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Lesson: Moments in 3D

Sample Question Videos

Worksheet • 15 Questions • 2 Videos

Q1:

The moment of the force ⃑ 𝐹 about the origin is 𝑀 𝑂 , where ⃑ 𝐹 = ⃑ 𝑖 βˆ’ 2 ⃑ 𝑗 βˆ’ ⃑ π‘˜ and 𝑀 = 2 0 ⃑ 𝑖 + 2 7 ⃑ 𝑗 βˆ’ 3 4 ⃑ π‘˜ 𝑂 . Given that the force passes through a point whose 𝑦 -coordinate is 4, find the π‘₯ and 𝑧 coordinates of the point.

  • A π‘₯ = 1 5 , 𝑧 = 1 2
  • B π‘₯ = βˆ’ 1 9 , 𝑧 = βˆ’ 8
  • C π‘₯ = 4 2 , 𝑧 = 2 8
  • D π‘₯ = βˆ’ 3 0 , 𝑧 = 2 4

Q2:

If the force ⃑ 𝐹 = π‘š ⃑ 𝑖 + 𝑛 ⃑ 𝑗 βˆ’ ⃑ π‘˜ is acting at a point whose position vector, with respect to the origin point, is ⃑ π‘Ÿ = 1 4 ⃑ 𝑖 βˆ’ ⃑ 𝑗 + 1 2 ⃑ π‘˜ , and the components of the moment of the force ⃑ 𝐹 about the π‘₯ -axis and the 𝑦 -axis are 73 and 242 units of moment, respectively, find the values of π‘š and 𝑛 .

  • A π‘š = 1 9 , 𝑛 = βˆ’ 6
  • B π‘š = 2 1 , 𝑛 = 6
  • C π‘š = 4 , 𝑛 = βˆ’ 2 0
  • D π‘š = 2 0 , 𝑛 = βˆ’ 7

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 𝐡 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 βˆ’ 1 , 1 3 4 + 3 7 8 + 7 5 6 i j k
  • C 1 , 1 3 4 βˆ’ 3 7 8 βˆ’ 7 5 6 i j k
  • D βˆ’ 2 5 2 βˆ’ 5 0 4 j k

Q5:

If the force ⃑ 𝐹 , where ⃑ 𝐹 = βˆ’ 2 ⃑ 𝑖 + 𝐿 ⃑ 𝑗 βˆ’ 9 ⃑ π‘˜ , is acting on the point 𝐴 ( 4 , 5 , βˆ’ 2 ) , and the moment 𝑀 𝐡 of the force about the point 𝐡 ( βˆ’ 4 , βˆ’ 4 , 3 ) is βˆ’ 9 1 ⃑ 𝑖 + 8 2 ⃑ 𝑗 + 2 ⃑ π‘˜ , 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 βˆ’ 8 6 7 ⃑ 𝑖 + 3 1 2 ⃑ 𝑗 + 2 2 0 ⃑ π‘˜
  • B βˆ’ 5 1 6 ⃑ 𝑖 + 6 8 8 ⃑ π‘˜
  • C βˆ’ 8 6 7 ⃑ 𝑖 + 6 2 4 ⃑ 𝑗 + 2 2 0 ⃑ π‘˜
  • D βˆ’ 3 5 1 ⃑ 𝑖 + 3 1 2 ⃑ 𝑗 βˆ’ 4 6 8 ⃑ π‘˜

Q7:

If a force ⃑ 𝐹 = 4 ⃑ 𝑖 + ⃑ 𝑗 βˆ’ ⃑ π‘˜ is acting at a point 𝐴 ( 1 2 , βˆ’ 1 2 , βˆ’ 4 ) , find the magnitude of the component of the moment of ⃑ 𝐹 about the 𝑦 -axis.

  • A4 units of moment
  • B16 units of moment
  • C72 units of moment
  • D60 units of moment

Q8:

The forces ⃑ 𝐹 = βˆ’ ⃑ 𝑖 + 5 ⃑ 𝑗 1 , ⃑ 𝐹 = βˆ’ 8 ⃑ 𝑖 + 2 ⃑ 𝑗 2 , and ⃑ 𝐹 = 8 ⃑ 𝑖 βˆ’ 2 ⃑ 𝑗 3 are acting at a point. If the moment vector of the resultant of these forces about the origin point is βˆ’ 1 0 ⃑ π‘˜ , find the intersection point of the line of action of the resultant with the 𝑦 -axis.

