Worksheet: Equivalent System of Forces to a Couple

In this worksheet, we will practice identifying the conditions for a system of coplanar forces to be equivalent to a couple and finding its moment.

Q1:

𝐴 𝐡 𝐢 𝐷 is a square, where the five forces, measured in newtons, are acting on it as shown in the figure. If the system of forces is equivalent to a couple, determine 𝐹  and 𝐹  .

  • A 𝐹 = 1 3  N , 𝐹 = 2 0  N
  • B 𝐹 = 4  N , 𝐹 = 1 1  N
  • C 𝐹 = 2 2  N , 𝐹 = 2 9  N
  • D 𝐹 = 2 2  N , 𝐹 = 1 1  N

Q2:

𝐴 𝐡 𝐢 𝐷 is a square having a side length of 85 cm. Forces of magnitudes 30, 55, 30, and 55 newtons are acting along the square’s sides, and two equal forces of magnitude 2 5 √ 2 newtons, are acting at 𝐴 and 𝐢 in the directions shown in the figure. Find the couple equivalent to the system.

Q3:

𝐴 𝐡 𝐢 𝐷 is a square having a side length of 50 cm. Forces of magnitudes 30, 60, 160, and 10 newtons are acting at οƒ  𝐴 𝐡 , οƒŸ 𝐡 𝐢 , οƒ  𝐢 𝐷 , and οƒ  𝐷 𝐴 , respectively, while two forces of magnitudes 4 0 √ 2 and 9 0 √ 2 newtons are acting at οƒ  𝐴 𝐢 and οƒ  𝐷 𝐡 , respectively. If the system is equivalent to a couple, find its moment considering the positive direction is 𝐷 𝐢 𝐡 𝐴 .

Q4:

𝐴 𝐡 𝐢 is a triangle, where 𝐡 𝐢 = 4 8 c m , and three forces of magnitudes 13, 13, and 24 newtons are acting along  𝐢 𝐴 ,  𝐴 𝐡 , and οƒͺ 𝐡 𝐢 respectively. If the system of forces is equivalent to a couple, determine the magnitude of its moment.

  • A 480 Nβ‹…cm
  • B 240 Nβ‹…cm
  • C 960 Nβ‹…cm
  • D 120 Nβ‹…cm

Q5:

In a triangle 𝐴 𝐡 𝐢 , 𝐴 𝐡 = 𝐡 𝐢 = 3 2 c m and π‘š ∠ 𝐡 = 1 2 0 ∘ . Forces of magnitudes 2, 2, and 2 √ 3 newtons are acting at  𝐴 𝐡 , οƒͺ 𝐡 𝐢 , and  𝐢 𝐴 , respectively. If the system is equivalent to a couple, find the magnitude of its moment considering the positive direction is 𝐴 𝐡 𝐢 .

  • A 3 2 √ 3 Nβ‹…cm
  • B 64 Nβ‹…cm
  • C 6 4 √ 3 Nβ‹…cm
  • D 32 Nβ‹…cm

Q6:

𝐴 𝐡 𝐢 is a triangle, where 𝐴 𝐡 = 8 c m , 𝐡 𝐢 = 3 c m , and π‘š ∠ 𝐡 = 6 0 ∘ , and forces of magnitudes 64 N, 24 N, and 56 N are acting along  𝐴 𝐡 , οƒͺ 𝐡 𝐢 , and  𝐢 𝐴 respectively. If the system of forces is equivalent to a couple, determine the magnitude of its moment.

  • A 4 8 √ 3 Nβ‹…cm
  • B 48 Nβ‹…cm
  • C 96 Nβ‹…cm
  • D 9 6 √ 3 Nβ‹…cm

Q7:

𝐴 𝐡 𝐢 𝐷 is a square of side length 60 cm, where 𝐸 ∈ οƒͺ 𝐢 𝐡 and 𝐹 ∈  𝐢 𝐷 , such that 𝐢 𝐸 = 𝐢 𝐹 = 1 8 0 c m . Forces of magnitudes 75, 5, 40, 40, and 3 5 √ 2 g-wt are acting at  𝐴 𝐡 , οƒͺ 𝐡 𝐢 ,  𝐢 𝐷 ,  𝐷 𝐴 , and οƒͺ 𝐸 𝐹 , respectively. If the system is equivalent to a couple, find its moment considering the positive direction is 𝐷 𝐢 𝐡 𝐴 .

