Worksheet: Lami's Theorem

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In this worksheet, we will practice solving problems about the equilibrium of a particle under the action of three coplanar forces using Lami's theorem.

Q1:

In the given figure, particle 𝐴 is in equilibrium under the effect of the forces shown which are in newtons. Find the force 𝐹.

  • A − 3 1 N
  • B62 N
  • C − 3 1 √ 3 N
  • D 3 1 √ 3 N

Q2:

A body weighing 12 N is attached to one end of a light, inextensible string. The other end of the string is fixed to a vertical wall. A horizontal force 𝐹, holds the body in equilibrium when the measure of the angle between the wall and the string is 30∘. Find 𝑇, the tension in the string and 𝐹, the horizontal force.

  • A 𝑇 = 8 √ 3 N , 𝐹 = 4 √ 3 N
  • B 𝑇 = 2 4 N , 𝐹 = 4 √ 3 N
  • C 𝑇 = 4 √ 3 N , 𝐹 = 8 √ 3 N
  • D 𝑇 = 8 √ 3 N , 𝐹 = 2 4 N

Q3:

A body weighing 𝑊 N is placed on a smooth plane inclined at 45∘ to the horizontal. If it is kept in equilibrium under the action of a horizontal force of magnitude 33 N, find the weight of the body 𝑊 and the reaction of the plane 𝑅.

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

Q4:

A body of weight 𝑊 is placed on a smooth plane inclined at 60∘ to the horizontal. The body is kept in equilibrium under the action of a force of magnitude 54 N, which acts up the slope parallel to the line of greatest slope. Find the magnitude of the reaction 𝑅 of the plane on the body and its weight 𝑊.

  • A 𝑊 = 2 7 √ 3 N , 𝑅 = 1 8 √ 3 N
  • B 𝑊 = 3 6 √ 3 N , 𝑅 = 1 8 √ 3 N
  • C 𝑊 = 1 0 8 N , 𝑅 = 5 4 √ 3 N
  • D 𝑊 = 1 8 √ 3 N , 𝑅 = 3 6 √ 3 N
  • E 𝑊 = 3 6 √ 3 N , 𝑅 = 3 6 √ 3 N

Q5:

A body weighing 143 N is placed on a smooth plane inclined at an angle 𝜃 to the horizontal. The body is kept in equilibrium by means of a force of 70 N which acts at an angle 𝜃 above the line of greatest slope of the plane. Find the magnitude of the normal reaction of the plane, giving your answer to the nearest two decimal places.

Q6:

A home-built swing is suspended by a rope passing through two pulleys, 𝐵 and 𝐶, whose ends meet at point 𝐴 where the swing is attached. The rope segments 𝐴𝐵 and 𝐴𝐶 make angles of 40∘ and 45∘ with the horizontal, respectively. When a child sits still on the seat, the tension in 𝐴𝐵 is 199 N. Given that, in this condition, the system is in equilibrium, find the combined weight of the child and the seat 𝑊 and the tension 𝑇 in the rope 𝐴𝐶, giving your answers to the nearest two decimal places.

  • A 𝑊 = 1 4 1 . 2 5 N , 𝑇 = 2 1 5 . 5 9 N
  • B 𝑊 = 2 8 0 . 3 6 N , 𝑇 = 1 5 3 . 0 3 N
  • C 𝑊 = 2 8 0 . 3 6 N , 𝑇 = 2 1 5 . 5 9 N
  • D 𝑊 = 1 4 1 . 2 5 N , 𝑇 = 1 5 3 . 0 3 N

Q7:

A sphere is resting on two laminae. The distance between the two points of contact is equal to the sphere’s radius. Determine the reaction of each lamina on the sphere, given that the weight of the sphere is 261 N.

  • A 𝑅 = 2 6 1 √ 3 N , 𝑅 = 2 6 1 N
  • B 𝑅 = 1 3 0 . 5 N , 𝑅 = 1 3 0 . 5 N
  • C 𝑅 = 2 6 1 N , 𝑅 = 1 3 0 . 5 N
  • D 𝑅 = 8 7 √ 3 N , 𝑅 = 1 3 0 . 5 N
  • E 𝑅 = 8 7 √ 3 N , 𝑅 = 8 7 √ 3 N

Q8:

In a square 𝐴𝐵𝐶𝐷, 𝑀 is the point of intersection of the two diagonals, 𝐸 is the midpoint of 𝐴𝐵, and 𝐹 is the midpoint of 𝐵𝐶. Three forces of magnitudes 𝐹, 𝐹, and 41 newtons are acting at 𝑀 in the directions 𝑀𝐸, 𝑀𝐹, and 𝑀𝐷, respectively. Given that the three forces are in equilibrium, find the values of 𝐹 and 𝐹.

