Lesson Explainer: Equilibrium of a Rigid Body under Coplanar Couples Mathematics

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 𝐴 and 𝐡 is given by οƒŸπ‘€=𝐴𝐡×⃑𝐹=𝐡𝐴×⃑𝐹, where ⃑𝐹=βˆ’βƒ‘πΉοŒ οŒ‘.

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

The moment οƒŸπ‘€ is a vector perpendicular to the plane defined by 𝐴𝐡 and the couple. In a 3D coordinate system with 𝐴𝐡 and the couple in the π‘₯𝑦-plane, οƒŸπ‘€ is along the 𝑧-axis. According to the right-hand rule for the cross product, it is pointing up (positive 𝑧-component) when the couple would produce a counterclockwise rotation and down (negative 𝑧-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 𝐴 and 𝐡, 𝑀, is then given by 𝑀=Β±||𝐴𝐡||β€–β€–βƒ‘πΉβ€–β€–πœƒ,sin where ⃑𝐹 is either force of the couple, ||𝐴𝐡|| is the length of the line segment 𝐴𝐡, and πœƒ is the angle between the line ⃖⃗𝐴𝐡 and ⃑𝐹. The sign of 𝑀 is positive for counterclockwise rotation and negative for clockwise rotation.

As ||𝐴𝐡||πœƒsin is the so-called perpendicular distance, π‘‘βŸ‚, between the two lines of action of the two forces constituting the couple, we have 𝑀=±𝐹𝑑,βŸ‚ where 𝐹 is the magnitude of ⃑𝐹.

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,

  1. the resultant force must be zero,
  2. 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, π‘€οŠ§ and π‘€οŠ¨, of two couples satisfy the equation 𝑀+𝑀=0. Which of the following is therefore true?

  1. The two couples are in equilibrium.
  2. The two couples are not in equilibrium.
  3. The two couples are equivalent to a force.
  4. 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 𝑀=𝑀.

This can be expressed as π‘€βˆ’π‘€=0.

This question states that 𝑀+𝑀=0, 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 𝑀=βˆ’π‘€.

This can also be expressed as 𝑀+𝑀=0, 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

𝐴𝐡 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 30∘ with the rod from point 𝐢. If a force 𝐹 is acting on the rod at point 𝐷 so that the rod is in a horizontal equilibrium position, find the magnitude of 𝐹, its direction πœƒ, and the length of 𝐢𝐷.

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 𝐴 and 𝐡 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 𝐹. 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 πœƒ=30𝐹=25.∘andN

Let us now calculate the moment of the couple acting at 𝐴 and 𝐡, π‘€οŒ οŒ‘. It is positive because it would produce a counterclockwise rotation,𝑀=𝐴𝐡⋅𝐹⋅90,sin where 𝐹 is the magnitude of either force in the couple acting at 𝐴 and 𝐡:𝑀=90β‹…7.5β‹…1=675β‹….Ncm

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 rod’s 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 rod’s 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 βˆ’675 Nβ‹…cm (clockwise).

The moment of the couple acting at 𝐢 and 𝐷, π‘€οŒ’οŒ£, is then 𝑀=βˆ’πΆπ·β‹…πΉβ‹…30βˆ’675=βˆ’πΆπ·β‹…25β‹…30675=𝐢𝐷252𝐢𝐷=675Γ—225=54.sinsincm

We have found that πœƒ=30∘, 𝐹=25N, and the length of 𝐢𝐷=54cm.

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

𝐴𝐡 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 𝐢 between 𝐴 and 𝐡, where 𝐴𝐢=2cm. 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 30∘ with the rod as shown in the figure below. Find the weight of the rod π‘Š and the magnitude of the reaction of the nail 𝑅.

Answer

Two couples act on the rod; one couple is due to the forces acting at 𝐴 and 𝐡, and the other couple is due to the weight of the rod acting at the rod’s midpoint and the reaction of the nail at 𝐢. The resultant of the forces is zero, and we have π‘Š=𝑅, where π‘Š is the magnitude of the weight, and 𝑅 is the magnitude of the reaction of the nail.

Let us calculate the moment of the couple, π‘€οŒ οŒ‘, acting at 𝐴 and 𝐡: 𝑀=𝐴𝐡⋅𝐹⋅30,sin where 𝐹 is the magnitude of either force in the couple acting at 𝐴 and 𝐡.

