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:

𝐴𝐡𝐢𝐷 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.

  • A𝐹=4N, 𝐹=11N
  • B𝐹=22N, 𝐹=29N
  • C𝐹=22N, 𝐹=11N
  • D𝐹=13N, 𝐹=20N

Q2:

𝐴𝐡𝐢𝐷 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 25√2 newtons, are acting at 𝐴 and 𝐢 in the directions shown in the figure. Find the couple equivalent to the system.

Q3:

𝐴𝐡𝐢𝐷 is a square having a side length of 50 cm. Forces of magnitudes 30, 60, 160, and 10 newtons are acting at 𝐴𝐡, οƒŸπ΅πΆ, 𝐢𝐷, and 𝐷𝐴, respectively, while two forces of magnitudes 40√2 and 90√2 newtons are acting at 𝐴𝐢 and 𝐷𝐡, respectively. If the system is equivalent to a couple, find its moment considering the positive direction is 𝐷𝐢𝐡𝐴.

Q4:

𝐴𝐡𝐢 is a triangle, where 𝐡𝐢=48cm, and three forces of magnitudes 13, 13, and 24 newtons are acting along 𝐢𝐴, 𝐴𝐡, and οƒͺ𝐡𝐢 respectively. If the system of forces is equivalent to a couple, determine the magnitude of its moment.

Q5:

In a triangle 𝐴𝐡𝐢, 𝐴𝐡=𝐡𝐢=32cm and π‘šβˆ π΅=120∘. Forces of magnitudes 2, 2, and 2√3 newtons are acting at 𝐴𝐡, οƒͺ𝐡𝐢, and 𝐢𝐴, respectively. If the system is equivalent to a couple, find the magnitude of its moment considering the positive direction is 𝐴𝐡𝐢.

  • A32√3 Nβ‹…cm
  • B32 Nβ‹…cm
  • C64√3 Nβ‹…cm
  • D64 Nβ‹…cm

Q6:

𝐴𝐡𝐢 is a triangle, where 𝐴𝐡=8cm, 𝐡𝐢=3cm, and π‘šβˆ π΅=60∘, and forces of magnitudes 64 N, 24 N, and 56 N are acting along 𝐴𝐡, οƒͺ𝐡𝐢, and 𝐢𝐴 respectively. If the system of forces is equivalent to a couple, determine the magnitude of its moment.

  • A96√3 Nβ‹…cm
  • B96 Nβ‹…cm
  • C48 Nβ‹…cm
  • D48√3 Nβ‹…cm

Q7:

𝐴𝐡𝐢𝐷 is a square of side length 60 cm, where 𝐸∈οƒͺ𝐢𝐡 and πΉβˆˆοƒ«πΆπ·, such that 𝐢𝐸=𝐢𝐹=180cm. Forces of magnitudes 75, 5, 40, 40, and 35√2 g-wt are acting at 𝐴𝐡, οƒͺ𝐡𝐢, 𝐢𝐷, 𝐷𝐴, and οƒͺ𝐸𝐹, respectively. If the system is equivalent to a couple, find its moment considering the positive direction is 𝐷𝐢𝐡𝐴.

Q8:

𝐴𝐡𝐢𝐷 is a rectangle, in which 𝐴𝐡=45cm, 𝐡𝐢=55cm, and 𝐷𝐸=28cm. Forces of magnitudes 225, 275, 265, and 135 newtons act along 𝐴𝐡, οƒͺ𝐡𝐢, οƒͺ𝐢𝐸, and 𝐸𝐴 respectively. If the system of forces is equivalent to a couple, determine the magnitude of the moment of the forces.

Q9:

𝐴𝐡𝐢𝐷 is a rectangle in which 𝐴𝐡=12cm, 𝐡𝐢=6cm and 𝑂 is midpoint of 𝐴𝐡. Forces of magnitudes 7 N, 2 N, 6 N, 18 N, 3√5 N, and 10√2 N are acting along οƒͺ𝐢𝐡, 𝐴𝐡, 𝐷𝐴, 𝐢𝐷, 𝐴𝐢, and 𝑂𝐢 respectively. If this system of forces is equivalent to a couple, find the norm of its moment.

