf{i} + \mathbf{j})\text{ N}$ and $\mathbf{F}$ act on a particle in equilibrium. Find $\mathbf{F}$ . Give the $\mathbf{i}$ and $\mathbf{j}$ components.

Question 2Calculator allowed3 marks

A particle is in equilibrium under a force of $5\text{ N}$ due north, a force of $3\text{ N}$ due east and a third force $\mathbf{F}$ . Find $|\mathbf{F}|$ to 3 significant figures.

Question 3Non-calculator3 marks

A person of mass $60\text{ kg}$ stands in a lift accelerating upwards at $1.2\text{ m s}^{-2}$ . Find the force the person exerts on the floor. Use $g = 9.8\text{ m s}^{-2}$ .

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