Worksheet: Resultant of Parallel Forces

In this worksheet, we will practice finding the resultant of a system of parallel coplanar forces and locating its point of action.

Q1:

Two parallel forces F and F are acting at two points 𝐴 and 𝐡 respectively in a perpendicular direction on ⃖⃗𝐴𝐡, where 𝐴𝐡=10cm. Their resultant Rij=βˆ’20βˆ’16 is acting at the point 𝐢 that belongs to ⃖⃗𝐴𝐡. Given that Fij=βˆ’30βˆ’24, determine F and the length of 𝐡𝐢.

  • A F i j  = βˆ’ 5 0 βˆ’ 4 0 , 𝐡 𝐢 = 2 5 c m
  • B F i j  = 1 0 + 8 , 𝐡 𝐢 = 1 5 c m
  • C F i j  = βˆ’ 5 0 βˆ’ 4 0 , 𝐡 𝐢 = 1 5 c m
  • D F i j  = 1 0 + 8 , 𝐡 𝐢 = 5 c m

Q2:

The points 𝐴, 𝐡, 𝐢, 𝐷, and 𝐸 are lying on the same straight line, where 2𝐴𝐡=𝐡𝐢=3𝐢𝐷=6𝐷𝐸=6cm. Four parallel forces of magnitudes 14, 19, 𝐹, and 20 newtons are acting at 𝐴, 𝐢, 𝐷, and 𝐸 respectively. If their resultant passes through point 𝐡, calculate the magnitude of force 𝐹, giving your answer in newtons.

  • A 𝐹 = βˆ’ 3 N
  • B 𝐹 = βˆ’ 1 3 . 5 N
  • C 𝐹 = 1 3 . 5 N
  • D 𝐹 = 3 N
  • E 𝐹 = βˆ’ 2 6 N

Q3:

Points 𝐴, 𝐡, 𝐢, 𝐷, and 𝐸 lying in the same straight line, such that 𝐴𝐡=8cm, 𝐡𝐢=18cm, 𝐢𝐷=12cm, and 𝐷𝐸=11cm. Five forces of magnitudes 40, 25, 20, 45, and 50 newtons are acting as shown in the figure. Determine their resultant 𝑅 and the distance π‘₯ between its line of action and point 𝐴.

  • A 𝑅 = 5 0 N , π‘₯ = 8 . 4 c m
  • B 𝑅 = 5 0 N , π‘₯ = 1 1 5 c m
  • C 𝑅 = βˆ’ 5 0 N , π‘₯ = βˆ’ 8 . 4 c m
  • D 𝑅 = 1 8 0 N , π‘₯ = 8 . 4 c m

Q4:

Three coplanar parallel forces having magnitudes of 6, 8, and 𝐹 newtons are acting at collinear points 𝐴, 𝐡, and 𝐢 respectively. 𝐴𝐡=10cm, and 𝐢 is not between 𝐴 and 𝐡. The first two forces are acting in opposite directions, and the resultant of the three forces has a magnitude of 6 N, acting in the direction of the second force, with its line of action intersecting 𝐴𝐡 at a point 𝐷, where 𝐴𝐷=60cm. Find the magnitude of 𝐹 and the length of 𝐡𝐢.

  • A 𝐹 = 4 N , 𝐡 𝐢 = 4 0 c m
  • B 𝐹 = 8 N , 𝐡 𝐢 = 5 5 c m
  • C 𝐹 = 4 N , 𝐡 𝐢 = 5 0 c m
  • D 𝐹 = 8 N , 𝐡 𝐢 = 4 5 c m

Q5:

𝐴 , 𝐡 , and 𝐢 are three points on one straight line, where 𝐴𝐡=6m, 𝐴𝐢=9m, and 𝐡∈𝐴𝐢. Forces of magnitudes 2 N and 2 N are acting vertically downwards at the two points 𝐴 and 𝐢, respectively, and a force having a magnitude of 7 N is acting vertically upwards at the point 𝐡. Find the magnitude and direction of the resultant 𝑅 and the distance π‘₯ of its point of action from the point 𝐴.

