Worksheet: Resultant of Coplanar Forces

In this worksheet, we will practice finding the resultant of a group of forces acting on a point.

Q1:

The diagram shows three coplanar forces acting at point 𝑀. Their magnitudes are 2 N, 2 N, and 8 N in the directions 𝑀𝐴, 𝑀𝐡, and 𝑀𝐢 respectively. Given that π‘šβˆ π΄π‘€π΅=60∘ and π‘šβˆ π΄π‘€πΆ=90∘, what is the magnitude of the resultant force? Give your answer to the nearest newton.

Q2:

The forces Fij=2+2, Fij=π‘Ž+9, and Fij=9+𝑏 act on a particle, where i and j are two perpendicular unit vectors. Given the forces’ resultant Rij=2βˆ’6, determine the values of π‘Ž and 𝑏.

  • A π‘Ž = βˆ’ 5 , 𝑏 = βˆ’ 1 3
  • B π‘Ž = 1 3 , 𝑏 = 5
  • C π‘Ž = βˆ’ 9 , 𝑏 = βˆ’ 1 7
  • D π‘Ž = 9 , 𝑏 = 1

Q3:

Three forces, Fij=(βˆ’5+10)N, Fij=(π‘Žβˆ’5)N, and Fij=(βˆ’4+𝑏)N act at a point. Their resultant is 6√2 N northwest. Determine the values of π‘Ž and 𝑏.

  • A π‘Ž = 3 , 𝑏 = 1
  • B π‘Ž = 1 , 𝑏 = 3
  • C π‘Ž = 2 1 , 𝑏 = 7
  • D π‘Ž = 7 , 𝑏 = 2 1

Q4:

Three forces, (5+10)ijN, (π‘Žβˆ’5)ijN, and (15+(𝑏+7))ijN act on a particle. Given that the resultant of the forces is (18+19)ijN, what are the values of π‘Ž and 𝑏?

  • A π‘Ž = 8 , 𝑏 = 2 1
  • B π‘Ž = βˆ’ 2 , 𝑏 = 2 1
  • C π‘Ž = βˆ’ 2 , 𝑏 = 7
  • D π‘Ž = βˆ’ 2 , 𝑏 = 1 4
  • E π‘Ž = 2 8 , 𝑏 = 7

Q5:

Determine the magnitude of the resultant of the forces shown in the figure that are measured in newtons.

  • A 5 5 √ 2 2 N
  • B 1 3 √ 1 3 N
  • C 3 7 √ 2 2 N
  • D 9 √ 2 N

Q6:

A body has a force of 10 newtons acting on it horizontally, 25 newtons acting on it vertically upward, and 5 newtons acting on it at an angle of 45∘ to the horizontal as shown in the figure. What is the magnitude of the single resultant force acting on the body, and at what angle to the horizontal does it act? Give your answers correct to one decimal place.

  • A 22.4 newtons, 7 3 . 2 ∘
  • B 31.6 newtons, 6 4 . 6 ∘
  • C 31.6 newtons, 2 5 . 4 ∘
  • D 31.6 newtons, 2 4 4 . 6 ∘
  • E 22.4 newtons, 1 6 . 8 ∘

Q7:

Four forces act on a particle as shown in the diagram. Determine 𝑅, the magnitude of their resultant, and find πœƒ, the angle between their resultant and the π‘₯-axis. Give your answer to the nearest minute if necessary.

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

Q8:

The diagram shows a system of three forces measured in newtons. Given that 𝐴𝐡=24cm and 𝐴𝐷=18cm, determine 𝑅, the magnitude of their resultant, and find πœƒ, the angle between their resultant and the positive π‘₯-axis. Give to the nearest minute.

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

Q9:

The diagram shows a square, 𝐴𝐡𝐢𝐷, of side 8 cm. The point 𝐸 is on 𝐡𝐢 where 𝐡𝐸=6cm. Forces of magnitudes 8 N, 20 N, 16√2 N, and 12 N act at 𝐴 as indicated on the diagram. Find the magnitude of their resultant.