  • A 𝐹 = 4 1 √ 2 2 , 𝐹 = 4 1
  • B 𝐹 = 4 1 √ 2 2 , 𝐹 = 4 1 √ 2 2
  • C 𝐹 = 4 1 √ 3 2 , 𝐹 = 4 1 2
  • D 𝐹 = 2 0 . 5 , 𝐹 = 2 0 . 5

Q9:

A weight of 7 N is suspended from the end of a string whose other end is fixed to a point on a vertical wall. A force acts on the weight perpendicular to the string. Given that the system is in equilibrium when the string is inclined at 30∘ to the wall, find the magnitude of the force 𝐹 and the tension in the string 𝑇.

  • A 𝐹 = 7 √ 3 2 N , 𝑇 = 7 2 N
  • B 𝐹 = 7 2 N , 𝑇 = 7 √ 3 2 N
  • C 𝐹 = 7 √ 3 3 N , 𝑇 = 1 4 √ 3 3 N
  • D 𝐹 = 1 4 √ 3 3 N , 𝑇 = 7 √ 3 3 N

Q10:

A force 𝐹 is acting perpendicular to a pendulum weighing 4.4 N such that the pendulum is held steady at an angle of 70∘ to the vertical. Find the magnitude of the force 𝐹 required to maintain this equilibrium, and find the resulting tension 𝑇 in the string, giving your answers to the nearest newton.

  • A 𝐹 = 1 N , 𝑇 = 4 N
  • B 𝐹 = 1 2 N , 𝑇 = 1 2 N
  • C 𝐹 = 4 N , 𝑇 = 1 N
  • D 𝐹 = 4 N , 𝑇 = 4 N

Q11:

A body of weight 118 N is placed on a smooth plane inclined to the horizontal at an angle whose cosine is 45. It is prevented from sliding by means of a horizontal force 𝐹. Find the magnitudes of the force 𝐹 and the reaction of the plane 𝑅.

  • A 𝐹 = 8 8 . 5 N , 𝑅 = 1 4 7 . 5 N
  • B 𝐹 = 9 4 . 4 N , 𝑅 = 1 4 7 . 5 N
  • C 𝐹 = 7 0 . 8 N , 𝑅 = 9 4 . 4 N
  • D 𝐹 = 8 8 . 5 N , 𝑅 = 9 4 . 4 N

Q12:

A body weighing 𝑊 is suspended by two strings. The first makes an angle 𝜃 with the vertical and then passes over a smooth pulley and has a weight of 13 N at its end. The other string makes an angle of 56∘ with the vertical and then passes over a smooth pulley and has a weight of 14 N at its end. Given that the system is in equilibrium in this state, find 𝑊 to the nearest newton and 𝜃 to the nearest minute.

  • A 𝑊 = 2 3 N , 𝜃 = 6 3 1 4 ∘
  • B 𝑊 = 2 3 N , 𝜃 = 5 2 5 8 ∘
  • C 𝑊 = 1 4 N , 𝜃 = 6 3 1 4 ∘
  • D 𝑊 = 1 4 N , 𝜃 = 5 2 5 8 ∘

Q13:

A body weighing 620 N is placed on a smooth plane inclined to the horizontal at an angle 𝜃, where sin𝜃=0.6. Given that the body is kept in equilibrium by means of a horizontal force 𝐹, determine the magnitude of 𝐹 and the reaction 𝑅 of the plane on the body.

  • A 𝐹 = 6 2 0 N , 𝑅 = 7 7 5 N
  • B 𝐹 = 6 2 0 N , 𝑅 = 9 9 2 N
  • C 𝐹 = 4 6 5 N , 𝑅 = 4 9 6 N
  • D 𝐹 = 4 6 5 N , 𝑅 = 7 7 5 N
  • E 𝐹 = 3 7 2 N , 𝑅 = 4 9 6 N

Q14:

A body weighing 18 N is placed on a smooth surface that is inclined at 60∘ to the horizontal. Given that the body is supported by a horizontal force 𝐹, determine 𝐹 and the reaction 𝑟 of the surface.