Q8:

𝐴 𝐡 𝐢 𝐷 is a rectangle, in which 𝐴 𝐡 = 4 5 c m , 𝐡 𝐢 = 5 5 c m , and 𝐷 𝐸 = 2 8 c m . Forces of magnitudes 225, 275, 265, and 135 newtons act along  𝐴 𝐡 , οƒͺ 𝐡 𝐢 , οƒͺ 𝐢 𝐸 , and  𝐸 𝐴 , respectively. If the system of forces is equivalent to a couple, Determine the magnitude of the moment of the forces.

Q9:

𝐴 𝐡 𝐢 𝐷 is a rectangle in which 𝐴 𝐡 = 1 2 c m , 𝐡 𝐢 = 6 c m and 𝑂 is midpoint of 𝐴 𝐡 . Forces of magnitudes 7 N, 2 N, 6 N, 18 N, 3 √ 5 N, and 1 0 √ 2 N are acting along οƒͺ 𝐢 𝐡 ,  𝐴 𝐡 ,  𝐷 𝐴 ,  𝐢 𝐷 ,  𝐴 𝐢 , and  𝑂 𝐢 respectively. If this system of forces is equivalent to a couple, find the norm of its moment.

  • A 24 Nβ‹…cm
  • B 72 Nβ‹…cm
  • C 84 Nβ‹…cm
  • D 60 Nβ‹…cm

Q10:

In a trapezium 𝐴 𝐡 𝐢 𝐷 , π‘š ∠ 𝐴 = π‘š ∠ 𝐡 = 9 0 ∘ , 𝐴 𝐷 = 2 7 c m , 𝐴 𝐡 = 3 5 c m , and 𝐡 𝐢 = 3 9 c m . Given that forces of magnitudes 54, 70, 78, and 74 newtons are acting along  𝐷 𝐴 ,  𝐴 𝐡 , οƒͺ 𝐡 𝐢 , and  𝐢 𝐷 , respectively, and that the system of forces is equivalent to a couple, find the magnitude of the moment of the forces.

Q11:

𝐴 𝐡 𝐢 𝐷 𝐸 is a regular pentagon whose side length is 16 cm. Five forces, each of magnitude 11 N, are acting at  𝐴 𝐡 , οƒͺ 𝐡 𝐢 ,  𝐢 𝐷 ,  𝐷 𝐸 , and  𝐸 𝐴 , respectively. If the system is equivalent to a couple, find the magnitude of its moment, considering the positive direction is 𝐴 𝐡 𝐢 𝐷 𝐸 , rounded to two decimal places.

Q12:

𝐸 𝐴 𝐡 𝐢 𝐷 is a pentagon in which π‘š ∠ 𝐸 = π‘š ∠ 𝐡 = π‘š ∠ 𝐢 = 9 0 ∘ , 𝐸 𝐴 = 2 4 c m , 𝐴 𝐡 = 𝐢 𝐷 = 1 9 c m , and 𝐡 𝐢 = 2 6 c m . Forces of magnitudes 72 N, 57 N, 78 N, 57 N, and 30 N are acting along  𝐸 𝐴 ,  𝐴 𝐡 , οƒͺ 𝐡 𝐢 ,  𝐢 𝐷 , and  𝐷 𝐸 respectively. If this system is equivalent to a couple, find its moment norm.

  • A 2,202 Nβ‹…cm
  • B 4,404 Nβ‹…cm
  • C 3,684 Nβ‹…cm
  • D 1,842 Nβ‹…cm

Q13:

𝐴 𝐡 𝐢 𝐷 𝐸 𝑂 is a regular hexagon of side length 8 cm, and forces of magnitudes 2, 13, and 11 newtons are acting at  𝐴 𝐡 ,  𝐢 𝑂 , and  𝐸 𝐷 respectively. If the system is equivalent to a couple, determine the magnitude of the moment of the forces.

