Worksheet: Equilibrium of a Rigid Body under Coplanar Couples

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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 M and M are in equilibrium, where Mk=50, then find the value of MM−=.

  • A 1 0 0 k
  • B − 5 0 k
  • C0
  • D 5 0 k

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:

In the figure shown, forces of magnitudes 13, 13, 44√3, 44√3, 289, and 289 newtons are acting on a rod. Given that the rod is in equilibrium and that 𝑥 is measured in centimeters, find the length of the rod.

Q4:

The forces Fij=2+7, Fij=𝑎−6, and Fij=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 𝑎 and 𝑏.

  • A 𝑎 = 4 , 𝑏 = − 9
  • B 𝑎 = − 8 , 𝑏 = 7
  • C 𝑎 = − 8 , 𝑏 = − 1
  • D 𝑎 = − 4 , 𝑏 = 7
  • E 𝑎 = − 8 , 𝑏 = − 9

Q5:

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

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

Q6:

Given that F, F, and F are three coplanar forces in equilibrium meeting at a point, where Fij=5−3 and Fij=4−14, find F.

  • A 9 − 1 7 i j
  • B i j + 1 1
  • C − − 1 1 i j
  • D − 9 + 1 7 i j

Q7:

Forces of magnitudes 𝑃 N, 𝑄 N, 16√3 N, and 24√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 = 2 4 , 𝑄 = 1 6
  • B 𝑃 = 8 , 𝑄 = 4 8
  • C 𝑃 = 1 6 , 𝑄 = 2 4
  • D 𝑃 = 3 2 , 𝑄 = 8
  • E 𝑃 = 8 , 𝑄 = 3 2

Q8:

𝐴 𝐵 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.

  • A 𝑊 = 2 4 √ 3 N , 𝑅 = 2 4 √ 3 N
  • B 𝑊 = 4 8 N , 𝑅 = 4 8 N
  • C 𝑊 = 4 8 √ 3 N , 𝑅 = 4 8 √ 3 N
  • D 𝑊 = 2 4 N , 𝑅 = 2 4 N

Q9:

𝐴 𝐵 𝐶 𝐷 is a rectangle, in which 𝐴𝐵=27cm, and 𝐵𝐶=18cm. Forces of magnitudes 𝐹, 14, 𝐹, and 14 newtons are acting along 𝐴𝐵, 𝐵𝐶, 𝐶𝐷, and 𝐷𝐴, respectively. If this system of forces is in equilibrium, determine the values of 𝐹 and 𝐹, such that the positive direction is 𝐷𝐶𝐵𝐴.

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

Q10:

A rod 𝐴𝐵 having a length of 72 cm is of negligible weight. 𝐶 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, 𝐶, 𝐷, and 𝐵 respectively. Given that the two forces at 𝐴 and 𝐵 are acting on the rod in an opposite direction to those at 𝐶 and 𝐷, and the rod is in equilibrium, determine the magnitude of the force that’s denoted by 𝐹.

Q11:

𝐴 𝐵 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.

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

Q12:

𝐴 𝐵 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.

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

Q13:

𝐴 𝐵 is a rod of negligible weight and length 54 cm. It is suspended horizontally by a pin at its midpoint. Forces of magnitude 68√3 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.

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

Q14:

The forces (−7+13)ij N, (𝑎+)ij N, and (−5+(𝑏−2))ij N are acting on a particle. Given that they are in equilibrium, what are the values of 𝑎 and 𝑏?

  • A 𝑎 = − 2 , 𝑏 = − 1 6
  • B 𝑎 = 1 2 , 𝑏 = − 1 4
  • C 𝑎 = 1 2 , 𝑏 = − 1 6
  • D 𝑎 = 1 2 , 𝑏 = − 1 2
  • E 𝑎 = 2 , 𝑏 = − 1 2

Q15:

The diagram shows a square, 𝐴𝐵𝐶𝐷, where 𝐸 is a point on 𝐴𝐷. Forces of magnitudes 6 N, 18√3 N, 33√2 N and 𝐹 N act at 𝐵 as indicated on the diagram. Given that the forces are in equilibrium, find 𝑚∠𝐴𝐵𝐸 and the value of 𝐹 to two decimal places.

  • A 𝑚 ∠ 𝐴 𝐵 𝐸 = 3 0 ∘ , 𝐹 = 1 7 . 4 1 N
  • B 𝑚 ∠ 𝐴 𝐵 𝐸 = 3 9 ∘ , 𝐹 = 6 4 . 1 8 N
  • C 𝑚 ∠ 𝐴 𝐵 𝐸 = 3 7 ∘ , 𝐹 = 1 . 8 2 N
  • D 𝑚 ∠ 𝐴 𝐵 𝐸 = 3 7 ∘ , 𝐹 = 4 8 . 5 9 N

Q16:

Coplanar forces of magnitudes 18 N, 5 N, 𝐹 N, 9 N, 𝐾 N, and 13 N are acting toward a particle, where the angle between each two consecutive forces is 60∘. 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 2 N , 𝐾 = 1 4 N
  • C 𝐹 = 2 7 N , 𝐾 = 1 4 N
  • D 𝐹 = 4 7 . 5 N , 𝐾 = 5 0 N

Q17:

A body weighing 61 kg-wt is placed on a smooth plane inclined at 30∘ to the horizontal. Given that it is maintained in a state of equilibrium by means of a force inclined at 60∘ above the horizontal, find the magnitude of the force 𝐹 and the reaction 𝑅 of the plane.

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

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