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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.

Q1:

Complete the following: If the couples 𝑀 and 𝑀 are in equilibrium, where 𝑀𝑀=5k, then 𝑀+𝑀=.

  • A10k
  • Bk
  • C0k
  • D10k
  • E5k

Q2:

The moments, 𝑀 and 𝑀, of two couples satisfy the equation 𝑀+𝑀=0. 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

Q3:

𝐴𝐵𝐶𝐷 is a square. The forces 𝐹, 2 N, 𝐹, and 2 N act along the directions 𝐴𝐵,𝐶𝐵,𝐶𝐷, and 𝐴𝐷 respectively. Find the value of 𝐹 for the system to be in an equilibrium state.

Q4:

𝐴𝐵 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 𝐶. 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 𝐶𝐷.

  • A𝐹=25N, 𝜃=60, 𝐶𝐷=27cm
  • B𝐹=7.5N, 𝜃=30, 𝐶𝐷=183cm
  • C𝐹=25N, 𝜃=30, 𝐶𝐷=183cm
  • D𝐹=25N, 𝜃=30, 𝐶𝐷=54cm
  • E𝐹=25N, 𝜃=60, 𝐶𝐷=54cm

Q5:

𝐴𝐵 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.

  • A823 kg-wt
  • B1623 kg-wt
  • C423 kg-wt
  • D1023 kg-wt

Q6:

𝐴𝐵 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 𝐶 between 𝐴 and 𝐵, where 𝐴𝐶=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=243N, 𝑅=243N
  • B𝑊=48N, 𝑅=48N
  • C𝑊=483N, 𝑅=483N
  • D𝑊=24N, 𝑅=24N

Q7:

In the figure below, 𝐴𝐵𝐶𝐷 is a rectangle in which 𝐴𝐵=5cm and 𝐵𝐶=4cm. The forces that are shown in the figure are in newtons, and the system is in equilibrium. Find the value of 𝐹+𝐹.

Q8:

𝐴𝐵𝐶𝐷𝐸𝐹 is a regular hexagon having a side length of 18 cm. Forces of magnitudes 18, 7, 18, and 7 newtons are acting along 𝐵𝐴, 𝐵𝐶, 𝐸𝐷, and 𝐸𝐹 respectively. Two other forces, each of magnitude 𝑃 newtons, are acting along 𝐶𝐷 and 𝐹𝐴. Find the value of 𝑃, given that the system is in equilibrium.

Q9:

𝐴𝐵𝐶𝐷 is a rectangle, in which 𝐴𝐵=27cm and 𝐵𝐶=18cm. Forces of magnitudes 𝐹, 14 N, 𝐹, and 14 N are acting along 𝐴𝐵, 𝐵𝐶, 𝐶𝐷, and 𝐷𝐴 respectively. If this system of forces is in equilibrium, find 𝐹 and 𝐹.

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

Q10:

𝐴𝐵 is a rod of negligible weight and length 54 cm. It is suspended horizontally by a pin at its midpoint. Forces of magnitude 683 N act on each end, one of them vertically upward at 𝐴 and the other vertically downward at 𝐵. The rod is pulled by a string, attached to it at point 𝐶, inclined at an angle of 60 to 𝐴𝐵. 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 𝐵𝐴. Assuming that there is no reaction at the pin, find the magnitude of 𝐹 and the length of 𝐷𝐶.

  • A𝐹=384N, 𝐷𝐶=19.12cm
  • B𝐹=192N, 𝐷𝐶=19.12cm
  • C𝐹=192N, 𝐷𝐶=38.25cm
  • D𝐹=384N, 𝐷𝐶=38.25cm

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

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