  • A 𝑅 = 3 N , downwards, π‘₯=2m
  • B 𝑅 = 7 N , upwards, π‘₯=6.86m
  • C 𝑅 = 7 N , downwards, π‘₯=9.86m
  • D 𝑅 = 3 N , upwards, π‘₯=8m

Q6:

𝐴 , 𝐡 , 𝐢 , 𝐷 , and 𝐸 are five points on the same straight line, where 𝐴𝐡=20cm, 𝐡𝐢=6cm, 𝐢𝐷=8cm, and 𝐷𝐸=5cm. Forces of magnitudes 4, 𝐹, and 10 newtons are acting vertically downwards at the points 𝐴, 𝐢, and 𝐸 respectively, and forces of magnitudes 7 and 𝐾 newtons are acting vertically upwards at 𝐡 and 𝐷 respectively. Given that the resultant of the forces is 3 N, and it is acting vertically downwards at the point 𝑁, where π‘βˆˆπ΄πΈ and 𝐴𝑁=14cm, determine the values of 𝐹 and 𝐾.

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

Q7:

Four parallel forces of magnitudes 6, 3, 8, and 2 kg-wt are acting perpendicularly in the same direction on the points 𝐴, 𝐡, 𝐢, and 𝐷 respectively. Given that the four points are on the same straight line, where 𝐴𝐡=𝐡𝐢=89cm and 𝐢𝐷=107cm, determine the magnitude and direction of the resultant force and the distance π‘₯ between the resultant’s point of action on the straight line and 𝐴.

  • A 𝑅 = 1 3 k g - w t , in the opposite direction to the forces, π‘₯=89cm
  • B 𝑅 = 1 3 k g - w t , in the same direction of the forces, π‘₯=89cm
  • C 𝑅 = 1 9 k g - w t , in the same direction of the forces, π‘₯=119cm
  • D 𝑅 = 1 9 k g - w t , in the opposite direction to the forces, π‘₯=119cm
  • E 𝑅 = 1 9 k g - w t , in the same direction of the forces, π‘₯=89cm

Q8:

Parallel forces F, F, F, and FοŠͺ act at the points 𝐴(βˆ’10,4), 𝐡(9,4), 𝐢(βˆ’7,7), and 𝐷(βˆ’3,1), respectively, where the forces are in equilibrium. Given that Fij=3+ and ||=2√10FN, F and F act in the opposite direction of F, find each of F, F, and FοŠͺ.

  • A F i j  = βˆ’ 3 βˆ’ , F i j  = βˆ’ 6 βˆ’ 2 , F i j οŠͺ = 6 + 2
  • B F i j  = 6 + 2 , F i j  = βˆ’ 3 βˆ’ , F i j οŠͺ = βˆ’ 6 βˆ’ 2
  • C F i j  = βˆ’ 6 βˆ’ 2 , F i j  = βˆ’ 3 βˆ’ , F i j οŠͺ = 6 + 2
  • D F i j  = βˆ’ 6 βˆ’ 2 , F i j  = βˆ’ 3 βˆ’ , F i j οŠͺ = βˆ’ 6 βˆ’ 2

Q9:

A force of 31 newtons is acting on a point 𝐴, while a parallel force of 𝐹 newtons is acting on a point 𝐡. The magnitude of the resultant of these two forces is 73 newtons. If the 31-newton force and the resultant are acting in opposite directions, what is the value of 𝐹?

Q10:

The given figure shows two parallel forces of magnitude 𝐹 N and 38 N and their resultant 𝑅. If 𝐴𝐡=0.8m and 𝑅=19N, determine 𝐹 and the length of 𝐡𝐢.

