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.

  • Aš‘„=āˆ’30, š‘§=24
  • Bš‘„=āˆ’19, š‘§=āˆ’8
  • Cš‘„=15, š‘§=12
  • Dš‘„=42, š‘§=28

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‘›.

  • Aš‘š=21, š‘›=6
  • Bš‘š=20, š‘›=āˆ’7
  • Cš‘š=4, š‘›=āˆ’20
  • Dš‘š=19, 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.

  • A640+2,640ij
  • B2,773+1,626ij
  • C640+1,626ij
  • D2,773+2,640ij

Q4:

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

  • A252+504jk
  • B1,134āˆ’378āˆ’756ijk
  • Cāˆ’1,134+378+756ijk
  • Dāˆ’252āˆ’504jk

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.

  • Aāˆ’516+688ik
  • Bāˆ’867+312+220ijk
  • Cāˆ’351+312āˆ’468ijk
  • Dāˆ’867+624+220ijk

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’10)
  • 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‘.

  • Aš‘=1, š‘=āˆ’1
  • Bš‘=48, š‘=9
  • Cš‘=6, š‘=āˆ’9
  • Dš‘=6, š‘=9

Q10:

FijļŠ§=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‘›.

  • Aš‘š=3, š‘›=āˆ’5
  • Bš‘š=0.5, š‘›=āˆ’2.5
  • Cš‘š=āˆ’2, š‘›=10
  • Dš‘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.

  • Aāˆš19 length units
  • B19 length units
  • C3 length units
  • Dāˆš17 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.

  • Aāˆ’92+69ij
  • Bāˆ’92+55ij
  • Cāˆ’55+69ij
  • D92āˆ’69ij

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.

  • A594āˆ’576+792ijk Nā‹…cm
  • B18+594+792ijk Nā‹…cm
  • Cāˆ’576+792ij Nā‹…cm
  • D18+792ij 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.

  • Aāˆ’48āˆš3+72āˆ’48ijk
  • B48āˆš3+72āˆ’48ijk
  • C72āˆ’48āˆš3+48ijk
  • Dāˆ’48āˆš3āˆ’72+48ijk

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.

  • A22+20ik
  • B180+240ik
  • C90+40ik
  • D360+120ik

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.

  • Aļƒ š‘€=āˆ’9ļŒ”ik, šæ=9āˆš437
  • Bļƒ š‘€=āˆ’9ļŒ”ik, šæ=āˆš417
  • Cļƒ š‘€=9+18+81ļŒ”ijk, šæ=āˆš417
  • Dļƒ š‘€=9+18+81ļŒ”ijk, šæ=9āˆš437

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.

  • A9+19āˆ’11ijk
  • Bāˆ’11+19+9ijk
  • Cāˆ’11+9+19ijk
  • D19āˆ’11+9ijk

Q18:

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

  • A224āˆ’102āˆ’99ijk
  • B231āˆ’85āˆ’92ijk
  • Cāˆ’99āˆ’102+224ijk
  • Dāˆ’92āˆ’85+231ijk

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.

  • AMijk=3+12āˆ’6, šæ=3āˆš302 length units
  • BMik=9+18, šæ=3āˆš302 length units
  • CMik=9+18, šæ=3āˆš142 length units
  • DMijk=3+12āˆ’6, šæ=3āˆš142 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.

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