In this explainer, we will learn how to investigate the equilibrium of a rigid body under the action of two or more coplanar couples.

A couple is a pair of parallel but not coincident forces of equal magnitudes and opposite directions.

### Definition: Moment of a Couple

The moment of a couple acting at and is given by where .

The moment due to a couple is independent of the point about which moments due to the couple are taken.

The moment is a vector perpendicular to the plane defined by and the couple. In a 3D coordinate system with and the couple in the -plane, is along the -axis. According to the right-hand rule for the cross product, it is pointing up (positive -component) when the couple would produce a counterclockwise rotation and down (negative -component) for a clockwise rotation.

This explainer deals with *coplanar* couples. The moments of coplanar couples are parallel (they are all perpendicular to the plane in which the couples lie). Therefore, they can be treated as scalar quantities that correspond to their components along the axis perpendicular to the plane defined by the couples.

This scalar moment of a couple acting at and , , is then given by where is either force of the couple, is the length of the line segment , and is the angle between the line and . The sign of is positive for counterclockwise rotation and negative for clockwise rotation.

As is the so-called perpendicular distance, , between the two lines of action of the two forces constituting the couple, we have where is the magnitude of .

We see that the particularity of a couple is that its resultant is zero, but it has a rotating effect on the body it acts on. Therefore, the body is not at equilibrium if it is subjected to a couple only.

Inversely, it means that there are two conditions for a body to be at equilibrium.

### Definition: Conditions for Equilibrium

For a body to be at equilibrium,

- the resultant force must be zero,
- the resultant moment of the forces acting on the body about all points must be zero.

If a body is submitted to two or more coplanar couples, then the resultant force acting on it is zero and, therefore, it is at equilibrium if the sum of the moments of the couples is zero.

Let us look at an example in which we consider the resultant moment of two couples.

### Example 1: Understanding the Relationship between Two Couples and the Equilibrium of a Body

The moments, and , of two couples satisfy the equation . Which of the following is therefore true?

- The two couples are in equilibrium.
- The two couples are not in equilibrium.
- The two couples are equivalent to a force.
- The two couples are equivalent.

### Answer

A couple consists of parallel forces of equal magnitude acting in opposite directions. The sum of these forces must be zero; hence, a couple cannot produce a net force. We can immediately, therefore, eliminate the option that the two couples are equivalent to a force, unless we mean a force of zero magnitude.

If the couples are equivalent, it must be the case that

This can be expressed as

This question states that which contradicts the stated condition for the equivalence of the couples.

If the couples are in equilibrium, the counterclockwise moment due to the couples equals the clockwise moment due to the couples, which can be expressed by

This can also be expressed as which is stated in the question to be the case; hence, we conclude that the couples are in equilibrium.

Let us now look at an example in which a system of two couples is analyzed.

### Example 2: Calculating an Unknown Force and Angle for Which a Body is in Equilibrium Under the Action of One or More Couples

is a rod having a length of 90 cm and a negligible weight. It is suspended horizontally by a pin at its midpoint. Two forces, each of a magnitude 7.5 N, are acting at its ends, as shown in the figure. It is also pulled by a string, whose tension is 25 N, in a direction making an angle of with the rod from point . If a force is acting on the rod at point so that the rod is in a horizontal equilibrium position, find the magnitude of , its direction , and the length of .

### Answer

The rod is said to be in equilibrium, which means that the resultant force must be zero and the sum of the moments of all forces must be zero.

Let us start with the resultant force. The two forces acting at and are a couple and their sum is zero. The weight of the rod is negligible and so is the reaction of the pin as well. We are left with the 25 N force and the force of magnitude . For the sum of two forces to be zero, they must form a couple, that is, be parallel, of equal magnitudes and opposite directions. Hence, we have

Let us now calculate the moment of the couple acting at and , . It is positive because it would produce a counterclockwise rotation, where is the magnitude of either force in the couple acting at and :

The rod is suspended by a pin, which would be the pivot if the rod were to turn. It is worth noting that, as the pin is at the rodβs midpoint, even if the weight were not negligible, it would have no moment with respect to the pivot since its line of action goes through the rodβs midpoint (provided the rod is unform). Hence, as the sum of the moments must be zero for the rod to be in equilibrium, the moment due to the remaining couple must be Nβ cm (clockwise).

The moment of the couple acting at and , , is then

We have found that , , and the length of .

Let us look at an example in which the weight of a rod contributes to the moments acting on it.

### Example 3: Calculating the Weight and Reaction Force given a Body Under the Action of a Couple

is a uniform rod with length 6 cm. It is free to rotate about a smooth nail in a small hole in the rod at a point between and , where . The rod is in equilibrium, laying horizontally, under the action of two forces, each of magnitude 8 N, acting at either end at an angle of with the rod as shown in the figure below. Find the weight of the rod and the magnitude of the reaction of the nail .

### Answer

Two couples act on the rod; one couple is due to the forces acting at and , and the other couple is due to the weight of the rod acting at the rodβs midpoint and the reaction of the nail at . The resultant of the forces is zero, and we have where is the magnitude of the weight, and is the magnitude of the reaction of the nail.