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

Q11:

Two parallel forces have magnitudes of 10 N and 20 N. The distance between their lines of action is 30 cm. If the two forces are acting in the same direction, find their resultant 𝑅 and the distance π‘₯ between its line of action and point 𝐴.

  • A 𝑅 = βˆ’ 3 0 N , π‘₯ = 4 5 c m
  • B 𝑅 = 1 0 N , π‘₯ = 3 0 c m
  • C 𝑅 = 3 0 N , π‘₯ = 1 0 c m
  • D 𝑅 = 3 0 N , π‘₯ = 2 0 c m
  • E 𝑅 = βˆ’ 1 0 N , π‘₯ = 3 0 c m

Q12:

Two parallel forces have magnitudes of 24 N and 60 N as shown in the figure. The distance between their lines of action is 90 cm. Given that the two forces are acting in opposite directions, determine their resultant 𝑅 and the distance π‘₯ between its line of action and point 𝐴.

  • A 𝑅 = βˆ’ 3 6 N , π‘₯ = 1 5 0 c m
  • B 𝑅 = βˆ’ 8 4 N , π‘₯ = 3 1 5 c m
  • C 𝑅 = 8 4 N , π‘₯ = 2 5 . 7 1 c m
  • D 𝑅 = 3 6 N , π‘₯ = 5 4 c m
  • E 𝑅 = βˆ’ 3 6 N , π‘₯ = 6 0 c m

Q13:

Given that FF//, where οƒŸπΉ=2οƒŸπΉοŠ§οŠ¨, and their resultant is acting at a point that is 16 cm away from F, determine the distance between the resultant’s line of action and F.

Q14:

The diagram shows a force of magnitude 𝐹 and a force of magnitude 12 N which are parallel and the resulting force of magnitude 𝑅. Given that 𝐴𝐡=72cm and 𝐴𝐢=48cm, determine the magnitudes of 𝐹 and 𝑅.

  • A 𝐹 = 2 4 N , 𝑅 = 3 6 N
  • B 𝐹 = 3 6 N , 𝑅 = 4 8 N
  • C 𝐹 = 6 N , 𝑅 = 1 8 N
  • D 𝐹 = 2 4 N , 𝑅 = 1 2 N
  • E 𝐹 = 3 6 N , 𝑅 = 2 4 N

Q15:

In the figure below, 𝐹 and 𝐹 are two parallel forces measured in newtons, where 𝑅 their resultant. If 𝑅=30N, 𝐴𝐡=36cm, and 𝐡𝐢=24cm, determine the magnitude of 𝐹 and 𝐹.

  • A 𝐹 = 1 5  N , 𝐹 = 4 5  N
  • B 𝐹 = 1 2  N , 𝐹 = 1 8  N
  • C 𝐹 = 4 8  N , 𝐹 = 1 8  N
  • D 𝐹 = 7 5  N , 𝐹 = 4 5  N

Q16:

Two parallel forces 𝐹 and 𝐹 have the same direction, and the distance between their lines of action is 90 cm. Given that the magnitude of their resultant is 49 N, and it is 60 cm away from 𝐹, find the magnitudes of the two forces, rounding your answer to two decimal places.

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

Q17:

𝐹  and 𝐹 are two parallel forces acting at the points 𝐴 and 𝐡 respectively, where 𝐴𝐡=52cm, and their resultant is acting at the point 𝐢, where πΆβˆˆβƒ–οƒ©οƒ©οƒ©οƒ©βƒ—π΄π΅. Given that 𝐡𝐢=12cm when the two forces are acting in the same direction, and their resultant is 28 N when they are acting in opposite directions, determine the magnitude of each of the two forces.

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

Q18:

The magnitude of the resultant of two parallel forces 𝐹 and 𝐹 is 192 N, where the magnitude of 𝐹 is 64 N, and its line of action is 57 cm away from that of the resultant’s. If the two forces have the same direction, determine the magnitude of 𝐹 and the distance between the lines of action of the two forces 𝑙.

