Lesson: Resultant of Coplanar Forces

In this lesson, we will learn how to solve problems involving the resultant of coplanar forces meeting at a point analytically.

Worksheet: Resultant of Coplanar Forces • 25 Questions

Q1:

The diagram shows three coplanar forces acting at point 𝑀 . Their magnitudes are 2 N, 2 N, and 8 N in the directions  𝑀 𝐴 ,  𝑀 𝐡 , and  𝑀 𝐢 respectively. Given that π‘š ∠ 𝐴 𝑀 𝐡 = 6 0 ∘ and π‘š ∠ 𝐴 𝑀 𝐢 = 9 0 ∘ , what is the magnitude of the resultant force? Give your answer to the nearest newton.

Q2:

The forces ⃑ 𝐹 = 2 ⃑ 𝑖 + 2 ⃑ 𝑗 1 , ⃑ 𝐹 = π‘Ž ⃑ 𝑖 + 9 ⃑ 𝑗 2 , and ⃑ 𝐹 = 9 ⃑ 𝑖 + 𝑏 ⃑ 𝑗 3 act on a particle, where ⃑ 𝑖 and ⃑ 𝑗 are two perpendicular unit vectors. Given the forces’ resultant ⃑ 𝑅 = 2 ⃑ 𝑖 βˆ’ 6 ⃑ 𝑗 , determine the values of π‘Ž and 𝑏 .

Q3:

Three forces, ⃑ 𝐹 = ο€Ί βˆ’ 5 ⃑ 𝑖 + 1 0 ⃑ 𝑗  1 N , ⃑ 𝐹 = ο€Ί π‘Ž ⃑ 𝑖 βˆ’ 5 ⃑ 𝑗  2 N , and ⃑ 𝐹 = ο€Ί βˆ’ 4 ⃑ 𝑖 + 𝑏 ⃑ 𝑗  3 N act at a point. Their resultant is 6 √ 2 N northwest. Determine the values of π‘Ž and 𝑏 .

Q4:

Three forces, ( 5 ⃑ 𝑖 + 1 0 ⃑ 𝑗 ) N, ( π‘Ž ⃑ 𝑖 βˆ’ 5 ⃑ 𝑗 ) N, and ( 1 5 ⃑ 𝑖 + ( 𝑏 + 7 ) ⃑ 𝑗 ) N act on a particle. Given that the resultant of the forces is ( 1 8 ⃑ 𝑖 + 1 9 ⃑ 𝑗 ) N, what are the values of π‘Ž and 𝑏 ?

Q5:

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

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 4 5 ∘ 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.

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.

Q8:

The diagram shows a system of three forces measured in newtons. Given that 𝐴 𝐡 = 2 4 c m and 𝐴 𝐷 = 1 8 c m , 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.

Q9:

The diagram shows a square, 𝐴 𝐡 𝐢 𝐷 , of side 8 cm. The point 𝐸 is on 𝐡 𝐢 where 𝐡 𝐸 = 6 c m . Forces of magnitudes 8 N, 20 N, 1 6 √ 2 N, and 12 N act at 𝐴 as indicated on the diagram. Find the magnitude of their resultant.

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  𝑀 𝐴 ,  𝑀 𝐡 , and  𝑀 𝐢 . 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.

Q11:

The diagram shows a regular hexagon, 𝐴 𝐡 𝐢 𝐷 𝐸 𝐹 , 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.

Q12:

The diagram shows a regular hexagon, 𝐴 𝐡 𝐢 𝐷 𝐸 𝑂 . Forces of magnitudes 59, 7 8 √ 3 , 𝐹 , 7 8 √ 3 , and 𝐾 act in the directions shown on the diagram. Given that the resultant of the forces acts along  𝑂 𝐢 , and its magnitude is 298 N, determine 𝐹 and 𝐾 .

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 .

Q14:

Four coplanar concurrent forces are acting at the point 𝑂 ; where s i n πœƒ = 4 5 . If the resultant of the forces forms an angle of 1 3 5 ∘ with the π‘₯ -axis and has a magnitude of 3 7 √ 2 N, find the values of 𝑃 and 𝑄 .

Q15:

Given that the forces 𝐹 , 2 6 √ 2 , 4 4 √ 2 , and 65 newtons are acting as shown in the figure, the magnitude of the resultant of the system of forces is 2 5 √ 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.

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 .

Q17:

𝐴 𝐡 𝐢 is a triangle with a right-angle at 𝐡 , where 𝐴 𝐡 = 3 2 c m , 𝐡 𝐢 = 2 4 c m , 𝐷 ∈ 𝐴 𝐢 , and 𝐡 𝐷 = 𝐷 𝐢 . Four forces having magnitudes of 2, 3, 19 and 14 newtons are acting at the point 𝐡 in the directions  𝐴 𝐡 , οƒͺ 𝐡 𝐢 ,  𝐢 𝐴 and  𝐡 𝐷 respectively. Find the resultant of these forces if it is acting in  𝐡 𝐷 .

Q18:

𝐴 𝐡 𝐢 𝐷 is a rectangle in which 𝐴 𝐡 = 5 c m , 𝐡 𝐢 = 1 2 c m , and 𝐸 ∈ 𝐡 𝐢 where 𝐡 𝐸 = 5 c m . Forces of magnitude 4, 13, 4 √ 2 , and 12 newtons are acting in the directions of  𝐴 𝐷 ,  𝐴 𝐢 ,  𝐴 𝐸 , and  𝐴 𝐡 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 𝐡 𝐢 , and 𝐹 is the midpoint of 𝐷 𝐢 . Five forces having magnitudes of 19 N, 4 √ 5 N, 2 0 √ 2 N, 1 1 √ 5 N, and 18 N are acting at the point 𝐴 in the directions of  𝐴 𝐡 ,  𝐴 𝐸 ,  𝐢 𝐴 ,  𝐴 𝐹 , and  𝐴 𝐷 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 3 4 ∘ , 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 anticlockwise from the first force and has a magnitude of 7 newtons. The third force acts 90 degrees anticlockwise from the second force and has a magnitude of 5 newtons. The fourth force acts 123 degrees anticlockwise from the third force and has a magnitude of 6 newtons. Find the magnitude of the resultant force acting on the particle.

Q22:

Forces , , and act as shown in the diagram. Find the magnitude of their resultant, giving your answer to two decimal places if needed.

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 3 0 ∘ to the horizontal. What single force is needed to produce a state of equilibrium on the body? Determine its direction.

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 4 5 ∘ 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.

Q25:

A body has three forces, in newtons, acting on it at angles of βˆ’ 1 3 5 ∘ , 0 ∘ and βˆ’ 1 5 0 ∘ 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.

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