# Lesson Explainer: Triangle of Forces Mathematics

In this explainer, we will learn how to solve problems about the equilibrium of a particle under the action of three forces meeting at a point using the resultant of forces or triangle of force method.

When two or more forces act on a rigid body and the body does not accelerate in any direction, that is to say, it either remains at rest or continues to move at a constant velocity, then the forces are said to be in equilibrium.

The simplest example of forces in equilibrium is two forces of equal magnitude acting on a body in opposite directions along the same line of action.

If a pair forces and are acting on a rigid body and that body is in equilibrium, then we know that a third force must be acting on the body, equal in magnitude and in the opposite direction to the resultant of and .

Recall that if we know the angle between and , then we can calculate the magnitude of their resultant by the formula

Let us see how we can apply this in an example. Suppose two forces of magnitudes 5 N and 4 N are acting on a rigid body and the angle between them is .

If the body is in equilibrium, then we can calculate force keeping it in equilibrium, as follows. The magnitude of the resultant is given by

We know that force is equal in magnitude and opposite in direction to the resultant , and so has a magnitude of 6.92 N, to two decimal places.

Let us try a slightly more difficult example using this idea.

### Example 1: Finding a Force Acting on a Body in Equilibrium Using the Resultant

Use the following diagram to find the tension in . Round your answer to two decimal places.

Recall that if a pair forces and are acting on a rigid body and that body is in equilibrium, then we know that a third force must be acting on the body, equal in magnitude and in the opposite direction to the resultant of and .

In this case, the pair of forces acting are the tensions and . The third force is the downward force of 10 N. We therefore know that the magnitude of the resultant of and is 10. Recall the equation relating the square of the magnitude of a resultant to the angle between two forces:

Thus, we have

Observe that in this set-up , and so

Looking at the diagram, we can calculate using the cosine rule:

Therefore,

We can substitute this expression for back into our equation for the tension:

Thus,

The tension in is therefore 9.05 N, to two decimal places.

When a rigid body is in equilibrium under the action of three coplanar forces meeting at a point, we can analyze the situation using a triangle of forces.

Considering forces now explicitly as vectors and representing them graphically as arrows with lengths proportional to their magnitudes, we can represent the addition of forces by placing the arrows head to tail, as shown in the following figure.

The resultant force, , of and is given by and is represented graphically as an arrow that has its tail at the same point as the tail of and its head at the same point as the head of , as shown in the following figure.

Let us define a force of equal magnitude to , acting in the opposite direction to it. If is added to , we obtain

We can see from this that

This is represented graphically in the following figure.

The three arrows representing the three forces are all joined head-to-tail, forming a triangle.

Let us suppose that these forces all act on a particle P, as shown in the following figure.

It has been established that the resultant of these forces is zero. If these are the only forces acting on P, the net force on it is zero, and P is in equilibrium.

Forces and need not be perpendicular to be two of the forces in a triangle of zero resultant force. The following figure shows an example of a force triangle where none of the forces are perpendicular to each other; the diagram also shows these forces acting on a particle P that is in equilibrium.

We can define a triangle of forces in equilibrium as follows.

### Definition: Triangle of Forces in Equilibrium

Three force vectors that form a triangle for which the directions of the forces are all either clockwise around the triangle or counterclockwise around the triangle have a zero resultant, and hence the forces are in equilibrium.

Let us look at an example where forces are graphically represented by a triangle of arrows.

### Example 2: Using a Triangle of Forces to Solve a Word Problem

Bassem is attempting a mechanics problem in which three coplanar forces , , and are acting on a body. He needs to determine whether the body is in equilibrium or not. He remembers his teacher saying something about checking whether he could arrange the forces into a triangle. So, he draws the shown figure.

Bassem concludes that the three forces are in equilibrium. Is he correct?

Which of the following best describes what he has done?

1. He has not paid attention to the direction of the forces. All the forces should meet head-to-tail. However, in his diagram, and meet head-to-head. Therefore, the forces do not actually form a triangle.
2. He has done nothing wrong.
3. He has used the wrong method; a force triangle is not a valid way of checking for equilibrium.
4. He has put the forces in the wrong order. He should have started with the force represented by the longest arrow and worked his way to the shortest.

To determine whether the forces are in equilibrium, we can refer to the figure in the question.

From the figure, we see that

The sum of the three forces is therefore given by

As is nonzero, the forces acting do not have zero resultant and so are not in equilibrium.

We can eliminate two of the options for the second part of the question immediately.

Something clearly has been done wrong as the forces are not in equilibrium despite forming a triangle, so it is incorrect to say that nothing has been done wrong.

The mistake made is not that a force triangle cannot be used to show that three forces are in equilibrium, as this is a valid method.

One of the remaining options states that

“He has put the forces in the wrong order. He should have started with the force represented by the longest arrow and worked his way to the shortest.”

