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Worksheet: Equivalence of Two Couples

Q1:

is a horizontal light rod having a length of 60 cm, where two forces, each of magnitude 45 N, are acting vertically at and in two opposite directions. Two other forces, each of magnitude 120 N, are acting in two opposite directions at points and of the rod, where . If they form a couple equivalent to the couple formed by the first two forces, find the size of the angle of inclination that the second two forces make with the rod.

  • A
  • B
  • C
  • D

Q2:

As shown in the figure, 𝐴 𝐡 𝐢 𝐷 is a square of side length 1 m. Two forces, each of magnitude 16 kg-wt, are acting along  𝐴 𝐡 and  𝐢 𝐷 , and another two forces, each of magnitude 𝐹 , are acting at 𝐡 and 𝐷 , where one of them makes an angle of 1 5 ∘ with οƒͺ 𝐡 𝐢 , and the other makes an angle of 1 5 ∘ with  𝐷 𝐴 . If the couple formed by the first two forces is equivalent to that formed by the other two, determine the magnitude of 𝐹 and round to two decimal places.

Q3:

is a parallelogram in which , , and the diagonal is perpendicular to . If two forces, each of magnitude 7 N, are acting along and , find the magnitude of each of the two forces and that act at and perpendicularly to the diagonal so that they form a couple equivalent to the couple by the two forces mentioned earlier.

  • A ,
  • B ,
  • C ,
  • D ,

Q4:

𝐴 𝐡 is a rod having a length of 6 m and negligible weight. Two equal forces, perpendicular to the rod and each with a magnitude of 70 N , are acting at its points of trisection in opposite directions. Given that these two forces are replaced by two other forces, each with a magnitude of 170 N , that are acting at the ends of the rod, such that they form a couple that is equivalent to the first, determine the angle of inclination of these two forces with the rod rounding the result to the nearest minutes.

  • A 2 4 1 9 β€² ∘
  • B 1 1 5 3 β€² ∘
  • C 8 2 7 β€² ∘
  • D 7 5 3 β€² ∘
  • E 7 8 7 β€² ∘

Q5:

𝐴 𝐡 𝐢 𝐷 is a rectangle, where 𝐴 𝐡 = 3 2 c m , and π‘š ∠ 𝐴 𝐷 𝐡 = 3 0 ∘ . Two forces, each with a magnitude of 16 newtons act along  𝐴 𝐡 and  𝐢 𝐷 , forming a couple. If, instead of these forces, two other forces, each with magnitude 𝐹 newtons, were to act outside the rectangle on 𝐡 and 𝐷 such that they made angles of 1 5 ∘ with 𝐡 𝐢 and 𝐷 𝐴 , respectively and formed an equivalent couple to the first two forces, find the value of 𝐹 .

  • A 8 √ 2
  • B 8 √ 3
  • C8
  • D 8 √ 6

Q6:

In a rectangle 𝐴 𝐡 𝐢 𝐷 , 𝐴 𝐡 = 1 8 c m and 𝐡 𝐢 = 2 4 c m . Two forces, each of magnitude 360 N, are acting along  𝐴 𝐡 and  𝐢 𝐷 . Two other forces, each of magnitude 𝐹 , are parallel to βƒ–     βƒ— 𝐡 𝐷 and acting on the points 𝐴 and 𝐢 . If the two couples are equivalent, find the value of 𝐹 .

Q7:

In a rectangle 𝐴 𝐡 𝐢 𝐷 , 𝐴 𝐡 = 7 2 c m and 𝐡 𝐢 = 9 6 c m . Two forces, each of magnitude 930 N, are acting along  𝐴 𝐡 and  𝐢 𝐷 . Two other forces, each of magnitude 𝐹 , are parallel to βƒ–     βƒ— 𝐡 𝐷 and acting on the points 𝐴 and 𝐢 . If the two couples are equivalent, find the value of 𝐹 .

Q8:

𝐴 𝐡 𝐢 𝐷 is a rectangle, where 𝐴 𝐡 = 8 c m and π‘š ∠ 𝐴 𝐷 𝐡 = 1 5 ∘ . Two forces of the same magnitude 8 newtons are acting along  𝐴 𝐡 and  𝐢 𝐷 , and another two forces of the same magnitude 𝐹 form a couple that is acting at 𝐡 and 𝐷 , where one of them makes an angle of 3 0 ∘ with βƒ–     βƒ— 𝐡 𝐢 . Determine the magnitude of 𝐹 so that the couple formed by the last two forces is equivalent to that formed by the first two forces.

  • A 4 √ 3 newtons
  • B 8 √ 6 newtons
  • C 8 √ 3 newtons
  • D 4 √ 6 newtons