Q10:

In a trapezoid 𝐴𝐡𝐢𝐷, π‘šβˆ π΄=π‘šβˆ π΅=90∘, 𝐴𝐷=27cm, 𝐴𝐡=35cm, and 𝐡𝐢=39cm. Given that forces of magnitudes 54, 70, 78, and 74 newtons are acting along 𝐷𝐴, 𝐴𝐡, οƒͺ𝐡𝐢, and 𝐢𝐷, respectively, and that the system of forces is equivalent to a couple, find the magnitude of the moment of the forces.

Q11:

𝐴𝐡𝐢𝐷𝐸 is a regular pentagon whose side length is 16 cm. Five forces, each of magnitude 11 N, are acting at 𝐴𝐡, οƒͺ𝐡𝐢, 𝐢𝐷, 𝐷𝐸, and 𝐸𝐴, respectively. If the system is equivalent to a couple, find the magnitude of its moment, considering the positive direction is 𝐴𝐡𝐢𝐷𝐸, rounded to two decimal places.

Q12:

𝐸𝐴𝐡𝐢𝐷 is a pentagon in which π‘šβˆ πΈ=π‘šβˆ π΅=π‘šβˆ πΆ=90∘, 𝐸𝐴=24cm, 𝐴𝐡=𝐢𝐷=19cm, and 𝐡𝐢=26cm. Forces of magnitudes 72 N, 57 N, 78 N, 57 N, and 30 N are acting along 𝐸𝐴, 𝐴𝐡, οƒͺ𝐡𝐢, 𝐢𝐷, and 𝐷𝐸 respectively. If this system is equivalent to a couple, find its moment norm.

Q13:

𝐴𝐡𝐢𝐷𝐸𝑂 is a regular hexagon of side length 8 cm, and forces of magnitudes 2, 13, and 11 newtons are acting at 𝐴𝐡, 𝐢𝑂, and 𝐸𝐷 respectively. If the system is equivalent to a couple, determine the magnitude of the moment of the forces.

  • A36 Nβ‹…cm
  • B52 Nβ‹…cm
  • C36√3 Nβ‹…cm
  • D52√3 Nβ‹…cm

Q14:

𝐴𝐡𝐢𝐷𝐻𝑂 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 𝐡𝐴, 𝐴𝑂, 𝐻𝑂, 𝐻𝐷, 𝐷𝐢, and οƒͺ𝐡𝐢 respectively. Find the magnitude of the moment of the couple that is equivalent to the system.

  • A98 Nβ‹…cm
  • B378 Nβ‹…cm
  • C63√3 Nβ‹…cm
  • D49√3 Nβ‹…cm

Q15:

If 𝐴𝐡𝐢𝐷𝐸𝑂 is a regular hexagon having a side length of 6 cm, where forces of magnitudes 20 N, 20 N, 13 N, 13 N, and 20√3 newtons are acting along 𝐴𝐡, οƒͺ𝐡𝐢, 𝐢𝑂, 𝐸𝐷, and 𝐢𝐴, respectively, and the system is equivalent to a couple, find its moment norm.

  • A21 Nβ‹…cm
  • B99 Nβ‹…cm
  • C99√3 Nβ‹…cm
  • D21√3 Nβ‹…cm

Q16:

𝐴𝐡𝐢𝐷 is a trapezoid, where π‘šβˆ π΄=π‘šβˆ π΅=90∘, 𝐴𝐡=24cm, 𝐴𝐷=11cm and 𝐡𝐢=18cm. 𝐸 and 𝑂 are the midpoints of 𝐴𝐡 and 𝐡𝐢 respectively. Forces of magnitude 77 N, 175 N, 220 N, and 10 N are acting along 𝐴𝐷, 𝐷𝐢, 𝐢𝐴, and 𝐸𝑂, respectively. If the system of forces is equivalent to a couple, determine the magnitude of the moment of the forces.

