Worksheet: Moments in 3D

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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 𝑀, where Fijk=−2− and 𝑀=20+27−34ijk. 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 = 1 5 , 𝑧 = 1 2
  • D 𝑥 = 4 2 , 𝑧 = 2 8

Q2:

If the force Fijk=𝑚+𝑛− is acting at a point whose position vector, with respect to the origin point, is rijk=14−+12, 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 1 , 𝑛 = 6
  • B 𝑚 = 2 0 , 𝑛 = − 7
  • C 𝑚 = 4 , 𝑛 = − 2 0
  • D 𝑚 = 1 9 , 𝑛 = − 6

Q3:

The forces F=5√673N and F=16√569N act along 𝐴𝐵 and 𝐴𝐶, respectively, as shown in the figure. Given that i, j, and k are a right system of unit vectors in the directions of 𝑥, 𝑦, and 𝑧, respectively, find the sum of the moments of the forces about point 𝑂 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 𝐸𝐵 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 1 , 1 3 4 − 3 7 8 − 7 5 6 i j k
  • B − 2 5 2 − 5 0 4 j k
  • C 2 5 2 + 5 0 4 j k
  • D − 1 , 1 3 4 + 3 7 8 + 7 5 6 i j k

Q5:

If the force F, where Fijk=−2+𝐿−9, is acting on the point 𝐴(4,5,−2), and the moment 𝑀 of the force about the point 𝐵(−4,−4,3) is −91+82+2ijk, 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-centimeters.

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

Q7:

If a force Fijk=6−7−8 is acting at a point 𝐴(5,−8,11), find the magnitude of the component of the moment of F about the 𝑦-axis.

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

Q8:

The forces Fij=−+5, Fij=−8+2, and Fij=8−2 are acting at a point. If the moment vector of the resultant of these forces about the origin point is −10k, find the intersection point of the line of action of the resultant with the 𝑦-axis.

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

Q9:

If the force Fijk=3+𝑏+𝑐 is acting at the point 𝐴(2,−14,10) 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 = 1 , 𝑐 = − 1
  • B 𝑏 = 4 8 , 𝑐 = 9
  • C 𝑏 = 6 , 𝑐 = − 9
  • D 𝑏 = 6 , 𝑐 = 9

Q10:

F i j = 𝑚 + and Fij=𝑛−5, 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 = 3 , 𝑛 = − 5
  • B 𝑚 = 0 . 5 , 𝑛 = − 2 . 5
  • C 𝑚 = − 2 , 𝑛 = 1 0
  • D 𝑚 = 0 . 5 , 𝑛 = 7 . 5

Q11:

If the force Fijk=𝑚−− is acting at a point 𝐴 whose position vector, with respect to the origin point, is rijk=−3+3−3, and the component of the moment of the force F about the 𝑦-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 √ 1 9 length units
  • B19 length units
  • C3 length units
  • D √ 1 7 length units

Q12:

In the figure shown, a force of magnitude 23√2 newtons acts at a point 𝐴, determine the moment vector of the force about the origin 𝑂 in N⋅m.

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

Q13:

A force having a magnitude of F=32√13newtons is acting on point 𝐵 in the direction of 𝐴𝐵 and another force having a magnitude of F=22√61newtons is acting on point 𝐶 in the direction of 𝐴𝐶 as shown in the figure. If i, j, and k are a right system of the unit vectors in the direction of 𝑥, 𝑦, and 𝑧, respectively, determine the vector sum of the moments of the forces about point 𝑂 in newton-centimeters.

  • A 5 9 4 − 5 7 6 + 7 9 2 i j k N⋅cm
  • B 1 8 + 5 9 4 + 7 9 2 i j k N⋅cm
  • C − 5 7 6 + 7 9 2 i j N⋅cm
  • D 1 8 + 7 9 2 i j 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-centimeters.

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

Q15:

In the figure, 𝐴𝐵 is a rod fixed to a vertical wall at end 𝐴. The other end 𝐵 is connected to a wire 𝐵𝐶, 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 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=−9−4− is acting at the point 𝐴(−3,2,4), find the moment 𝑀 of the force F about the point 𝐵(6,7,5) and then calculate the length 𝐿 of the perpendicular segment from 𝐵 to the line of action of the force.

  • A 𝑀 = − 9 i k , 𝐿 = 9 √ 4 3 7
  • B 𝑀 = − 9 i k , 𝐿 = √ 4 1 7
  • C 𝑀 = 9 + 1 8 + 8 1 i j k , 𝐿 = √ 4 1 7
  • D 𝑀 = 9 + 1 8 + 8 1 i j k , 𝐿 = 9 √ 4 3 7

Q17:

If a force F is acting at the point 𝐴(9,−6,−1), where the moment of F about the origin is 85+90+225ijk, find F.

  • A 9 + 1 9 − 1 1 i j k
  • B − 1 1 + 1 9 + 9 i j k
  • C − 1 1 + 9 + 1 9 i j k
  • D 1 9 − 1 1 + 9 i j k

Q18:

In the figure, if the forces Fijk=−7−+3 and Fijk=−7+8−6 are acting on the point 𝐴, where 𝐹 and 𝐹 are measured in newtons, determine the moment vector of the resultant about the point 𝑂 in newton-centimeters.

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

Q19:

Find the moment M of the force F about the origin point, given that Fijk=−2++, and is acting at a point 𝐴 whose position vector is rijk=6+6−3 with respect to the origin point, then determine the length 𝐿 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 − 6 , 𝐿 = 3 √ 3 0 2 length units
  • B M i k = 9 + 1 8 , 𝐿 = 3 √ 3 0 2 length units
  • C M i k = 9 + 1 8 , 𝐿 = 3 √ 1 4 2 length units
  • D M i j k = 3 + 1 2 − 6 , 𝐿 = 3 √ 1 4 2 length units

Q20:

If Fijk=−19+𝐿+2 acts at a point 𝐴(−3,5,−3), and the moment of F about the origin point is equal to 4+63+101ijk, find the value of 𝐿.

  • A − 1
  • B − 2
  • C31
  • D − 4

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