What are the conditions of equilibrium for concurrent forces and how is Lami's theorem used?

In one line

Concurrent coplanar forces are in equilibrium when the sums of their x and y components are both zero; for exactly three such forces, Lami's theorem gives a quick shortcut.

Explained simply

A lamp hanging still from two strings is in equilibrium: the strings' pulls and the lamp's weight cancel out exactly. Since all three forces meet at the knot, nothing can rotate it, so we only need to make sure the pulls cancel.

Technically, for coplanar concurrent forces: \( \Sigma F_x = 0 \) and \( \Sigma F_y = 0 \). Lami's theorem: if three coplanar concurrent forces are in equilibrium, each force is proportional to the sine of the angle between the other two.

How it works

1. Draw the free body diagram of the point (knot, ring, centre of a sphere).
2. Mark every force with its direction; assume unknown tensions pull away from the point.
3. If exactly three forces act, find the angle between each pair and apply Lami's theorem.
4. Otherwise, write \( \Sigma F_x = 0 \) and \( \Sigma F_y = 0 \) and solve the two equations (at most two unknowns).
5. A negative answer means the assumed direction is reversed.

Key formula

Lami: \( \dfrac{P}{\sin\alpha} = \dfrac{Q}{\sin\beta} = \dfrac{R}{\sin\gamma} \), where \( \alpha \) is the angle between Q and R, \( \beta \) between R and P, and \( \gamma \) between P and Q (all forces drawn away from the common point). Forces in N.

Quick example

A 100 N lamp hangs from strings at 30° (left) and 60° (right) to the horizontal. Angle between the strings is 90°; between the left string and W is 120°; between the right string and W is 150°. Lami: \( T_1/\sin150^\circ = T_2/\sin120^\circ = 100/\sin90^\circ \), so \( T_1 = 50 \) N and \( T_2 = 86.6 \) N. Check: \( 86.6\cos60^\circ = 50\cos30^\circ = 43.3 \) N horizontally.

Common mistakes

• Using the angle between a force and the horizontal instead of the angle between the other two forces in Lami's theorem.
• Applying Lami's theorem to four forces or to non-concurrent forces.
• Drawing one force towards the point and the others away, which gives wrong angles.

How MU asks it

Asked as 5 to 6-mark numericals: strings and pulleys, spheres resting in a channel or against walls, a body held on a smooth incline. The short-theory version (3 marks) is 'State the conditions of equilibrium for concurrent forces'. Show the FBD, the angles, the equation and boxed answers with units.