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

In this worksheet, we will practice investigating the equilibrium of a rigid body under the action of two or more coplanar couples.


Complete the following: If the couples 𝑀īŠ§ and 𝑀īŠ¨ are in equilibrium, where 𝑀−𝑀=5īŠ§īŠ¨k, then 𝑀+𝑀=īŠ§īŠ¨.

  • A10k
  • Bk
  • C0k
  • D−10k
  • E−5k


The moments, 𝑀īŠ§ and 𝑀īŠ¨, of two couples satisfy the equation 𝑀+𝑀=0īŠ§īŠ¨. Which of the following is therefore true?

  • Athe two couples are not in equilibrium
  • Bthe two couples are in equilibrium
  • Cthe two couples are equivalent
  • Dthe two couples are equivalent to a force


𝐴đĩđļ𝐷 is a square. The forces 𝐹, 2 N, 𝐹, and 2 N act along the directions īƒĢ𝐴đĩ,īƒĒđļđĩ,īƒĢđļ𝐷, and īƒĢ𝐴𝐷 respectively. Find the value of 𝐹 for the system to be in an equilibrium state.


𝐴đĩ 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 is 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 đļ. 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 𝜃, and the length of đļ𝐷.

  • A𝐹=25N, 𝜃=60∘, đļ𝐷=27cm
  • B𝐹=7.5N, 𝜃=30∘, đļ𝐷=18√3cm
  • C𝐹=25N, 𝜃=30∘, đļ𝐷=18√3cm
  • D𝐹=25N, 𝜃=30∘, đļ𝐷=54cm
  • E𝐹=25N, 𝜃=60∘, đļ𝐷=54cm


𝐴đĩ 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.

  • A8√23 kg-wt
  • B16√23 kg-wt
  • C4√23 kg-wt
  • D10√23 kg-wt


𝐴đĩ 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 đļ between 𝐴 and đĩ, where 𝐴đļ=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 𝑊 and the magnitude of the reaction of the nail 𝑅.

  • A𝑊=24√3N, 𝑅=24√3N
  • B𝑊=48N, 𝑅=48N
  • C𝑊=48√3N, 𝑅=48√3N
  • D𝑊=24N, 𝑅=24N


In the figure below, 𝐴đĩđļ𝐷 is a rectangle in which 𝐴đĩ=5cm and đĩđļ=4cm. The forces that are shown in the figure are in newtons, and the system is in equilibrium. Find the value of 𝐹+𝐹īŠ§īŠ¨.


𝐴đĩđļ𝐷𝐸𝐹 is a regular hexagon having a side length of 18 cm. Forces of magnitudes 18, 7, 18, and 7 newtons are acting along īƒĢđĩ𝐴, īƒĒđĩđļ, īƒĢ𝐸𝐷, and īƒĒ𝐸𝐹 respectively. Two other forces, each of magnitude 𝑃 newtons, are acting along īƒĢđļ𝐷 and īƒĢ𝐹𝐴. Find the value of 𝑃, given that the system is in equilibrium.


𝐴đĩđļ𝐷 is a rectangle, in which 𝐴đĩ=27cm and đĩđļ=18cm. Forces of magnitudes 𝐹īŠ§, 14 N, 𝐹īŠ¨, and 14 N are acting along īƒĢ𝐴đĩ, īƒĒđĩđļ, īƒĢđļ𝐷, and īƒĢ𝐷𝐴 respectively. If this system of forces is in equilibrium, find 𝐹īŠ§ and 𝐹īŠ¨.

  • A𝐹=7īŠ§N, 𝐹=14īŠ¨N
  • B𝐹=14īŠ§N, 𝐹=14īŠ¨N
  • C𝐹=9.33īŠ§N, 𝐹=21īŠ¨N
  • D𝐹=21īŠ§N, 𝐹=21īŠ¨N


𝐴đĩ 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 đĩ. The rod is pulled by a string, attached to it at point đļ, inclined at an angle of 60∘ to 𝐴đĩ. 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 đĩ𝐴. Assuming that there is no reaction at the pin, find the magnitude of 𝐹 and the length of 𝐷đļ.

  • A𝐹=384N, 𝐷đļ=19.12cm
  • B𝐹=192N, 𝐷đļ=19.12cm
  • C𝐹=192N, 𝐷đļ=38.25cm
  • D𝐹=384N, 𝐷đļ=38.25cm

This lesson includes 93 additional questions and 474 additional question variations for subscribers.

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