Lesson Worksheet: Lami’s Theorem Mathematics

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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.

  • Aβˆ’31 N
  • B62 N
  • Cβˆ’31√3 N
  • D31√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.

  • A⃑𝑇=8√3N, ⃑𝐹=4√3N
  • B⃑𝑇=24N, ⃑𝐹=4√3N
  • C⃑𝑇=4√3N, ⃑𝐹=8√3N
  • D⃑𝑇=8√3N, ⃑𝐹=24N

Q3:

In the figure, a horizontal force of magnitude 890 N is acting on a particle at 𝐢 which is attached to two strings connected to 𝐴 and 𝐡 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.

  • A𝑇=269N, 𝑇=639N
  • B𝑇=740N, 𝑇=1240N
  • C𝑇=740N, 𝑇=639N
  • D𝑇=639N, 𝑇=219N
  • E𝑇=219N, 𝑇=269N

Q4:

A body weighing π‘Š 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 π‘Š and the reaction of the plane 𝑅.

  • Aπ‘Š=33N, 𝑅=33√2N
  • Bπ‘Š=33N, 𝑅=22√3N
  • Cπ‘Š=33√22N, 𝑅=33√2N
  • Dπ‘Š=33√22N, 𝑅=22√3N

Q5:

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 π‘Ÿ, where π‘Ÿ is the reaction of the plane on the body.

  • A𝐹=90√3kg-wt, π‘Ÿ=180kg-wt
  • B𝐹=90kg-wt, π‘Ÿ=30√3kg-wt
  • C𝐹=30√3kg-wt, π‘Ÿ=30√3kg-wt
  • D𝐹=180kg-wt, π‘Ÿ=90√3kg-wt

Q6:

A light string is tied from one end to a point on the surface of a homogenous sphere and the other end is attached to a point on a vertical smooth wall. The sphere is resting against the wall and weighs 33 N, and the string inclines to the vertical by 30∘. Find the tension 𝑇 in the string and the reaction 𝑅 of the wall.

  • A𝑇=66N, 𝑅=57.16N
  • B𝑇=22√3N, 𝑅=11√3N
  • C𝑇=22√3N, 𝑅=57.16N
  • D𝑇=33N, 𝑅=16.5N

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.

  • AοƒŸπ‘…=261√3N, οƒŸπ‘…=261N
  • BοƒŸπ‘…=130.5N, οƒŸπ‘…=130.5N
  • CοƒŸπ‘…=261N, οƒŸπ‘…=130.5N
  • DοƒŸπ‘…=87√3N, οƒŸπ‘…=130.5N
  • EοƒŸπ‘…=87√3N, οƒŸπ‘…=87√3N

Q8:

A weight of 90 g-wt is suspended by two inextensible strings. The first is inclined at an angle πœƒ 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 πœƒ and the magnitude of the tension 𝑇 in the second string.

  • Aπœƒ=60∘, 𝑇=45g-wt
  • Bπœƒ=60∘, 𝑇=90√3g-wt
  • Cπœƒ=60∘, 𝑇=45√3g-wt
  • Dπœƒ=30∘, 𝑇=45√3g-wt
  • Eπœƒ=30∘, 𝑇=45g-wt

Q9:

As shown in the figure, a block weighing π‘Š N is suspended from a string connected to two other pieces of string, each of which passes over a smooth pulley. Given that two bodies 𝐴 and 𝐡, weighing 50 and 48newtons, respectively, are attached to the other ends of the strings and that the system is in equilibrium, determine πœƒ to the nearest minute and π‘Š to the nearest two decimal places.

  • Aπ‘Š=67.82N, πœƒ=4810∘
  • Bπ‘Š=67.82N, πœƒ=4234∘
  • Cπ‘Š=65.11N, πœƒ=4726∘
  • Dπ‘Š=65.11N, πœƒ=4234∘
  • Eπ‘Š=67.82N, πœƒ=4726∘

Q10:

A body weighing 85 N is placed on a smooth plane inclined at 45∘ to the horizontal. The body is kept in equilibrium by an inextensible string fixed to a point on a vertical wall at the top of the slope. Given that the tension in the string is of magnitude 62 N, find the measure of the angle πœƒ the string makes with the horizontal, giving your answer to the nearest minute, and the magnitude of the reaction 𝑅 of the plane on the body, stating your answer to the nearest two decimal places.

  • Aπœƒ=5912β€²βˆ˜, 𝑅=44.89N
  • Bπœƒ=5912β€²βˆ˜, 𝑅=62.46N
  • Cπœƒ=7547β€²βˆ˜, 𝑅=32.74N
  • Dπœƒ=4515β€²βˆ˜, 𝑅=61.54N

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