Worksheet: Triangle of Forces

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

Q1:

Adel is attempting a mechanics problem in which three coplanar forces , , and 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.

Adel 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, and meet head-to-head. Therefore, the forces do not actually form a triangle.

Q2:

Three coplanar forces , , and are acting on a body in equilibrium. Their triangle of forces forms a right triangle as shown.

Given that and , find the magnitude of .

  • A 18 newtons
  • B newtons
  • CThere is not enough information to find the magnitude of ; you need to know the angles.
  • D 12 newtons
  • E 8 newtons

Q3:

A body weighing 6.4 N is suspended by two strings 𝐴 𝐶 and 𝐵 𝐶 of lengths 2.1 cm and 2.8 cm, respectively. Given that the two strings are fixed from the top on the same horizontal line and perpendicular to each other, find the tensions in the two strings 𝑇 𝐴 𝐶 and 𝑇 𝐵 𝐶 .

  • A 𝑇 = 1 0 . 6 7 𝐴 𝐶 N , 𝑇 = 3 . 8 4 𝐵 𝐶 N
  • B 𝑇 = 5 . 1 2 𝐴 𝐶 N , 𝑇 = 8 . 5 3 𝐵 𝐶 N
  • C 𝑇 = 1 0 . 6 7 𝐴 𝐶 N , 𝑇 = 8 . 5 3 𝐵 𝐶 N
  • D 𝑇 = 5 . 1 2 𝐴 𝐶 N , 𝑇 = 3 . 8 4 𝐵 𝐶 N

Q4:

A weight of 740 N is suspended from two strings of lengths 24 cm and 70 cm, respectively, which are connected to two points 74 cm apart on the same horizontal level. Find the magnitudes of tensions 𝑇 1 in the first string and 𝑇 2 in the second string.

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

Q5:

A string of length 78 cm is connected to a fixed point on the ceiling. On the other end hangs a body weighing 420 N. Determine the magnitude of the horizontal force 𝐹 required to maintain the body at a distance of 30 cm from the ceiling and the tension 𝑇 in the string.

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

Q6:

A weight of 24 g-wt is connected to the end of a string. The other end of the string is attached to the ceiling of a room. The weight is pulled by a horizontal force until the string is inclined at 6 0 ∘ to the vertical. Given that it is in equilibrium in this position, find the magnitudes of the horizontal force 𝐹 and the tension in the string 𝑇 .

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

Q7:

A string of length 94 cm fixes a body of weight 24 N to the ceiling. Determine the magnitude of the force 𝐹 acting perpendicular to the string required to maintain the body at a distance of 47 cm from the ceiling and the tension 𝑇 in the string.

  • A 𝐹 = 2 4 N , 𝑇 = 1 2 N
  • B 𝐹 = 1 2 N , 𝑇 = 1 2 √ 3 N
  • C 𝐹 = 1 2 √ 3 N , 𝑇 = 8 √ 3 N
  • D 𝐹 = 1 2 √ 3 N , 𝑇 = 1 2 N

Q8:

A string of length 30 cm is attached to two points 𝐴 and 𝐵 on the ceiling 27 cm apart. A horizontal force 𝐹 acts on a smooth ring through which the string passes such that the system is in equilibrium when the ring is situated vertically below 𝐵 and the string is taut. Given that the weight of the ring is 486 g-wt, find the force 𝐹 and the tension in the string 𝑇 .

  • A 𝑇 = 4 3 9 . 8 3 g - , 𝐹 = 8 7 7 . 2 3 g -
  • B 𝑇 = 2 4 3 . 6 7 g - , 𝐹 = 2 5 . 5 8 g -
  • C 𝑇 = 4 8 3 . 3 1 g - , 𝐹 = 4 3 7 . 4 g -
  • D 𝑇 = 4 3 9 . 8 3 g - w t , 𝐹 = 4 3 7 . 4 g -

Q9:

A body is under the effect of three forces of magnitudes 𝐹 , 𝐹 , 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 and 𝐹 .

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

Q10:

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

Q11:

A body weighing 240 N is attached at point 𝐵 by a string whose other end is fixed to a point 𝐴 on a vertical wall. The length of the string 𝐴 𝐵 is 30 cm. The body is also pulled by a horizontal string attached from point 𝐵 until point 𝐵 is 18 cm away from the wall. Determine the tensions 𝑇 in the horizontal string and 𝑇 in string 𝐴 𝐵 .

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

Q12:

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

  • A 5 ∶ 1 3 ∶ 1 2
  • B 1 2 ∶ 5 ∶ 1 3
  • C 1 3 ∶ 1 2 ∶ 5
  • D 5 ∶ 1 2 ∶ 1 3

Q13:

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

Q14:

A body weighing 85 N is placed on a smooth plane inclined at 4 5 ∘ 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 𝜃 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.