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Worksheet: Resultant of Two Forces

In this worksheet, we will practice finding the resultant of two forces acting on one point and finding the direction of the resultant.

Q1:

Two forces act on a particle. One of the forces has a magnitude of 14 N, but the magnitude of the other force is unknown. Given that the resultant force points in the direction of the angle bisector of the two forces, find the unknown magnitude.

Q2:

The angle between two forces of equal magnitude is 6 0 ∘ , and the magnitude of their resultant is 7 1 √ 3 N. What is the magnitude of the forces?

Q3:

Two forces, both of magnitude 𝐹 N, make an angle of 6 0 ∘ with each other. The magnitude of their resultant is 12 N more than that of two forces, both of magnitude 1 4 𝐹 N, which make an angle of 1 2 0 ∘ with each other. Determine the value of 𝐹 giving your answer to two decimal places.

Q4:

Forces of magnitudes 22 N and 42 N act at a point. What is the maximum possible value of their resultant?

Q5:

Two perpendicular forces of magnitudes ( 6 𝐹 βˆ’ 5 ) N and ( 𝐹 + 1 5 ) N act on a particle. If their resultant bisects the angle between them, what is the value of 𝐹 ?

Q6:

Two forces of magnitudes 35 N and 91 N are acting at a particle. Given that the resultant is perpendicular to the first force, find the magnitude of the resultant.

  • A 90 N
  • B 1 4 √ 6 1 N
  • C 126 N
  • D 84 N

Q7:

Two perpendicular forces of magnitudes 88 N and 44 N act at a point. Their resultant makes an angle πœƒ with the 88 N force. Find the value of s i n πœƒ .

  • A 1 2
  • B 2 √ 5 5
  • C2
  • D √ 5 5

Q8:

The resultant of two perpendicular forces 𝐹 = 6 1 N and 𝐹 = 1 7 2 N , makes an angle πœƒ with 𝐹 1 . Find 𝑅 the magnitude of the resultant, and determine the angle πœƒ , giving your answer to the nearest minute.

  • A 𝑅 = 5 √ 1 3 N , πœƒ = 1 9 2 6 β€² ∘
  • B 𝑅 = 2 3 N , πœƒ = 7 0 3 4 β€² ∘
  • C 𝑅 = 2 3 N , πœƒ = 1 9 2 6 β€² ∘
  • D 𝑅 = 5 √ 1 3 N , πœƒ = 7 0 3 4 β€² ∘
  • E 𝑅 = √ 2 3 N , πœƒ = 1 9 2 6 β€² ∘

Q9:

Two perpendicular forces, 𝐹 1 and 𝐹 2 , act at a point. Their resultant, 𝑅 , has magnitude 188 N and makes an angle of 6 0 ∘ with 𝐹 1 . Find the magnitudes of 𝐹 1 and 𝐹 2 .

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

Q10:

Two forces having magnitudes of 55 and 𝐹 newtons are acting on a particle. The first force is acting towards east, and the second force is acting in a direction inclined 2 3 ∘ to the north of west. The resultant is acting in a direction inclined 6 7 ∘ to the north of east. Determine the magnitudes of 𝐹 and the resultant 𝑅 rounded to two decimal places.

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

Q11:

Forces 𝐹 1 and 𝐹 2 act at a point. The magnitude of 𝐹 2 is 5 N more than that of 𝐹 1 . Their resultant is perpendicular to 𝐹 1 and has magnitude 5 √ 2 3 N. Find the magnitude of each force.

  • A 𝐹 = 1 4 . 2 7 1 N , 𝐹 = 1 9 . 2 7 2 N
  • B 𝐹 = 6 0 1 N , 𝐹 = 5 5 2 N
  • C 𝐹 = 1 9 . 2 7 1 N , 𝐹 = 1 4 . 2 7 2 N
  • D 𝐹 = 5 5 1 N , 𝐹 = 6 0 2 N

Q12:

Two forces 𝐹 1 and 𝐹 2 are acting at a point. The value of 𝐹 1 is 3 N more than 𝐹 2 . If the magnitude of their resultant is 39 N and is perpendicular to the smaller force, find the magnitudes of 𝐹 2 and 𝐹 1 and the measure of the angle πœƒ between them correct to the nearest minute.

  • A 𝐹 = 5 0 4 2 N , 𝐹 = 5 0 7 1 N , πœƒ = 1 7 3 4 6 β€² ∘
  • B 𝐹 = 2 5 5 0 2 N , 𝐹 = 2 5 5 3 1 N , πœƒ = 1 7 7 1 3 β€² ∘
  • C 𝐹 = 1 6 8 2 N , 𝐹 = 1 7 1 1 N , πœƒ = 1 6 9 1 5 β€² ∘
  • D 𝐹 = 2 5 2 2 N , 𝐹 = 2 5 5 1 N , πœƒ = 1 7 1 1 2 β€² ∘

Q13:

The measure of the angle between two forces is 1 2 0 ∘ and the magnitude of their resultant is 79 N. Find their magnitudes, given that they have a difference of 51 N.

  • A 10 N, 61 N
  • B 35 N, 86 N
  • C 14 N, 65 N
  • D 40 N, 91 N

Q14:

Two forces of magnitudes 𝐹 N and 72 N ( 𝐹 < 7 2 ) act at a point. The angle between them is πœƒ , where t a n πœƒ = βˆ’ √ 3 3 , and their resultant is perpendicular to 𝐹 . Find 𝑅 , the magnitude of their resultant, and find the value of 𝐹 .

