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Worksheet: Triangle of Forces

In this worksheet, we will practice solving problems about the equilibrium of a rigid body using the triangle of forces method.

Q1:

In the figure, three forces of magnitudes , , and newtons meet at a point. The lines of action of the forces are parallel to the sides of the right triangle. Given that the system is in equilibrium, find .

  • A
  • B
  • C
  • D

Q2:

Noah is attempting a mechanics problem in which three coplanar forces F 1 , F 2 , and F 3 are acting on a body. He needs to determine whether the body is in equilibrium or not. He remembers his teacher saying something about checking whether he could arrange the forces into a triangle. So, he draws the shown figure.

Noah concludes that the three forces are in equilibrium. Is he correct?

  • ANo
  • BYes

Which of the following best describes what he has done?

  • AHe has put the forces in the wrong order. He should have started with the force represented by the longest arrow and worked his way to the shortest.
  • BHe has used the wrong method; a force triangle is not a valid way of checking for equilibrium.
  • CHe has done nothing wrong.
  • DHe has not paid attention to the direction of the forces. All the forces should meet head-to-tail. However, in his diagram, F 2 and F 3 meet head-to-head. Therefore, the forces do not actually form a triangle.

Q3:

A body of weight 𝑊 is attached to a wall by a string of length 25 cm. It is held in equilibrium by the effect of a horizontal force of magnitude 93 g-wt that keeps the body 15 cm away from the wall. Determine 𝑇 and 𝑊 .

  • A 𝑇 = 5 5 . 8 g - , 𝑊 = 6 9 . 7 5 g -
  • B 𝑇 = 1 5 5 g - , 𝑊 = 6 9 . 7 5 g -
  • C 𝑇 = 5 5 . 8 g - , 𝑊 = 1 1 6 . 2 5 g -
  • D 𝑇 = 1 5 5 g - w t , 𝑊 = 1 2 4 g -

Q4:

A body weighing 1 170 g-wt is suspended from a wall by a string. Another string attached to the first one is pulled horizontally over a smooth pulley by a weight of 465 g-wt attached to its end. Find the tension 𝑇 of first string, giving your answer to the nearest integer, and the angle 𝜃 the string makes with the vertical, stating your answer to the nearest minute if necessary.

  • A 𝑇 = 1 0 7 4 g - , 𝜃 = 6 6 3 5 ′ ∘
  • B 𝑇 = 1 2 5 9 g - , 𝜃 = 6 8 2 0 ′ ∘
  • C 𝑇 = 1 0 7 4 g - , 𝜃 = 2 3 2 5 ′ ∘
  • D 𝑇 = 1 2 5 9 g - w t , 𝜃 = 2 1 4 0 ′ ∘

Q5:

𝐴 𝐵 is a light string whose two ends are fixed to two horizontal points on a horizontal line. Two weights 𝐾 and 40 g-wt are suspended from the string at points 𝐶 and 𝐷 respectively. This system of forces is in equilibrium when 𝐶 𝐷 is horizontal, and the two parts 𝐴 𝐶 and 𝐵 𝐷 are inclined to the horizontal at angles of 3 0 ∘ and 3 0 ∘ respectively. Determine the tensions in the strings: 𝑇 1 in 𝐴 𝐶 , 𝑇 2 in 𝐶 𝐷 , and 𝑇 3 in 𝐷 𝐵 . Also, find the value of 𝐾 .

  • A 𝑇 = 8 0 √ 3 1 g - , 𝑇 = 4 0 √ 3 2 g - , 𝑇 = 8 0 √ 3 3 g - , 𝐾 = 4 0 g -
  • B 𝑇 = 8 0 1 g - , 𝑇 = 4 0 2 g - , 𝑇 = 8 0 3 g - , 𝐾 = 4 0 √ 3 g -
  • C 𝑇 = 8 0 √ 3 3 1 g - , 𝑇 = 4 0 2 g - , 𝑇 = 8 0 √ 3 3 3 g - , 𝐾 = 4 0 √ 3 g -
  • D 𝑇 = 8 0 1 g - w t , 𝑇 = 4 0 √ 3 2 g - , 𝑇 = 8 0 3 g - , 𝐾 = 4 0 g -

Q6:

A body weighing 240 N is attached to a string whose other end is fixed to a point 𝐴 on a vertical wall. Another string is attached to the first at point 𝐵 which is 30 cm away from point 𝐴 . It is pulled horizontally until point 𝐵 is 18 cm away from the wall. Determine the tensions 𝑇 1 in the horizontal string and 𝑇 2 in 𝐴 𝐵 .

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

Q7:

A uniform rod of length 50 cm and weight 143 N is freely suspended at its ends from the ceiling by means of two perpendicular strings. Given that the length of one of the strings is 30 cm, determine the tension in each string.

  • A 85.8 N, 178.75 N
  • B 238.33 N, 178.75 N
  • C 238.33 N, 114.4 N
  • D 85.8 N, 114.4 N

Q8:

A body is under the effect of three forces of magnitudes 𝐹 1 , 𝐹 2 , and 36 newtons, acting in the directions 𝐴 𝐵 , 𝐵 𝐶 , and 𝐴 𝐶 , respectively, where △ 𝐴 𝐵 𝐶 is a triangle such that 𝐴 𝐵 = 4 c m , 𝐵 𝐶 = 6 c m , and 𝐴 𝐶 = 6 c m . Given that the system is in equilibrium, find 𝐹 1 and 𝐹 2 .

  • A 𝐹 = 5 4 1 N , 𝐹 = 3 6 2 N
  • B 𝐹 = 3 6 1 N , 𝐹 = 2 4 2 N
  • C 𝐹 = 2 4 1 N , 𝐹 = 5 4 2 N
  • D 𝐹 = 2 4 1 N , 𝐹 = 3 6 2 N

Q9:

A body weighing 85 N is placed on a smooth plane inclined at 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 gradient. Given that the tension in the string is of magnitude 62 N, find the size of the angle the string makes with the horizontal, giving your answer to the nearest minute, and the magnitude of the reaction of the plane on the body, stating your answer to the nearest two decimal places.