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Worksheet: Equilibrium of a Rigid Body under Coplanar Couples

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

Q1:

If the couples and are in equilibrium, where , then find the value of .

  • A0
  • B
  • C
  • D

Q2:

The forces F i j 1 = 2 + 7 , F i j 2 = π‘Ž βˆ’ 6 , and F i j 3 = 6 + ( 𝑏 + 8 ) act on a particle, where i and j are two perpendicular unit vectors. Given that the system is in equilibrium, determine the values of π‘Ž and 𝑏 .

  • A π‘Ž = 4 , 𝑏 = βˆ’ 9
  • B π‘Ž = βˆ’ 8 , 𝑏 = 7
  • C π‘Ž = βˆ’ 8 , 𝑏 = βˆ’ 1
  • D π‘Ž = βˆ’ 8 , 𝑏 = βˆ’ 9
  • E π‘Ž = βˆ’ 4 , 𝑏 = 7

Q3:

Given that , , and are three coplanar forces in equilibrium meeting at a point, where and , find .

  • A
  • B
  • C
  • D

Q4:

𝐴 𝐡 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 𝐴 𝐢 = 2 c m . 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 3 0 ∘ 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 .

  • A π‘Š = 2 4 √ 3 N , 𝑅 = 2 4 √ 3 N
  • B π‘Š = 4 8 N , 𝑅 = 4 8 N
  • C π‘Š = 4 8 √ 3 N , 𝑅 = 4 8 √ 3 N
  • D π‘Š = 2 4 N , 𝑅 = 2 4 N

Q5:

Forces of magnitudes 𝑃 N, 𝑄 N, 1 6 √ 3 N, and 2 4 √ 3 N act at the point 𝑂 as shown in the diagram. Given that the forces are in equilibrium, determine the values of 𝑃 and 𝑄 .

  • A 𝑃 = 8 , 𝑄 = 4 8
  • B 𝑃 = 1 6 , 𝑄 = 2 4
  • C 𝑃 = 3 2 , 𝑄 = 8
  • D 𝑃 = 8 , 𝑄 = 3 2
  • E 𝑃 = 2 4 , 𝑄 = 1 6

Q6:

The forces N, N, and N are acting on a particle. Given that they are in equilibrium, what are the values of and ?

  • A ,
  • B ,
  • C ,
  • D ,
  • E ,

Q7:

Three forces are acting on a particle. Two of them are given by ( 2 + + 3 ) i j k newtons and ( βˆ’ 3 + 2 ) i j k newtons. Given that the particle remains at rest, find the third force.

  • A ( βˆ’ 2 + 3 βˆ’ 6 ) i j k newtons
  • B ( 3 βˆ’ 2 + 5 ) i j k newtons
  • C ( βˆ’ βˆ’ 4 βˆ’ ) i j k newtons
  • D ( βˆ’ 3 + 2 βˆ’ 5 ) i j k newtons
  • E ( + 4 + ) i j k newtons

Q8:

The diagram shows a square, 𝐴 𝐡 𝐢 𝐷 , where 𝐸 is a point on 𝐴 𝐷 . Forces of magnitudes 6 N, 1 8 √ 3 N, 3 3 √ 2 N and 𝐹 N act at 𝐡 as indicated on the diagram. Given that the forces are in equilibrium, find π‘š ∠ 𝐴 𝐡 𝐸 and the value of 𝐹 to two decimal places.

  • A π‘š ∠ 𝐴 𝐡 𝐸 = 3 7 ∘ , 𝐹 = 1 . 8 2 N
  • B π‘š ∠ 𝐴 𝐡 𝐸 = 3 7 ∘ , 𝐹 = 4 8 . 5 9 N
  • C π‘š ∠ 𝐴 𝐡 𝐸 = 3 9 ∘ , 𝐹 = 6 4 . 1 8 N
  • D π‘š ∠ 𝐴 𝐡 𝐸 = 3 0 ∘ , 𝐹 = 1 7 . 4 1 N

Q9:

𝐴 𝐡 is a rod having a length of 50 cm and a negligible weight. Two coplanar 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.

  • A 1 6 √ 2 3 kg-wt
  • B 4 √ 2 3 kg-wt
  • C 1 0 √ 2 3 kg-wt
  • D 8 √ 2 3 kg-wt

Q10:

A rod 𝐴 𝐡 having a length of 72 cm is of negligible weight. 𝐢 and 𝐷 are two points on the rod that are 42 cm and 60 cm away from the end 𝐴 respectively. Forces of magnitudes 380, 𝐹 , 380, and 𝐹 newtons are acting perpendicularly to the rod at the points 𝐴 , 𝐢 , 𝐷 , and 𝐡 respectively. Given that the two forces at 𝐴 and 𝐡 are acting on the rod in an opposite direction to those at 𝐢 and 𝐷 , and the rod is in equilibrium, determine the magnitude of the force that’s denoted by 𝐹 .