  • A 𝑅 = 7 2 N , 𝐹 = 3 6 √ 3
  • B 𝑅 = 3 6 √ 3 N , 𝐹 = 3 6
  • C 𝑅 = 3 6 √ 3 N , 𝐹 = 3 6 √ 3
  • D 𝑅 = 3 6 N , 𝐹 = 3 6 √ 3

Q15:

Two forces with an angle of 𝛼 between them, where t a n 𝛼 = βˆ’ √ 3 3 , are acting at a point. If the magnitude of the greater force is 122 N, and the resultant of the two forces is perpendicular to the smaller force, determine the magnitudes of the smaller force 𝐹 and the resultant 𝑅 .

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

Q16:

Forces of magnitudes 4 N and 2 8 √ 3 N act at a point. The measure of the angle between them is 3 0 ∘ . If a third force of magnitude 8 N is applied at the same point, what are the maximum and minimum possible values of the magnitude of 𝑅 , the resultant of the three forces?

  • A 𝑅 = 6 0 m a x N , 𝑅 = 5 2 m i n N
  • B 𝑅 = 5 2 m a x N , 𝑅 = 4 4 m i n N
  • C 𝑅 = 4 0 m a x N , 𝑅 = 2 4 m i n N
  • D 𝑅 = 6 0 m a x N , 𝑅 = 4 4 m i n N

Q17:

Two forces, both of magnitude 𝐹 N, act at the same point. The magnitude of their resultant is 90 N. When the direction of one of the forces is reversed, the magnitude of their resultant is 90 N. Determine the value of 𝐹 .

  • A 𝐹 = 9 0 √ 2
  • B 𝐹 = 9 0
  • C 𝐹 = 3 √ 5
  • D 𝐹 = 4 5 √ 2

Q18:

Two forces act together to produce a resultant. The magnitude of the smaller force is 1 6 that of the larger. If the smaller force is increased by 11 N and the larger force is doubled, then the resultant remains in the same direction as before. Find the magnitude of the two original forces.

  • A 3 3 5 N, 1 9 8 5 N
  • B 1 1 1 0 N, 3 3 5 N
  • C 66 N, 396 N
  • D 11 N, 66 N

Q19:

Given that ⃑ 𝐹 β«½ ⃑ 𝐹 1 2 and their resultant is ⃑ 𝑅 = 2 ⃑ 𝐹 1 , where ⃑ 𝐹 = βˆ’ 8 ⃑ 𝑖 βˆ’ 7 ⃑ 𝑗 2 , determine ⃑ 𝐹 1 .

  • A βˆ’ 2 4 ⃑ 𝑖 βˆ’ 2 1 ⃑ 𝑗
  • B βˆ’ 1 6 ⃑ 𝑖 βˆ’ 1 4 ⃑ 𝑗
  • C 8 ⃑ 𝑖 + 7 ⃑ 𝑗
  • D βˆ’ 8 ⃑ 𝑖 βˆ’ 7 ⃑ 𝑗

Q20:

A force of 20 N is acting due east on a cube. We can represent every 2 N of this force by a directed line of length 7 cm. If an additional force of magnitude 2 N acting in the same direction is added to the previous force, determine the magnitude 𝐹 of the force acting on the cube and the length 𝐿 of the directed line segment that represents this force.

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

Q21:

The angle between forces 𝐹 = 9 1 N and 𝐹 = 3 1 2 N is 1 2 0 ∘ . Find 𝑅 , the magnitude of the resultant force, giving your answer to 2 decimal places if necessary. Find πœƒ , the measure of the angle between the resultant and 𝐹 1 , giving your answer to the nearest minute.

  • A 𝑅 = 3 9 . 0 5 N , πœƒ = 1 5 6 3 7 β€² ∘
  • B 𝑅 = 3 6 . 3 5 N , πœƒ = 1 0 3 3 7 β€² ∘
  • C 𝑅 = 2 7 . 6 2 N , πœƒ = 1 5 6 3 7 β€² ∘
  • D 𝑅 = 2 7 . 6 2 N , πœƒ = 1 0 3 3 7 β€² ∘

Q22:

The angle between forces 𝐹 1 and 𝐹 2 is 1 1 2 ∘ , and the measure of the angle between their resultant and 𝐹 2 is 5 6 ∘ . If the magnitude of 𝐹 1 is 28 N, what is the magnitude of 𝐹 2 ?

Q23:

Forces of magnitudes 7 𝐹 N and 8 𝐹 N act on a particle. Given that their resultant is perpendicular to the first force, find the measure of the angle between them to the nearest minute.

  • A 7 1 1 β€² ∘
  • B 2 8 5 7 β€² ∘
  • C 4 8 4 9 β€² ∘
  • D 1 5 1 3 β€² ∘

Q24:

Forces and have magnitudes of 14 N and 18 N respectively. The cosine of the angle between the forces is . Find , the magnitude of the resultant force giving your answer to 2 decimal places, and find , the measure of the angle between the resultant and , giving your answer to the nearest degree.

  • A ,
  • B ,
  • C ,
  • D ,
  • E ,

Q25:

Two forces 𝐹 = 3 9 1 N and 𝐹 = 3 9 √ 2 2 N act at a point. The measure of the angle between them is 4 5 ∘ . Determine 𝑅 , the magnitude of their resultant, and find πœƒ , the measure of the angle between their resultant and 𝐹 1 . Give your answer to the nearest minute.

  • A 𝑅 = 7 8 N , πœƒ = 1 8 2 6 ∘ β€²
  • B 𝑅 = 3 9 √ 5 N , πœƒ = 1 8 2 6 ∘ β€²
  • C 𝑅 = 7 8 N , πœƒ = 2 6 3 4 ∘ β€²
  • D 𝑅 = 3 9 √ 5 N , πœƒ = 2 6 3 4 ∘ β€²

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