Q17:

𝐴𝐡𝐢𝐷 is a quadrilateral in which 𝐴𝐡=𝐴𝐷=8cm, 𝐡𝐢=𝐢𝐷=13cm, and π‘šβˆ π΅π΄π·=120∘. Forces act on the directed line segments 𝐴𝐡, οƒͺ𝐡𝐢, 𝐢𝐷, and 𝐷𝐴. If the system is reduced to a couple having a moment of 42√3 Nβ‹…cm in the direction of 𝐴𝐡𝐢𝐷, find the magnitude of 𝐹 and 𝐹.

  • A𝐹=145N, 𝐹=9120N
  • B𝐹=807N, 𝐹=1307N
  • C𝐹=807N, 𝐹=9120N
  • D𝐹=145N, 𝐹=1307N

Q18:

𝐴𝐡𝐢𝐷 is a trapezoid in which 𝐴𝐷βˆ₯𝐡𝐢, 𝐴𝐡 is perpendicular to them, 𝐸 is the projection of 𝐷 on 𝐡𝐢, 𝐡𝐢=16cm, 𝐴𝐡=12cm, and 𝐴𝐷=11cm. Forces of magnitudes 51141341010,,,,and newtons are acting along 𝐢𝐴, 𝐴𝐷, 𝐷𝐢, 𝐸𝐷, and 𝐴𝐡 respectively. If the system is equivalent to a couple, find the magnitude of the couple moment.

Q19:

𝐴𝐡𝐢𝐷 is an isosceles trapezoid in which 𝐴𝐷βˆ₯𝐡𝐢, 𝐴𝐷=15cm, 𝐴𝐡=𝐷𝐢=17cm, and 𝐡𝐢=31cm. Forces having magnitudes of 51, 79, 51, and 31 newtons are acting in the directions of 𝐴𝐡, οƒͺ𝐡𝐢, 𝐢𝐷, and 𝐷𝐴, respectively. If the system is equivalent to a couple, find the magnitude of its moment considering the positive direction is 𝐷𝐢𝐡𝐴.

Q20:

𝐴𝐡𝐢𝐷 is a rectangle, where 𝐴𝐡=π‘₯cm, 𝐡𝐢=2π‘₯cm, and 𝐸 and 𝑂 are the midpoints of 𝐴𝐷 and 𝐡𝐢, respectively. Forces of magnitudes 8 N, 8 N, 31√2 N, and 23√2 N are acting along 𝐸𝐴, 𝐴𝐡, οƒͺ𝐡𝐸, and 𝐷𝑂, 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.

  • A8π‘₯ Nβ‹…cm
  • B31π‘₯ Nβ‹…cm
  • C15π‘₯ Nβ‹…cm
  • D23π‘₯ Nβ‹…cm

Q21:

The sides of an equilateral triangle 𝐴𝐡𝐢, 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.

  • A576√3 Nβ‹…cm
  • B288 Nβ‹…cm
  • C288√3 Nβ‹…cm
  • D576 Nβ‹…cm

Q22:

𝐴𝐡 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 𝐴𝐡, determine the algebraic sum of the moments of these forces about the point 𝐴.

Q23:

Three forces 𝐴𝐡, οƒͺ𝐡𝐢 and 𝐢𝐴 are represented by the sides of a right-angled triangle 𝐴𝐡𝐢 where 𝐡 is a right-angle. 1 cm on the triangle represents 40 N of force, and 𝐴𝐡=19cm and 𝐡𝐢=40cm. Find the magnitude of the resulting couple.

Q24:

𝐴𝐡𝐢𝐷 is a quadrilateral, where 𝐴𝐡=18cm, 𝐡𝐢=24cm, 𝐢𝐷=𝐷𝐴=17cm, and π‘šβˆ π΄π΅πΆ=90∘. 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 𝐴𝐡, οƒͺ𝐡𝐢, and 𝐢𝐴 respectively. Given that 𝐴𝐡=𝐴𝐢=36cm and 𝐡𝐢=24cm, determine the magnitude of the resultant couple rounded to the nearest hundredth.

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