Thus, 𝑀=6β‹…8β‹…30𝑀=24β‹….sinNcm

The couple of the weight of the rod, acting at the midpoint of the uniform rod (let us call this point 𝐷), and the reaction of the nail at 𝐢 would produce a clockwise rotation. These two forces are perpendicular to the rod. Hence, the moment due to this couple acting at 𝐢 and 𝐷, π‘€οŒ’οŒ£, is given by 𝑀=βˆ’πΆπ·β‹…πΉβ‹…90,sin where 𝐹 is the magnitude of either the weight acting at 𝐷 or the reaction of the nail acting at 𝐢. Point 𝐷 is located 3 cm from 𝐴, that is, 1 cm from 𝐢. Hence, 𝐢𝐷=1cm.

This gives us 𝑀=βˆ’1⋅𝐹⋅1𝑀=βˆ’πΉ.

The sum of the moments must be zero for the rod to be at equilibrium; therefore, we have 𝑀+𝑀=0,24βˆ’πΉ=0,𝐹=24,π‘Š=𝑅=24.NN

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

𝐴𝐡 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 𝐹. Determine the value of 𝐹 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 𝐢 the point at which the downward force of magnitude 𝐹 acts and point 𝐷 the point at which the downward force of magnitude 2 kg-wt acts.

Hence, with π‘€οŒ οŒ£, the moment of the 2 kg-wt couple, acting at 𝐴 and 𝐷 and π‘€οŒ’οŒ‘, the moment of the 𝐹 couple, acting at 𝐢 and 𝐡, we have 𝑀+𝑀=0.

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, 1=𝑔kg-wtN, where 𝑔 is acceleration due to gravity.

The 2 kg-wt couple would produce a clockwise rotation, while the 𝐹 couple would produce a counterclockwise rotation. Therefore, we have βˆ’π΄π·β‹…πΉβ‹…90+𝐢𝐡⋅𝐹⋅45=0,sinsin where 𝐹 is the magnitude of either force in the couple acting at 𝐴 and 𝐷 and 𝐹 is the magnitude of either force in the couple acting at 𝐢 and 𝐡.

Thus, βˆ’40β‹…2β‹…1+30β‹…πΉβ‹…βˆš22=0𝐹=8015√2𝐹=8√23.kg-wt

The value of 𝐹 that makes the rod in equilibrium is 8√23 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

𝐴𝐡 is a rod of negligible weight and length 54 cm. It is suspended horizontally by a pin at its midpoint. Forces of magnitude 68√3 N act on each end, one of them vertically upward at 𝐴 and the other vertically downward at 𝐡. The rod is pulled by a string, attached to it at point 𝐢, inclined at an angle of 60∘ to 𝐴𝐡. The tension in the string has a magnitude of 192 N. The rod is kept in horizontal equilibrium by a fourth force 𝐹 acting on the rod at point 𝐷 with an angle of 60∘ to 𝐡𝐴. Assuming that there is no reaction at the pin, find the magnitude of 𝐹 and the length of 𝐷𝐢.

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 𝐹. 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 𝐹 must produce a rotation in the opposite direction of that produced by the moment of the 68√3 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 68√3 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 𝐹=192.N

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 π‘€οŒ οŒ‘, the moment of the couple, acting at 𝐴 and 𝐡 and π‘€οŒ£οŒ’, the moment of the couple, acting at 𝐢 and 𝐷, we have 𝑀+𝑀=0.

The 68√3 N couple would produce a clockwise rotation, while the 192 N couple would produce a counterclockwise rotation. Therefore, we have βˆ’π΄π΅β‹…πΉβ‹…90+𝐷𝐢⋅𝐹⋅60=0,sinsin where 𝐹 is the magnitude of either force in the couple acting at 𝐴 and 𝐡 and 𝐹 is the magnitude of either force in the couple acting at 𝐷 and 𝐢.

Thus, βˆ’54β‹…68√3β‹…1+𝐷𝐢⋅192β‹…βˆš32=0𝐷𝐢=54β‹…68√396√3𝐷𝐢=9β‹…174𝐷𝐢=38.25.cm

The magnitude of 𝐹 is 192 N and the length of 𝐷𝐢 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 𝑀.
  • 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 𝐴 and 𝐡, the scalar moment, 𝑀, is then given by 𝑀=Β±||𝐴𝐡||β€–β€–βƒ‘πΉβ€–β€–πœƒ=±𝐹𝑑,οŒ οŒ‘βŸ‚sin where ⃑𝐹 is either force of the couple, ||𝐴𝐡|| is the length of the line segment 𝐴𝐡, πœƒ is the (geometric) angle between the line ⃖⃗𝐴𝐡 and ⃑𝐹, 𝐹 is the magnitude of ⃑𝐹, and π‘‘βŸ‚ is the perpendicular distance between the two lines of action of the two forces of the couple. The sign of 𝑀 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.

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