Let us calculate the moment of the couple, , acting at and : where is the magnitude of either force in the couple acting at and .

Thus,

The couple of the weight of the rod, acting at the midpoint of the uniform rod (let us call this point ), and the reaction of the nail at would produce a clockwise rotation. These two forces are perpendicular to the rod. Hence, the moment due to this couple acting at and , , is given by where is the magnitude of either the weight acting at or the reaction of the nail acting at . Point is located 3 cm from , that is, 1 cm from . Hence, .

This gives us

The sum of the moments must be zero for the rod to be at equilibrium; therefore, we have

The reaction of the nail on the rod has equal magnitude to the weight of the rod, and this magnitude is 24 N.

Let us now look at an example of two couples acting on a body where neither couple consists of forces that act at opposite ends of the body.

### Example 4: Calculating the Value of a Couple for Which a Body is in Equilibrium

is a rod having a length of 50 cm and a negligible weight. Two coplanar pairs of forces are acting on the rod as shown in the figure. The first couple consists of two forces acting perpendicularly to the rod, each of magnitude 2 kg-wt, and the second couple consists of two forces, each of magnitude . Determine the value of that makes the rod in equilibrium.

### Answer

For the rod to be at equilibrium, both the resultant of the forces and the sum of the moments of the forces must be zero.

The rod has negligible weight; therefore, only two couples act on it. The resultant of the forces is thus zero since each couple is made of forces of equal magnitudes but opposite directions.

Recall that the moment due to a couple is independent of the point about which moments due to the couple are taken. For the rod to be at equilibrium, the sum of the moments must be zero.

Let us call point the point at which the downward force of magnitude acts and point the point at which the downward force of magnitude 2 kg-wt acts.

Hence, with , the moment of the 2 kg-wt couple, acting at and and , the moment of the couple, acting at and , we have

Note that the kilogram-weight is a unit of force; 1 kg-wt is the force equivalent to the weight of 1 kg of mass. Hence, , where is acceleration due to gravity.

The 2 kg-wt couple would produce a clockwise rotation, while the couple would produce a counterclockwise rotation. Therefore, we have where is the magnitude of either force in the couple acting at and and is the magnitude of either force in the couple acting at and .

Thus,

The value of that makes the rod in equilibrium is kg-wt.

Let us look at an example in which no figure is supplied.

### Example 5: Solving a Problem Involving a Rod in Equilibrium Under the Action of One or More Couples

is a rod of negligible weight and length 54 cm. It is suspended horizontally by a pin at its midpoint. Forces of magnitude N act on each end, one of them vertically upward at and the other vertically downward at . The rod is pulled by a string, attached to it at point , inclined at an angle of to . The tension in the string has a magnitude of 192 N. The rod is kept in horizontal equilibrium by a fourth force acting on the rod at point with an angle of to . Assuming that there is no reaction at the pin, find the magnitude of and the length of .

### Answer

It is always good to start such a question by drawing a diagram. The only thing that we need to determine when drawing the diagram is the orientation of the force with magnitude 192 N and that of force . Since the rod is at equilibrium, the sum of the moments must be zero. It means that the moment of the force with magnitude 192 N and that of force must produce a rotation in the opposite direction of that produced by the moment of the N forces.

The first condition for the rod to be at equilibrium is that the resultant of forces is zero. The weight and the reaction at the pin are negligible. The forces form a couple as they are parallel and in opposite directions (and have the same magnitude). Their sum is zero. The sum of the other two forces must therefore be zero, which means that they are parallel, in opposite directions, and of equal magnitude. Hence, we have

The second condition for the rod to be at equilibrium is that the sum of the moments of all forces is zero. We have here two coplanar couples acting on the rod. Recall that the moment of a couple is independent of the point about which moments due to the couple are taken. With , the moment of the couple, acting at and and , the moment of the couple, acting at and , we have

The N couple would produce a clockwise rotation, while the 192 N couple would produce a counterclockwise rotation. Therefore, we have where is the magnitude of either force in the couple acting at and and is the magnitude of either force in the couple acting at and .

Thus,

The magnitude of is 192 N and the length of is 38.25 cm.

Let us summarize what we have learned in this explainer.

### Key Points

- A couple has a zero net force but it has a rotating effect on the body it acts on, described by its moment .
- The moment of a couple is independent of the point about which moments due to the couple are taken.
- There are two conditions for a body to be at equilibrium:
- The resultant force must be zero.
- The resultant moment of the forces acting on the body about all points must be zero.

- The moments of coplanar couples can be treated as scalar quantities that correspond to their components along the axis perpendicular to the plane defined by the couples. For a couple acting at and , the scalar moment, , is then given by where is either force of the couple, is the length of the line segment , is the (geometric) angle between the line and , is the magnitude of , and is the perpendicular distance between the two lines of action of the two forces of the couple. The sign of is positive for counterclockwise rotation and negative for clockwise rotation.
- For a body that is acted on by multiple coplanar couples to be in equilibrium, the sum of the scalar moments of all couples must be zero.