  • A 𝐹 = 3 6 √ 3 N , 𝑟 = 3 6 N
  • B 𝐹 = 1 8 √ 3 N , 𝑟 = 3 6 √ 2 N
  • C 𝐹 = 1 8 √ 3 N , 𝑟 = 3 6 N
  • D 𝐹 = 3 6 √ 3 N , 𝑟 = 3 6 √ 2 N

Q15:

A particle is in equilibrium under the action of three coplanar forces of magnitudes 𝐹 N, 15 N, and 25 N. Given that the last two forces are perpendicular to each other, determine the value of 𝐹.

  • A 5 √ 3 4 N
  • B20 N
  • C17.49 N
  • D 5 √ 1 7 N

Q16:

The diagram shows a body weighing 6.1 N suspended in equilibrium by two light, inextensible strings, 𝐵𝐶 and 𝐴𝐶. Given that 𝐵𝐶=2.4cm, 𝐴𝐶=3.2cm, and the strings are perpendicular, find the tensions 𝑇 and 𝑇.

  • A 𝑇 = 4 . 8 8 N , 𝑇 = 8 . 1 3 N
  • B 𝑇 = 1 0 . 1 7 N , 𝑇 = 3 . 6 6 N
  • C 𝑇 = 1 0 . 1 7 N , 𝑇 = 8 . 1 3 N
  • D 𝑇 = 4 . 8 8 N , 𝑇 = 3 . 6 6 N

Q17:

A weight of 90 g-wt is suspended by two inextensible strings. The first is inclined at an angle 𝜃 to the vertical, and the second is at 30∘ to the vertical. If the magnitude of the tension in the first string is 45 g-wt, find 𝜃 and the magnitude of the tension 𝑇 in the second string.

  • A 𝜃 = 6 0 ∘ , 𝑇 = 4 5 g - w t
  • B 𝜃 = 6 0 ∘ , 𝑇 = 9 0 √ 3 g - w t
  • C 𝜃 = 6 0 ∘ , 𝑇 = 4 5 √ 3 g - w t
  • D 𝜃 = 3 0 ∘ , 𝑇 = 4 5 √ 3 g - w t
  • E 𝜃 = 3 0 ∘ , 𝑇 = 4 5 g - w t

Q18:

A weight of 81 g-wt is suspended from two perpendicular strings. The tension in the first string is 21 g-wt, and it makes an angle 𝜃 with the vertical, while the tension in the second string is 46 g-wt, and it makes an angle 𝜃 with the vertical. Given that the system is in equilibrium, find 𝜃 and 𝜃, giving your answer to the nearest minute.

  • A 𝜃 = 1 5 2 ′ ∘ , 𝜃 = 7 4 5 8 ′ ∘
  • B 𝜃 = 5 5 2 4 ′ ∘ , 𝜃 = 3 4 3 6 ′ ∘
  • C 𝜃 = 7 4 5 8 ′ ∘ , 𝜃 = 1 5 2 ′ ∘
  • D 𝜃 = 3 4 3 6 ′ ∘ , 𝜃 = 5 5 2 4 ′ ∘

Q19:

A light string forms figure 𝐴𝐵𝐶𝐷, where 𝐴 and 𝐷 are fixed points on the horizontal line 𝐴𝐷. A body weighing 42 g-wt is suspended from 𝐵, and a horizontal force of magnitude 𝐹 pulls the string at 𝐶 until 𝐶𝐷 is vertical. Given that 𝐵𝐶 makes an angle of 60∘ with the horizontal and 𝐴𝐵⟂𝐵𝐶, find the tensions 𝑇 and 𝑇 and the force 𝐹.

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

Q20:

In the figure, a horizontal force of magnitude 890 N is acting on a particle at 𝐶 which is attached to two strings connected to 𝐴 and 𝐵 respectively. Given that the particle is in equilibrium and the two strings and the particle all lie in the same vertical plane, find the tension in the two strings to the nearest newton.

