Worksheet: Moments in 3D

In this worksheet, we will practice finding the moment of vector forces acting on a body about a point in 3D.

Q1:

The moment of the force F about the origin is š‘€ļŒ®, where Fijk=āˆ’2āˆ’ and š‘€=20+27āˆ’34ļŒ®ijk. Given that the force passes through a point whose š‘¦-coordinate is 4, find the š‘„ and š‘§ coordinates of the point.

  • A š‘„ = āˆ’ 3 0 , š‘§ = 2 4
  • B š‘„ = āˆ’ 1 9 , š‘§ = āˆ’ 8
  • C š‘„ = 1 5 , š‘§ = 1 2
  • D š‘„ = 4 2 , š‘§ = 2 8

Q2:

If the force Fijk=š‘š+š‘›āˆ’ is acting at a point whose position vector, with respect to the origin point, is rijk=14āˆ’+12, and the components of the moment of the force F about the š‘„-axis and the š‘¦-axis are 73 and 242 units of moment, respectively, find the values of š‘š and š‘›.

  • A š‘š = 2 1 , š‘› = 6
  • B š‘š = 2 0 , š‘› = āˆ’ 7
  • C š‘š = 4 , š‘› = āˆ’ 2 0
  • D š‘š = 1 9 , š‘› = āˆ’ 6

Q3:

The forces FļŠ§=5āˆš673N and FļŠØ=16āˆš569N act along ļƒ«š“šµ and ļƒ«š“š¶, respectively, as shown in the figure. Given that i, j, and k are a right system of unit vectors in the directions of š‘„, š‘¦, and š‘§, respectively, find the sum of the moments of the forces about point š‘‚ in newton-metres.

  • A 6 4 0 + 2 , 6 4 0 i j
  • B 2 , 7 7 3 + 1 , 6 2 6 i j
  • C 6 4 0 + 1 , 6 2 6 i j
  • D 2 , 7 7 3 + 2 , 6 4 0 i j

Q4:

In the figure, a force of magnitude 42 newtons is acting along diagonal šøšµ in a cuboid whose dimensions are 18 cm, 18 cm, and 9 cm. Determine the vector moment of the force about š‘‡ in newton-centimetres.

  • A 1 , 1 3 4 āˆ’ 3 7 8 āˆ’ 7 5 6 i j k
  • B āˆ’ 2 5 2 āˆ’ 5 0 4 j k
  • C 2 5 2 + 5 0 4 j k
  • D āˆ’ 1 , 1 3 4 + 3 7 8 + 7 5 6 i j k

Q5:

If the force F, where Fijk=āˆ’2+šæāˆ’9, is acting on the point š“(4,5,āˆ’2), and the moment š‘€ļŒ” of the force about the point šµ(āˆ’4,āˆ’4,3) is āˆ’91+82+2ijk, determine the value of šæ.

Q6:

In the figure, determine the sum of the moment vectors of the forces 86 and 65 newtons about š‘‚ in newton-centimeters.

  • A āˆ’ 8 6 7 + 3 1 2 + 2 2 0 i j k
  • B āˆ’ 3 5 1 + 3 1 2 āˆ’ 4 6 8 i j k
  • C āˆ’ 5 1 6 + 6 8 8 i k
  • D āˆ’ 8 6 7 + 6 2 4 + 2 2 0 i j k

Q7:

If a force Fijk=6āˆ’7āˆ’8 is acting at a point š“(5,āˆ’8,11), find the magnitude of the component of the moment of F about the š‘¦-axis.

  • A106 units of moment
  • B141 units of moment
  • C260 units of moment
  • D13 units of moment

Q8:

The forces FijļŠ§=āˆ’+5, FijļŠØ=āˆ’8+2, and FijļŠ©=8āˆ’2 are acting at a point. If the moment vector of the resultant of these forces about the origin point is āˆ’10k, find the intersection point of the line of action of the resultant with the š‘¦-axis.

  • A ( 0 , āˆ’ 1 0 )
  • B ( āˆ’ 2 , 0 )
  • C ( 0 , 5 )
  • D ( āˆ’ 1 , 0 )

Q9:

If the force Fijk=3+š‘+š‘ is acting at the point š“(2,āˆ’14,10) and the two components of the moment of F about the š‘¦-axis and the š‘§-axis are 12 and 54 respectively, find the values of š‘ and š‘.

  • A š‘ = 1 , š‘ = āˆ’ 1
  • B š‘ = 4 8 , š‘ = 9
  • C š‘ = 6 , š‘ = āˆ’ 9
  • D š‘ = 6 , š‘ = 9

Q10:

F i j ļŠ§ = š‘š + and FijļŠØ=š‘›āˆ’5, where FļŠ§ and FļŠØ are two forces acting at the points š“(3,1) and šµ(āˆ’1,āˆ’1) respectively. The sum of moments about the point of origin equals zero. The sum of the moments about the point š¶(1,2) also equals zero. Determine the values of š‘š and š‘›.

  • A š‘š = 3 , š‘› = āˆ’ 5
  • B š‘š = 0 . 5 , š‘› = āˆ’ 2 . 5
  • C š‘š = āˆ’ 2 , š‘› = 1 0
  • D š‘š = 0 . 5 , š‘› = 7 . 5

Q11:

If the force Fijk=š‘šāˆ’āˆ’ is acting at a point š“ whose position vector, with respect to the origin point, is rijk=āˆ’3+3āˆ’3, and the component of the moment of the force F about the š‘¦-axis is 9 moment units, find the length of the perpendicular segment drawn from the origin point to the line of action of F.