  • A 5 2 √ 3 Nβ‹…cm
  • B 52 Nβ‹…cm
  • C 36 Nβ‹…cm
  • D 3 6 √ 3 Nβ‹…cm

Q14:

𝐴 𝐡 𝐢 𝐷 𝐻 𝑂 is a regular hexagon whose side length is 7 cm. Forces of magnitudes of 9, 8, 10, 9, 8, and 10 newtons are acting along  𝐡 𝐴 ,  𝐴 𝑂 ,  𝐻 𝑂 ,  𝐻 𝐷 ,  𝐷 𝐢 , and οƒͺ 𝐡 𝐢 respectively. Find the magnitude of the moment of the couple that is equivalent to the system.

  • A 6 3 √ 3 Nβ‹…cm
  • B 4 9 √ 3 Nβ‹…cm
  • C 378 Nβ‹…cm
  • D 98 Nβ‹…cm

Q15:

If 𝐴 𝐡 𝐢 𝐷 𝐸 𝑂 is a regular hexagon having a side length of 6 cm, where forces of magnitudes 20 N, 20 N, 13 N, 13 N, and 2 0 √ 3 newtons are acting along  𝐴 𝐡 , οƒͺ 𝐡 𝐢 ,  𝐢 𝑂 ,  𝐸 𝐷 , and  𝐢 𝐴 , respectively, and the system is equivalent to a couple, find its moment norm.

  • A 9 9 √ 3 Nβ‹…cm
  • B 99 Nβ‹…cm
  • C 2 1 √ 3 Nβ‹…cm
  • D 21 Nβ‹…cm

Q16:

𝐴 𝐡 𝐢 𝐷 is a trapezium, where π‘š ∠ 𝐴 = π‘š ∠ 𝐡 = 9 0 ∘ , 𝐴 𝐡 = 2 4 c m , 𝐴 𝐷 = 1 1 c m and 𝐡 𝐢 = 1 8 c m . 𝐸 and 𝑂 are the midpoints of 𝐴 𝐡 and 𝐡 𝐢 respectively. Forces of magnitude 77 N, 175 N, 220 N, and 10 N are acting along  𝐴 𝐷 ,  𝐷 𝐢 ,  𝐢 𝐴 , and  𝐸 𝑂 , respectively. If the system of forces is equivalent to a couple, determine the magnitude of the moment of the forces.

Q17:

𝐴 𝐡 𝐢 𝐷 is a quadrilateral in which 𝐴 𝐡 = 𝐴 𝐷 = 8 c m , 𝐡 𝐢 = 𝐢 𝐷 = 1 3 c m , and π‘š ∠ 𝐡 𝐴 𝐷 = 1 2 0 ∘ . Forces act on the directed line segments  𝐴 𝐡 , οƒͺ 𝐡 𝐢 ,  𝐢 𝐷 , and  𝐷 𝐴 . If the system is reduced to a couple having a moment of 4 2 √ 3 Nβ‹…cm in the direction of 𝐴 𝐡 𝐢 𝐷 , find the magnitude of 𝐹  and 𝐹  .

  • A 𝐹 = 8 0 7  N , 𝐹 = 9 1 2 0  N
  • B 𝐹 = 1 4 5  N , 𝐹 = 1 3 0 7  N
  • C 𝐹 = 8 0 7  N , 𝐹 = 1 3 0 7  N
  • D 𝐹 = 1 4 5  N , 𝐹 = 9 1 2 0  N

Q18:

𝐴 𝐡 𝐢 𝐷 is a trapezium in which 𝐴 𝐷 βˆ₯ 𝐡 𝐢 , 𝐴 𝐡 is perpendicular to them, 𝐸 is the projection of 𝐷 on 𝐡 𝐢 , 𝐡 𝐢 = 1 6 c m , 𝐴 𝐡 = 1 2 c m , and 𝐴 𝐷 = 1 1 c m . Forces of magnitudes 5 1 1 4 1 3 4 1 0 1 0 , , , , a n d newtons are acting along  𝐢 𝐴 ,  𝐴 𝐷 ,  𝐷 𝐢 ,  𝐸 𝐷 , and  𝐴 𝐡 respectively. If the system is equivalent to a couple, find the magnitude of the couple moment.