  • A32 N
  • B 8 √ 2 N
  • C16 N
  • D 4 √ 1 0 N
  • E 4 0 √ 2 N

Q10:

𝐴 𝐡 𝐢 is an equilateral triangle, where 𝑀 is the point of intersection of its medians. Three forces of magnitudes 48, 26, and 42 newtons are acting at 𝑀 in the directions of 𝑀𝐴, 𝑀𝐡, and 𝑀𝐢. Determine the magnitude of the resultant 𝑅 and its angle of inclination πœƒ with the positive direction of the π‘₯-axis rounded to the nearest minute.

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

Q11:

The diagram shows a regular hexagon, 𝐴𝐡𝐢𝐷𝐸𝐹, whose diagonals intersect at the point 𝑀. The 6 forces shown acting at 𝑀 are measured in newtons. Find 𝑅 the magnitude of their resultant and πœƒ, the angle between their resultant and the positive π‘₯-axis. Round your value of πœƒ to the nearest minute if necessary.

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

Q12:

The diagram shows a regular hexagon, 𝐴𝐡𝐢𝐷𝐸𝑂. Forces of magnitudes 59, 78√3, 𝐹, 78√3, and 𝐾 act in the directions shown on the diagram. Given that the resultant of the forces acts along 𝑂𝐢, and its magnitude is 298 N, determine 𝐹 and 𝐾.

  • A 𝐹 = 5 N , 𝐾 = 5 9 √ 3 N
  • B 𝐹 = 5 N , 𝐾 = 5 9 N
  • C 𝐹 = 5 √ 3 N , 𝐾 = 5 9 N
  • D 𝐹 = 5 √ 3 N , 𝐾 = 5 9 √ 3 N

Q13:

Forces of magnitudes 𝐹, 16, 𝐾, 18, 9√3 newtons act at a point in the directions shown on the diagram. Their resultant, 𝑅, has a magnitude of 20 N. Find the values of 𝐹 and 𝐾.

  • A 𝐹 = 3 6 , 𝐾 = 2 7 √ 3
  • B 𝐹 = 5 4 , 𝐾 = 9 √ 3
  • C 𝐹 = 2 0 , 𝐾 = 1 1 √ 3
  • D 𝐹 = 1 2 , 𝐾 = 1 9 √ 3

Q14:

Four coplanar concurrent forces are acting at the point 𝑂; where sinπœƒ=45. If the resultant of the forces forms an angle of 135∘ with the π‘₯-axis and has a magnitude of 37√2 N, find the values of 𝑃 and 𝑄.

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

Q15:

Given that the forces 𝐹, 26√2, 44√2, and 65 newtons are acting as shown in the figure, the magnitude of the resultant of the system of forces is 25√2 N, and the resultant force makes an angle πœƒ with the force 𝐹, determine the magnitude of 𝐹, also the angle πœƒ approximated to the nearest minute.

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

Q16:

Coplanar forces of magnitudes 𝐹 N, 8√3 N, √3 N, and 9√3 N act on a particle as shown in the diagram. Given that the magnitude of their resultant is 9√3 N, determine the value of 𝐹.

  • A300
  • B 1 0 √ 3
  • C √ 3
  • D 9 √ 3

Q17:

𝐴 𝐡 𝐢 is a triangle with a right angle at 𝐡, where 𝐴𝐡=32cm, 𝐡𝐢=24cm, 𝐷∈𝐴𝐢, and 𝐡𝐷=𝐷𝐢. Four forces of magnitudes 2, 3, 19, and 14 newtons are acting at point 𝐡 in the directions 𝐴𝐡, οƒͺ𝐡𝐢, 𝐢𝐴, and 𝐡𝐷, respectively. Find the magnitude of the resultant of these forces.