  • A 𝑇 = 2 6 9 N , 𝑇 = 6 3 9 N
  • B 𝑇 = 7 4 0 N , 𝑇 = 1 , 2 4 0 N
  • C 𝑇 = 7 4 0 N , 𝑇 = 6 3 9 N
  • D 𝑇 = 6 3 9 N , 𝑇 = 2 1 9 N
  • E 𝑇 = 2 1 9 N , 𝑇 = 2 6 9 N

Q21:

In the figure, a force of magnitude 390 N is acting on a particle at 𝐶 making an angle of 73∘ with the horizontal. Two strings are connected to the particle at 𝐶, and their other ends are attached to 𝐴 and 𝐵 on the same horizontal line. Given that the particle is in equilibrium, find the tension in each string to the nearest newton.

  • A 𝑇 = 1 3 8 N , 𝑇 = 2 5 7 N
  • B 𝑇 = 2 5 7 N , 𝑇 = 1 4 7 N
  • C 𝑇 = 1 4 7 N , 𝑇 = 1 3 8 N
  • D 𝑇 = 2 2 8 N , 𝑇 = 5 9 3 N
  • E 𝑇 = 2 2 8 N , 𝑇 = 2 5 7 N

Q22:

A body, placed on a smooth plane inclined at 30∘ to the horizontal, is kept in equilibrium by means of two forces. One of them, of magnitude of 22 N, acts up the plane parallel to the line of greatest slope. The second, of magnitude 83 N, acts upward at 60∘ to the line of the greatest slope of the plane. Find the weight of the body 𝑊 and the reaction of the plane 𝑅.

  • A 𝑊 = 8 3 2 N , 𝑅 = 1 2 7 2 N
  • B 𝑊 = 1 2 7 N , 𝑅 = 2 2 √ 3 N
  • C 𝑊 = 2 2 √ 3 N , 𝑅 = 1 2 7 N
  • D 𝑊 = 1 2 7 2 N , 𝑅 = 8 3 2 N

Q23:

A body of weight 90 kg-wt is placed on a smooth plane that is inclined at 30∘ to the horizontal. If the body is held in equilibrium by means of a force 𝐹 that acts at an angle of 30∘ above the plane, determine the magnitudes of 𝐹 and 𝑟, where 𝑟 is the reaction of the plane on the body.

  • A 𝐹 = 9 0 √ 3 k g - w t , 𝑟 = 1 8 0 k g - w t
  • B 𝐹 = 9 0 k g - w t , 𝑟 = 3 0 √ 3 k g - w t
  • C 𝐹 = 3 0 √ 3 k g - w t , 𝑟 = 3 0 √ 3 k g - w t
  • D 𝐹 = 1 8 0 k g - w t , 𝑟 = 9 0 √ 3 k g - w t

Q24:

A body of weight 𝑊 is suspended from two strings. The first makes an angle 𝜃 with the vertical and then passes over a smooth pulley and has a weight of 7.3 N at the end. The second string makes an angle of 37∘ with the vertical and then passes over another smooth pulley and has a weight of 4.4 N at the end. Given that the system is in equilibrium, find the values of 𝜃 to the nearest minute and 𝑊 to the nearest two decimal places.

  • A 𝜃 = 6 1 1 4 ′ ∘ , 𝑊 = 0 . 7 9 N
  • B 𝜃 = 8 6 5 0 ′ ∘ , 𝑊 = 6 . 0 7 N
  • C 𝜃 = 4 9 5 0 ′ ∘ , 𝑊 = 5 . 5 9 N
  • D 𝜃 = 2 1 1 6 ′ ∘ , 𝑊 = 1 0 . 3 2 N

Q25:

A car was stuck in the mud. A 20-meter-long cable was stretched tightly from the car to a tree. A person grasped the cable from the middle and pulled with a force of 100 newtons perpendicular to the stretched cable. The center of the cable moved 2 meters and remained still. What was the tension in the cable, knowing that it is extensible?

  • A 𝑇 = 5 0 √ 2 6 N
  • B 𝑇 = 1 0 √ 6 2 6 N
  • C 𝑇 = 2 0 √ 6 2 6 N
  • D 𝑇 = 1 0 0 √ 2 6 N
  • E 𝑇 = 5 0 √ 6 2 6 1 3 N

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