Equilibrium of forces: triangle of forces and Lami's theorem: WASSCE Physics

A signboard hanging from two wires outside a shop in Kissy is perfectly still: every force on it is balanced. Engineers who design bridges, cranes and roofs spend their lives making sure forces add up to zero. This lesson shows you how.

Learning objectives

By the end of this lesson you should be able to:

state the conditions for equilibrium of a particle under concurrent forces
apply the triangle of forces and Lami's theorem
solve equilibrium problems by resolving forces
find tensions in strings supporting a hanging load

Equilibrium

Definition

A body is in equilibrium when it is at rest or moving with uniform velocity, so that it has no acceleration. This happens when the resultant force on it is zero (and, for a large body, the resultant moment is zero).

Concurrent forces are forces whose lines of action pass through one point. For a body under concurrent forces, equilibrium needs only one condition:

\sum F_x = 0 \quad\text{and}\quad \sum F_y = 0

In words: the sum of the forces in any direction is zero — upward forces equal downward forces, leftward forces equal rightward forces.

Resultant and equilibrant

The resultant is the single force that has the same effect as all the forces together.
The equilibrant is the single force that balances them: equal in size to the resultant, opposite in direction.

The triangle of forces

Definition

Triangle of forces: if three forces acting at a point are in equilibrium, they can be represented in magnitude and direction by the three sides of a triangle taken in order (nose to tail).

The converse is also true: if three forces can be drawn as a closed triangle, they are in equilibrium. More generally, any number of forces in equilibrium form a closed polygon (the polygon of forces).

Lami's theorem

Three forces F1, F2 and F3 acting at a point O and balancing; alpha is the angle between F2 and F3, beta between F3 and F1, gamma between F1 and F2

Figure 1. Lami's theorem: each force is paired with the angle between the other two. WASSCE Tutor diagram.

Definition

Lami's theorem: if three concurrent forces are in equilibrium, each force is proportional to the sine of the angle between the other two:

\dfrac{F_1}{\sin\alpha} = \dfrac{F_2}{\sin\beta} = \dfrac{F_3}{\sin\gamma}

\begin{aligned}&\dfrac{F_1}{\sin\alpha} \\ &= \dfrac{F_2}{\sin\beta} \\ &= \dfrac{F_3}{\sin\gamma}\end{aligned}

Worked examples

Worked example

1. A lamp of weight 40 N hangs from the midpoint of a wire. Each half of the wire makes 30° with the horizontal. Find the tension in the wire.

By symmetry both tensions are equal, $T$ . Resolve vertically: the upward components balance the weight.

2T\sin 30° = 40 \Rightarrow 2T(0.5) = 40 \Rightarrow T = 40\ \text{N}

\begin{gathered}2T\sin 30° = 40 \\[6pt] \Rightarrow 2T(0.5) = 40 \\[6pt] \Rightarrow T = 40\ \text{N}\end{gathered}

Horizontally, $T\cos 30°$ to the left balances $T\cos 30°$ to the right automatically.

Common mistake

The flatter the wire, the larger the tension. If each half made only 5° with the horizontal, $T = \dfrac{40}{2\sin 5°} = 229$ N. A washing line can never be pulled perfectly straight when anything hangs on it: the tension would have to be infinite.

Worked example

2. A weight of 100 N is supported by two strings. One makes 30° with the vertical and the other 60° with the vertical, on opposite sides. Find the tensions $T_1$ (30° string) and $T_2$ (60° string).

The angle between the two strings is $30° + 60° = 90°$ . The angle between $T_1$ and the weight (straight down) is $180° - 30° = 150°$ ; between $T_2$ and the weight, $180° - 60° = 120°$ .

By Lami's theorem:

\dfrac{T_1}{\sin 120°} = \dfrac{T_2}{\sin 150°} = \dfrac{100}{\sin 90°}

\begin{aligned}&\dfrac{T_1}{\sin 120°} \\ &= \dfrac{T_2}{\sin 150°} \\ &= \dfrac{100}{\sin 90°}\end{aligned}

T_1 = 100\sin 120° = 86.6\ \text{N} \qquad T_2 = 100\sin 150° = 50\ \text{N}

\begin{gathered}\begin{aligned}T_1 &= 100\sin 120° \\ &= 86.6\ \text{N}\end{aligned} \\[6pt] \begin{aligned}T_2 &= 100\sin 150° \\ &= 50\ \text{N}\end{aligned}\end{gathered}

Check by resolving horizontally: $T_1\sin 30° = 43.3$ and $T_2\sin 60° = 43.3$ . ✓ The string nearer the vertical carries more of the load.

Worked example

3. A box of weight 50 N is pulled aside by a horizontal force $P$ until the rope it hangs from makes 37° with the vertical. Find $P$ and the tension in the rope.

Resolve vertically: $T\cos 37° = 50 \Rightarrow T = \dfrac{50}{0.799} = 62.6$ N.

Resolve horizontally: $P = T\sin 37° = 62.6 \times 0.602 = 37.7$ N.

(Quick check: $\tan 37° = \dfrac{P}{50}$ , so $P = 50 \times 0.754 = 37.7$ N. ✓)

A body on a smooth inclined plane

On a smooth (frictionless) slope at angle θ, a block can be held still by a force $P$ up the slope. Resolving along and perpendicular to the slope:

P = mg\sin\theta \qquad R = mg\cos\theta

For a 20 N block on a smooth 30° slope: $P = 20 \times 0.5 = 10$ N, and the normal reaction $R = 20 \times 0.866 = 17.3$ N.

Key points

Equilibrium: zero acceleration, zero resultant force ( $\sum F_x = 0$ , $\sum F_y = 0$ ).
The equilibrant is equal and opposite to the resultant.
Three forces in equilibrium form a closed triangle of forces.
Lami's theorem: $\dfrac{F_1}{\sin\alpha} = \dfrac{F_2}{\sin\beta} = \dfrac{F_3}{\sin\gamma}$ , each angle being between the other two forces.
Resolving in two perpendicular directions solves almost every problem.

WASSCE examination focus

WAEC tests this

Expect to state the conditions of equilibrium, state Lami's theorem or the triangle of forces, and calculate tensions in strings holding a load. Draw a clear force diagram with angles marked. If you use Lami's theorem, show which angle goes with which force; if you resolve, write both the vertical and horizontal equations.

Quick check

What is the resultant force on a body in equilibrium?
A 60 N weight hangs from the middle of a wire whose halves each make 30° with the horizontal. Find the tension.
A 30 N block rests on a smooth 30° slope, held by a force up the slope. Find that force.

Answers: 1. Zero. 2. $2T\sin 30° = 60 \Rightarrow T = 60$ N. 3. $30\sin 30° = 15$ N.

Equilibrium of forces: triangle of forces and Lami's theorem

Learning objectives

Equilibrium

Resultant and equilibrant

The triangle of forces

Lami's theorem

Worked examples

A body on a smooth inclined plane

Key points

WASSCE examination focus

Quick check

Keep going with equilibrium of forces

More lessons in Equilibrium of Forces