Worksheet: Moments in 3D

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In this worksheet, we will practice finding and interpreting the three components of the moment of a force in the i, j, and k directions.

Q1:

The moment of the force F about the origin is 𝑀 , where F i j k = − 2 − and 𝑀 = 2 0 + 2 7 − 3 4 i j k . Given that the force passes through a point whose 𝑦 -coordinate is 4, find the 𝑥 and 𝑧 coordinates of the point.

  • A 𝑥 = − 3 0 , 𝑧 = 2 4
  • B 𝑥 = − 1 9 , 𝑧 = − 8
  • C 𝑥 = 4 2 , 𝑧 = 2 8
  • D 𝑥 = 1 5 , 𝑧 = 1 2

Q2:

If the force ⃑ 𝐹 = 𝑚 ⃑ 𝑖 + 𝑛 ⃑ 𝑗 − ⃑ 𝑘 is acting at a point whose position vector, with respect to the origin point, is ⃑ 𝑟 = 1 4 ⃑ 𝑖 − ⃑ 𝑗 + 1 2 ⃑ 𝑘 , and the components of the moment of the force ⃑ 𝐹 about the 𝑥 -axis and the 𝑦 -axis are 73 and 242 units of moment, respectively, find the values of 𝑚 and 𝑛 .

  • A 𝑚 = 2 0 , 𝑛 = − 7
  • B 𝑚 = 2 1 , 𝑛 = 6
  • C 𝑚 = 4 , 𝑛 = − 2 0
  • D 𝑚 = 1 9 , 𝑛 = − 6

Q3:

The forces and act along and , respectively, as shown in the figure. Given that , , and are a right system of unit vectors in the directions of , , and , respectively, find the sum of the moments of the forces about point in newton-meters.

  • A
  • B
  • C
  • D

Q4:

In the figure, a force of magnitude 42 newtons is acting along diagonal 𝐸 𝐵 in a cuboid whose dimensions are 18 cm, 18 cm, and 9 cm. Determine the vector moment of the force about 𝑇 in newton-centimetres.

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

Q5:

If the force ⃑ 𝐹 , where ⃑ 𝐹 = − 2 ⃑ 𝑖 + 𝐿 ⃑ 𝑗 − 9 ⃑ 𝑘 , is acting on the point 𝐴 ( 4 , 5 , − 2 ) , and the moment 𝑀 𝐵 of the force about the point 𝐵 ( − 4 , − 4 , 3 ) is − 9 1 ⃑ 𝑖 + 8 2 ⃑ 𝑗 + 2 ⃑ 𝑘 , determine the value of 𝐿 .

Q6:

In the figure, determine the sum of the moment vectors of the forces 86 and 65 newtons about 𝑂 in newton-centimetres.

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

Q7:

If a force F i j k = 6 − 7 − 8 is acting at a point 𝐴 ( 5 , − 8 , 1 1 ) , find the magnitude of the component of the moment of F about the 𝑦 -axis.

  • A13 units of moment
  • B141 units of moment
  • C260 units of moment
  • D106 units of moment

Q8:

The forces F i j = − + 5 , F i j = − 8 + 2 , and F i j = 8 − 2 are acting at a point. If the moment vector of the resultant of these forces about the origin point is − 1 0 k , find the intersection point of the line of action of the resultant with the 𝑦 -axis.

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

Q9:

If the force F i j k = 3 + 𝑏 + 𝑐 is acting at the point 𝐴 ( 2 , − 1 4 , 1 0 ) and the two components of the moment of F about the 𝑦 -axis and the 𝑧 -axis are 12 and 54 respectively, find the values of 𝑏 and 𝑐 .

  • A 𝑏 = 6 , 𝑐 = − 9
  • B 𝑏 = 4 8 , 𝑐 = 9
  • C 𝑏 = 1 , 𝑐 = − 1
  • D 𝑏 = 6 , 𝑐 = 9

Q10:

F = 𝑚 i + j and F = 𝑛 i − 5 j , where F and F are two forces acting at the points 𝐴 ( 3 , 1 ) and 𝐵 ( − 1 , − 1 ) respectively. The sum of moments about the point of origin equals zero. The sum of the moments about the point 𝐶 ( 1 , 2 ) also equals zero. Determine the values of 𝑚 and 𝑛 .

  • A 𝑚 = 0 . 5 , 𝑛 = 7 . 5
  • B 𝑚 = 0 . 5 , 𝑛 = − 2 . 5
  • C 𝑚 = − 2 , 𝑛 = 1 0
  • D 𝑚 = 3 , 𝑛 = − 5

Q11:

If the force ⃑ 𝐹 = 𝑚 ⃑ 𝑖 + 3 ⃑ 𝑗 − 3 ⃑ 𝑘 is acting at a point 𝐴 whose position vector, with respect to the origin point, is ⃑ 𝑟 = − 6 ⃑ 𝑖 − 2 ⃑ 𝑗 + 4 ⃑ 𝑘 , and the component of the moment of the force ⃑ 𝐹 about the 𝑦 -axis is − 3 0 moment units, find the length of the perpendicular segment drawn from the origin point to the line of action of ⃑ 𝐹 .

  • A20 length units
  • B 2 √ 4 2 3 length units
  • C56 length units
  • D 2 √ 1 4 length units

Q12:

In the figure shown, a force of magnitude 2 3 √ 2 newtons acts at a point 𝐴 , determine the moment vector of the force about the origin 𝑂 in N⋅m.

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

Q13:

A force having a magnitude of newtons is acting on point in the direction of and another force having a magnitude of newtons is acting on point in the direction of as shown in the figure. If , , and are a right system of the one vectors in the direction of , , and , respectively, determine the vector sum of the moments of the forces about point in newton-centimeters.

  • A N⋅cm
  • B N⋅cm
  • C N⋅cm
  • D N⋅cm

Q14:

Given that a force of magnitude 6 N is acting on 𝐶 as in the figure, determine its moment vector about 𝐴 in newton-centimetres.

  • A 4 8 √ 3 i + 7 2 j − 4 8 k
  • B 7 2 i − 4 8 √ 3 j + 4 8 k
  • C − 4 8 √ 3 i − 7 2 j + 4 8 k
  • D − 4 8 √ 3 i + 7 2 j − 4 8 k

Q15:

In the figure, 𝐴 𝐵 is a rod fixed to a vertical wall at end 𝐴 . The other end 𝐵 is connected to a wire 𝐵 𝐶 , where 𝐶 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 𝐴 in newton-metres.

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

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