Q11:

𝐴 𝐡 𝐢 𝐷 is a rectangle, in which 𝐴 𝐡 = 2 7 c m , and 𝐡 𝐢 = 1 8 c m . Forces of magnitudes 𝐹  , 14, 𝐹  , and 14 newtons are acting along  𝐴 𝐡 , οƒͺ 𝐡 𝐢 ,  𝐢 𝐷 , and  𝐷 𝐴 , respectively. If this system of forces is in equilibrium, determine the values of 𝐹  and 𝐹  , such that the positive direction is 𝐷 𝐢 𝐡 𝐴 .

  • A 𝐹 = 9 . 3 3  N , 𝐹 = 2 1  N
  • B 𝐹 = 1 4  N , 𝐹 = 1 4  N
  • C 𝐹 = 7  N , 𝐹 = 1 4  N
  • D 𝐹 = 2 1  N , 𝐹 = 2 1  N

Q12:

A body weighing 61 kg-wt is placed on a smooth plane inclined at 3 0 ∘ to the horizontal. Given that it is maintained in a state of equilibrium by means of a force inclined at 6 0 ∘ above the horizontal, find the magnitude of the force 𝐹 and the reaction 𝑅 of the plane.

  • A 𝐹 = 6 1 √ 6 6 k g - w t , 𝑅 = 6 1 √ 3 3 k g - w t
  • B 𝐹 = 6 1 √ 3 3 k g - w t , 𝑅 = 1 2 2 3 k g - w t
  • C 𝐹 = 6 1 √ 6 6 k g - w t , 𝑅 = 6 1 √ 6 6 k g - w t
  • D 𝐹 = 6 1 √ 3 3 k g - w t , 𝑅 = 6 1 √ 3 3 k g - w t

Q13:

Coplanar forces of magnitude 18 N, 5 N, 𝐹 N, 9 N, 𝐾 N, and 13 N are acting towards a particle, where the angle between each two consecutive forces is 6 0 ∘ . Find the magnitudes of 𝐹 and 𝐾 for the system to be in equilibrium.

  • A 𝐹 = 1 7 N , 𝐾 = 4 N
  • B 𝐹 = 2 7 N , 𝐾 = 1 4 N
  • C 𝐹 = 4 7 . 5 N , 𝐾 = 5 0 N
  • D 𝐹 = 2 2 N , 𝐾 = 1 4 N

Q14:

The moments, 𝑀 1 and 𝑀 2 , of two couples satisfy the equation 𝑀 + 𝑀 = 0 1 2 . Which of the following is therefore true?

  • A the two couples are equivalent to a force
  • B the two couples are not in equilibrium
  • C the two couples are equivalent
  • D the two couples are in equilibrium

Q15:

In the shown figure, forces of magnitudes 13, 13, 4 4 √ 3 , 4 4 √ 3 , 289, and 289 newtons are acting on a rod. Given that the rod is in equilibrium, and π‘₯ is measured in centimeters, find the length of the rod.

  • A 3 cm
  • B 15 cm
  • C 30 cm
  • D 21 cm

Q16:

𝐴 𝐡 is a rod of negligible weight, and length 54 cm. It is suspended horizontally by a pin at its midpoint. Forces of magnitude 6 8 √ 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 6 0 ∘ to 𝐴 𝐡 . 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 6 0 ∘ to 𝐡 𝐴 . Find the magnitude of 𝐹 and the length of 𝐷 𝐢 .

  • A 𝐹 = 1 9 2 N , 𝐷 𝐢 = 1 9 . 1 2 c m
  • B 𝐹 = 3 8 4 N , 𝐷 𝐢 = 3 8 . 2 5 c m
  • C 𝐹 = 3 8 4 N , 𝐷 𝐢 = 1 9 . 1 2 c m
  • D 𝐹 = 1 9 2 N , 𝐷 𝐢 = 3 8 . 2 5 c m

Q17:

𝐴 𝐡 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 3 0 ∘ 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 𝐢 𝐷 .

  • A 𝐹 = 7 . 5 N , πœƒ = 3 0 ∘ , 𝐢 𝐷 = 1 8 √ 3 c m
  • B 𝐹 = 2 5 N , πœƒ = 3 0 ∘ , 𝐢 𝐷 = 1 8 √ 3 c m
  • C 𝐹 = 2 5 N , πœƒ = 6 0 ∘ , 𝐢 𝐷 = 2 7 c m
  • D 𝐹 = 2 5 N , πœƒ = 3 0 ∘ , 𝐢 𝐷 = 5 4 c m
  • E 𝐹 = 2 5 N , πœƒ = 6 0 ∘ , 𝐢 𝐷 = 5 4 c m

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