This can be understood to mean that we should join the arrows head-to-tail in order of length, as shown in the following figure.

Clearly is nonzero if this is done, and so the forces are not in equilibrium, so this could not show that the forces are in equilibrium.

The final remaining option is that

“He has not paid attention to the direction of the forces. All the forces should meet head-to-tail. However, in his diagram, and meet head-to-head. Therefore, the forces do not actually form a triangle.”

It is true that and meet head-to-head, and it is true that, in a triangle of forces, all the arrows must meet head-to-tail. If the arrow representing is reversed, a triangle of forces with zero resultant is formed, as shown in the following figure.

This option identifies what was done incorrectly.

It is important to note that, by changing the direction of so that its head is at the tail of rather than , has been removed and replaced by another force that has the same magnitude as but in the opposite direction. The three forces shown in the question cannot be in equilibrium.

Now let us look at an example where a triangle of forces is used to determine the magnitude of an unknown force.

### Example 3: Using a Triangle of Forces to Solve an Equilibrium Problem

Three coplanar forces , , and are acting on a body in equilibrium. Their triangle of forces forms a right triangle as shown.

Given that and , find the magnitude of .

For the body that forces , , and act on to be in equilibrium, it must be true that

The triangle formed by the forces is right-angled, so using Pythagoras’s theorem, we see that

Rearranging to make the subject, we obtain

Substituting the magnitudes of the forces, we find that

The negative square root of 144 is not considered, as magnitudes of vectors are necessarily positive valued.

Sometimes when we are solving triangle of forces problems, we are given the lengths of the sides of the triangle. Since the forces are proportional to the side lengths of the triangle, we can form the following relationship.

### Definition: The Triangle of Forces Rule

A triangle of forces that are in equilibrium can be represented on a diagram as shown below.

Since the magnitudes of the forces are proportional to the side lengths of the triangle, we can form the following relationship:

Let us now look at an example where three forces act at a point.

### Example 4: Finding the Ratio between Three Forces Acting Parallel to the Sides of a Right Triangle Given That the System Is in Equilibrium

In the figure, three forces of magnitudes , , and newtons meet at a point. The lines of action of the forces are parallel to the sides of the right triangle. Given that the system is in equilibrium, find .

The length of the hypotenuse of the right triangle, , is not known, but using Pythagoras’s theorem, we see that

To form a triangle of forces with zero resultant, the magnitudes of the forces must be in the same ratio as that of the lengths of the sides of the triangle.

Comparing the side lengths of the triangle, we see that and

Hence, we see that

Let us now look at an example involving determining a force in an isosceles triangle of forces rather than a right triangle of forces.

### Example 5: Equilibrium of a System of Three Forces Acting through a Triangle

A body is under the effect of three forces of magnitudes , , and 36 newtons, acting in the directions , , and , respectively, where is a triangle such that , , and . Given that the system is in equilibrium, find and .

To form a triangle of forces with zero resultant, the magnitudes of the forces must be in the same ratio as that of the lengths of the sides of triangle , which is shown in the following figure.

The triangle of forces that corresponds to this triangle is shown in the following figure.

The lengths of and are equal, so the magnitude of is 36 N.

The lengths of and are related as follows:

Hence, the magnitude of is given by

Let us look at an example where a problem involving the equilibrium of a suspended object is solved using a triangle of forces.

### Example 6: Finding the Tension in the Strings That Keep a Uniform Rod in Equilibrium

A uniform rod of length 50 cm and weight 143 N is freely suspended at its ends from the ceiling by means of two perpendicular strings attached to the same point on the ceiling. Given that the length of one of the strings is 30 cm, determine the tension in each string.

The rod and the strings are represented qualitatively in the following figure.

The rod and strings form a right triangle. Using Pythagoras's theorem, we can determine , the length of the unknown side:

The forces acting are the weight of the rod and the tensions in the strings. These forces are in equilibrium, so they can be drawn acting at the same point. This is represented qualitatively in the following figure, where corresponds to the tension in the 40 cm string and corresponds to the tension in the 30 cm string.

These forces can form a triangle of forces, as shown in the following figure.

As the forces are in equilibrium, we see that

It is stated that hence,

From the triangle formed by the rod and strings, we can know that the sides of the triangle that correspond to the strings have lengths of 40 cm and 30 cm. The ratio of the lengths of the sides of this triangle equals the ratio of the forces corresponding to the tensions in the strings, and so we see that

Substituting the expression for into we obtain

As

Now that we have seen a variety of examples, let us recap some key points of the explainer.

### Key Points

• For two forces and that have a resultant , it is the case that where , , and form a triangle of forces that has a zero resultant.
• A body that is acted on only by forces that form a triangle of forces is in equilibrium. Arrows representing the forces in the triangle must all meet head-to-tail.
• The ratios of the lengths of the sides of a triangle are equal to the ratios of the magnitudes of the forces in a triangle of forces.