  • A 𝐹 = 2 5 6  N , 𝑙 = 4 2 . 7 5 c m
  • B 𝐹 = 1 2 8  N , 𝑙 = 8 5 . 5 c m
  • C 𝐹 = 1 2 8  N , 𝑙 = 1 7 1 c m
  • D 𝐹 = 2 5 6  N , 𝑙 = 1 7 1 c m

Q19:

The magnitude of the resultant of two parallel forces 𝐹 and 𝐹 equals 61 N. The magnitude of 𝐹 equals 112 N and the distance between 𝐹 and the line of action of the resultant is 17 cm. If 𝐹 and the resultant have opposite directions, find the magnitude of the second force 𝐹 and the distance between the lines of action of the forces 𝑙, rounding this answer to two decimal places.

  • A 𝐹 = 5 1  N , 𝑙 = 9 . 2 6 c m
  • B 𝐹 = 5 1  N , 𝑙 = 2 0 . 3 3 c m
  • C 𝐹 = 1 7 3  N , 𝑙 = 5 . 9 9 c m
  • D 𝐹 = 1 7 3  N , 𝑙 = 2 . 1 1 c m

Q20:

Two vertical forces are acting on a horizontal light rod 𝐴𝐡. The smaller force 𝐹 has magnitude 108 N and acts on end 𝐴. The other force acts on end 𝐡. Determine the length of the rod given that the magnitude of the forces’ resultant equals 84 N, and the distance between its line of action and 𝐡 is 18 cm.

Q21:

The figure below shows the two parallel forces 48 N and 32 N and their resultant 𝑅. If 𝐴𝐡=75cm, determine 𝑅 and the length of 𝐴𝐢.

  • A 𝐴 𝐢 = 2 5 c m , 𝑅 = 1 6 N
  • B 𝐴 𝐢 = 3 0 c m , 𝑅 = 8 0 N
  • C 𝐴 𝐢 = 4 5 c m , 𝑅 = 8 0 N
  • D 𝐴 𝐢 = 5 0 c m , 𝑅 = 1 6 N

Q22:

Two parallel forces of magnitude 3𝐹 and 7𝐹 are acting at the two points 𝐴 and 𝐡 respectively, where 𝐴 and 𝐡 are 35 cm apart. Find the distance 𝐴𝐢 given that the resultant is acting at the point 𝐢 between 𝐴 and 𝐡.

Q23:

Two like forces of magnitudes 12 and 8 newtons are acting at the two points 𝐴 and 𝐡 respectively, where 𝐴𝐡=49cm. If another force having a magnitude of 𝐹, and in the same direction, is added to the first force at 𝐴, then the resultant will move 14 units. Find the magnitude of 𝐹.

Q24:

Two parallel forces of magnitudes 19 N and 44 N and of the same direction are acting at the points 𝐴 and 𝐡 of a rigid body. If the second force is transferred a distance 𝐿 parallel to its line of action in the direction of 𝐡𝐴, the resultant of the forces will be transferred a distance π‘₯ parallel to its line of action as well. Determine the distance π‘₯.

  • A π‘₯ = 4 4 6 3 𝐿
  • B π‘₯ = 1 9 6 3 𝐿
  • C π‘₯ = 6 3 1 9 𝐿
  • D π‘₯ = 1 9 4 4 𝐿

Q25:

Given that the two parallel forces Fij=2+ and Fij=βˆ’4βˆ’2 are acting at 𝐴(βˆ’3,βˆ’5) and 𝐡(5,3) respectively, determine their resultant R, and find its point of action.

  • A R i j = βˆ’ 6 βˆ’ 3 , acts at (3,1)
  • B R i j = βˆ’ 2 βˆ’ , acts at (13,11)
  • C R i j = βˆ’ 6 βˆ’ 3 , acts at (5,7)
  • D R i j = βˆ’ 2 βˆ’ , acts at (5,7)

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