Q18:

𝐴 𝐡 𝐢 𝐷 is a rectangle in which 𝐴𝐡=5cm, 𝐡𝐢=12cm, and 𝐸∈𝐡𝐢 where 𝐡𝐸=5cm. Forces of magnitude 4, 13, 4√2, and 12 newtons are acting in the directions of 𝐴𝐷, 𝐴𝐢, 𝐴𝐸, and 𝐴𝐡 respectively. Find the magnitude of their resultant.

Q19:

𝐴 𝐡 𝐢 𝐷 is a square whose side length is 5 cm, where 𝐸 is the midpoint of 𝐡𝐢, and 𝐹 is the midpoint of 𝐷𝐢. Five forces having magnitudes of 19 N, 4√5 N, 20√2 N, 11√5 N, and 18 N are acting at the point 𝐴 in the directions of 𝐴𝐡, 𝐴𝐸, 𝐢𝐴, 𝐴𝐹, and 𝐴𝐷 respectively. Find the magnitude of the resultant of these forces.

Q20:

Three coplanar forces, each of magnitude 12 N act at a point. The angle between the first and second forces is equal to the angle between the second and third forces. Given that the measure of this angle is 34∘, find the magnitude of the resultant of the three forces. Give your answer to the nearest newton.

Q21:

Four coplanar forces are acting on a particle. The first force has a magnitude of 20 newtons. The second force acts 32 degrees counterclockwise from the first force and has a magnitude of 7 newtons. The third force acts 90 degrees counterclockwise from the second force and has a magnitude of 5 newtons. The fourth force acts 123 degrees counterclockwise from the third force and has a magnitude of 6 newtons. Find the magnitude of the resultant force acting on the particle.

Q22:

Forces 𝐹=14N, 𝐹=6N, and 𝐹=10N act as shown in the diagram. Find the magnitude of their resultant, giving your answer to two decimal places.

Q23:

Suppose a body has a force of 3 pounds acting on it to the left, 4 pounds acting on it upward, and 2 pounds acting on it 30∘ to the horizontal. What single force is needed to produce a state of equilibrium on the body? Determine its direction.

  • A 5.158 newtons, 1 6 5 . 7 7 ∘ south of east
  • B 6.884 newtons, 2 8 4 . 2 3 ∘ south of east
  • C 5.158 newtons, 7 5 . 7 7 ∘ south of east
  • D 6.884 newtons, 1 6 5 . 7 7 ∘ south of east
  • E 5.158 newtons, 1 0 4 . 2 3 ∘ south of east

Q24:

The condition of equilibrium is when the sum of the forces acting on a body is the zero vector. Suppose a body has a force of 2 pounds acting on it from the right, 5 pounds acting on it downward, and 3 pounds acting on it from 45∘ above the horizontal from the right as shown in the figure. Find the magnitude and direction of the single force needed to produce a state of equilibrium on the body, giving your answers correct to one decimal place if necessary.

  • A 8.2 newtons, 5 9 . 9 ∘ from the horizontal
  • B 2.9 newtons, 3 0 . 1 ∘ from the horizontal
  • C 8.2 newtons, 2 3 9 . 9 ∘ from the horizontal
  • D 2.9 newtons, 2 3 9 . 9 ∘ from the horizontal
  • E 8.2 newtons, 2 1 0 . 1 ∘ from the horizontal

Q25:

A body has three forces, in newtons, acting on it at angles of βˆ’135∘, 0∘ and βˆ’150∘ as shown in the figure. What is the magnitude of the resultant force in newtons? Taking counterclockwise as positive, what is the angle of inclination of the reaction force from the horizontal? Give your answers correct to one decimal place.

  • A 19.4 newtons, 3 8 . 3 ∘
  • B 37.8 newtons, 2 1 8 . 3 ∘
  • C 19.4 newtons, 5 1 . 7 ∘
  • D 19.4 newtons, 2 3 1 . 7 ∘
  • E 37.8 newtons, 2 1 2 . 2 ∘

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