  • A āˆš 1 9 length units
  • B19 length units
  • C3 length units
  • D āˆš 1 7 length units

Q12:

In the figure shown, a force of magnitude 23āˆš2 newtons acts at a point š“, determine the moment vector of the force about the origin š‘‚ in Nā‹…m.

  • A āˆ’ 9 2 + 6 9 i j
  • B āˆ’ 9 2 + 5 5 i j
  • C āˆ’ 5 5 + 6 9 i j
  • D 9 2 āˆ’ 6 9 i j

Q13:

A force having a magnitude of FļŠ§=32āˆš13newtons is acting on point šµ in the direction of ļƒ«š“šµ and another force having a magnitude of FļŠØ=22āˆš61newtons is acting on point š¶ in the direction of ļƒ«š“š¶ as shown in the figure. If i, j, and k are a right system of the unit vectors in the direction of š‘„, š‘¦, and š‘§, respectively, determine the vector sum of the moments of the forces about point š‘‚ in newton-centimeters.

  • A 5 9 4 āˆ’ 5 7 6 + 7 9 2 i j k Nā‹…cm
  • B 1 8 + 5 9 4 + 7 9 2 i j k Nā‹…cm
  • C āˆ’ 5 7 6 + 7 9 2 i j Nā‹…cm
  • D 1 8 + 7 9 2 i j Nā‹…cm

Q14:

Given that a force of magnitude 6 N is acting on š¶ as in the figure, determine its moment vector about š“ in newton-centimeters.

  • A āˆ’ 4 8 āˆš 3 + 7 2 āˆ’ 4 8 i j k
  • B 4 8 āˆš 3 + 7 2 āˆ’ 4 8 i j k
  • C 7 2 āˆ’ 4 8 āˆš 3 + 4 8 i j k
  • D āˆ’ 4 8 āˆš 3 āˆ’ 7 2 + 4 8 i j k

Q15:

In the figure, š“šµ is a rod fixed to a vertical wall at end š“. The other end šµ is connected to a wire šµš¶, where š¶ is fixed to a different point on the same vertical wall. If the tension in the wire equals 65 N, calculate the moment of the tension about point š“ in newton-metres.

  • A 2 2 + 2 0 i k
  • B 1 8 0 + 2 4 0 i k
  • C 9 0 + 4 0 i k
  • D 3 6 0 + 1 2 0 i k

Q16:

If the force Fijk=āˆ’9āˆ’4āˆ’ is acting at the point š“(āˆ’3,2,4), find the moment š‘€ļŒ” of the force F about the point šµ(6,7,5) and then calculate the length šæ of the perpendicular segment from šµ to the line of action of the force.

  • A ļƒ  š‘€ = āˆ’ 9 ļŒ” i k , šæ = 9 āˆš 4 3 7
  • B ļƒ  š‘€ = āˆ’ 9 ļŒ” i k , šæ = āˆš 4 1 7
  • C ļƒ  š‘€ = 9 + 1 8 + 8 1 ļŒ” i j k , šæ = āˆš 4 1 7
  • D ļƒ  š‘€ = 9 + 1 8 + 8 1 ļŒ” i j k , šæ = 9 āˆš 4 3 7

Q17:

If a force F is acting at the point š“(9,āˆ’6,āˆ’1), where the moment of F about the origin is 85+90+225ijk, find F.

  • A 9 + 1 9 āˆ’ 1 1 i j k
  • B āˆ’ 1 1 + 1 9 + 9 i j k
  • C āˆ’ 1 1 + 9 + 1 9 i j k
  • D 1 9 āˆ’ 1 1 + 9 i j k

Q18:

In the figure, if the forces FijkļŠ§=āˆ’7āˆ’+3 and FijkļŠØ=āˆ’7+8āˆ’6 are acting on the point š“, where š¹ļŠ§ and š¹ļŠØ are measured in newtons, determine the moment vector of the resultant about the point š‘‚ in newton-centimeters.

  • A 2 2 4 āˆ’ 1 0 2 āˆ’ 9 9 i j k
  • B 2 3 1 āˆ’ 8 5 āˆ’ 9 2 i j k
  • C āˆ’ 9 9 āˆ’ 1 0 2 + 2 2 4 i j k
  • D āˆ’ 9 2 āˆ’ 8 5 + 2 3 1 i j k

Q19:

Find the moment M of the force F about the origin point, given that Fijk=āˆ’2++, and is acting at a point š“ whose position vector is rijk=6+6āˆ’3 with respect to the origin point, then determine the length šæ of the perpendicular segment drawn from the origin point to the line of action of the force F.

  • A M i j k = 3 + 1 2 āˆ’ 6 , šæ = 3 āˆš 3 0 2 length units
  • B M i k = 9 + 1 8 , šæ = 3 āˆš 3 0 2 length units
  • C M i k = 9 + 1 8 , šæ = 3 āˆš 1 4 2 length units
  • D M i j k = 3 + 1 2 āˆ’ 6 , šæ = 3 āˆš 1 4 2 length units

Q20:

If Fijk=āˆ’19+šæ+2 acts at a point š“(āˆ’3,5,āˆ’3), and the moment of F about the origin point is equal to 4+63+101ijk, find the value of šæ.

  • A āˆ’ 1
  • B āˆ’ 2
  • C31
  • D āˆ’ 4

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