Q19:

𝐴 𝐡 𝐢 𝐷 is an isosceles trapezium in which 𝐴 𝐷 βˆ₯ 𝐡 𝐢 , 𝐴 𝐷 = 1 5 c m , 𝐴 𝐡 = 𝐷 𝐢 = 1 7 c m , and 𝐡 𝐢 = 3 1 c m . Forces having magnitudes of 51, 79, 51, and 31 newtons are acting in the directions of  𝐴 𝐡 , οƒͺ 𝐡 𝐢 ,  𝐢 𝐷 , and  𝐷 𝐴 , respectively. If the system is equivalent to a couple, find the magnitude of its moment considering the positive direction is 𝐷 𝐢 𝐡 𝐴 .

Q20:

𝐴 𝐡 𝐢 𝐷 is a rectangle, where 𝐴 𝐡 = π‘₯ c m , 𝐡 𝐢 = 2 π‘₯ c m , and 𝐸 and 𝑂 are the midpoints of 𝐴 𝐷 and 𝐡 𝐢 , respectively. Forces of magnitudes 8 N, 8 N, 3 1 √ 2 N, and 2 3 √ 2 N are acting along  𝐸 𝐴 ,  𝐴 𝐡 , οƒͺ 𝐡 𝐸 , and  𝐷 𝑂 , respectively. Given that this system of forces is equivalent to a couple, determine the magnitude of its moment in terms of π‘₯ , giving your answer in Nβ‹…cm.

  • A 2 3 π‘₯ Nβ‹…cm
  • B 8 π‘₯ Nβ‹…cm
  • C 3 1 π‘₯ Nβ‹…cm
  • D 1 5 π‘₯ Nβ‹…cm

Q21:

The sides of an equilateral triangle 𝐴 𝐡 𝐢 , taken the same way round, completely represent three forces with a drawing scale of 4 cm to 8 N. If the length of a side of the triangle is 24 cm, find the magnitude of the resulting couple, giving an exact answer in Nβ‹…cm.

  • A 2 8 8 √ 3 Nβ‹…cm
  • B 576 Nβ‹…cm
  • C 5 7 6 √ 3 Nβ‹…cm
  • D 288 Nβ‹…cm

Q22:

𝐴 𝐡 is a rod having a length of 105 cm and negligible weight. Forces of magnitudes 214 N, 67 N, 115 N, and 176 N are acting on the rod as shown in the figure. Given that 𝐢 and 𝐷 are the points of trisection of 𝐴 𝐡 , determine the algebraic sum of the moments of these forces about the point 𝐴 .

Q23:

Three forces  𝐴 𝐡 , οƒͺ 𝐡 𝐢 and  𝐢 𝐴 are represented by the sides of a right-angled triangle 𝐴 𝐡 𝐢 where 𝐡 is a right-angle. 1 cm on the triangle represents 40 N of force, and 𝐴 𝐡 = 1 9 c m and 𝐡 𝐢 = 4 0 c m . Find the magnitude of the resulting couple.

Q24:

𝐴 𝐡 𝐢 𝐷 is a quadrilateral, where 𝐴 𝐡 = 1 8 c m , 𝐡 𝐢 = 2 4 c m , 𝐢 𝐷 = 𝐷 𝐴 = 1 7 c m , and π‘š ∠ 𝐴 𝐡 𝐢 = 9 0 ∘ . Given that four forces, measured in newtons, are acting on the quadrilateral as shown in the given figure, determine the magnitude of the moment of the equivalent couple.

Q25:

Three forces of magnitudes 15, 10, and 15 newtons are acting along  𝐴 𝐡 , οƒͺ 𝐡 𝐢 , and  𝐢 𝐴 respectively. Given that 𝐴 𝐡 = 𝐴 𝐢 = 3 6 c m and 𝐡 𝐢 = 2 4 c m , determine the magnitude of the resultant couple rounded to the nearest hundredth.

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