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

Q9:

If the force ⃑ 𝐹 = 7 ⃑ 𝑖 + 𝑏 ⃑ 𝑗 + 𝑐 ⃑ π‘˜ is acting at the point 𝐴 ( βˆ’ 7 , βˆ’ 5 , 4 ) and the two components of the moment of ⃑ 𝐹 about the 𝑦 -axis and the 𝑧 -axis are βˆ’ 7 and 98 respectively, find the values of 𝑏 and 𝑐 .

  • A 𝑏 = βˆ’ 9 , 𝑐 = βˆ’ 5
  • B 𝑏 = βˆ’ 1 9 , 𝑐 = βˆ’ 5
  • C 𝑏 = 1 5 , 𝑐 = βˆ’ 7
  • D 𝑏 = βˆ’ 9 , 𝑐 = 5

Q10:

⃑ 𝐹 = π‘š ⃑ 𝑖 + ⃑ 𝑗 1 and ⃑ 𝐹 = 𝑛 ⃑ 𝑖 βˆ’ 5 ⃑ 𝑗 2 , where ⃑ 𝐹 1 and ⃑ 𝐹 2 are two forces acting at the points 𝐴 ( 3 , 1 ) and 𝐡 ( βˆ’ 1 , βˆ’ 1 ) respectively. The sum of moments about the point of origin equals zero. The sum of the moments about the point 𝐢 ( 1 , 2 ) also equals zero. Determine the values of π‘š and 𝑛 .

  • A π‘š = 3 , 𝑛 = βˆ’ 5
  • B π‘š = 0 . 5 , 𝑛 = βˆ’ 2 . 5
  • C π‘š = βˆ’ 2 , 𝑛 = 1 0
  • D π‘š = 0 . 5 , 𝑛 = 7 . 5

Q11:

If the force ⃑ 𝐹 = π‘š ⃑ 𝑖 + 3 ⃑ 𝑗 βˆ’ 3 ⃑ π‘˜ is acting at a point 𝐴 whose position vector, with respect to the origin point, is ⃑ π‘Ÿ = βˆ’ 6 ⃑ 𝑖 βˆ’ 2 ⃑ 𝑗 + 4 ⃑ π‘˜ , and the component of the moment of the force ⃑ 𝐹 about the 𝑦 -axis is βˆ’ 3 0 moment units, find the length of the perpendicular segment drawn from the origin point to the line of action of ⃑ 𝐹 .

  • A 2 √ 1 4 length units
  • B 2 √ 4 2 3 length units
  • C56 length units
  • D20 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 Nβ‹…m.

  • A βˆ’ 9 2 + 6 9 i j
  • B 9 2 βˆ’ 6 9 i j
  • C βˆ’ 5 5 + 6 9 i j
  • D βˆ’ 9 2 + 5 5 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 Nβ‹…cm
  • B Nβ‹…cm
  • C Nβ‹…cm
  • D Nβ‹…cm

Q14:

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

  • A βˆ’ 4 8 √ 3 ⃑ 𝑖 + 7 2 ⃑ 𝑗 βˆ’ 4 8 ⃑ π‘˜
  • B 7 2 ⃑ 𝑖 βˆ’ 4 8 √ 3 ⃑ 𝑗 + 4 8 ⃑ π‘˜
  • C βˆ’ 4 8 √ 3 ⃑ 𝑖 βˆ’ 7 2 ⃑ 𝑗 + 4 8 ⃑ π‘˜
  • D 4 8 √ 3 ⃑ 𝑖 + 7 2 ⃑ 𝑗 βˆ’ 4 8 ⃑ π‘˜

Q15:

In the figure, 𝐴 𝐡 is a rod fixed to a vertical wall at end 𝐴 . The other end 𝐡 is connected to a wire 𝐡 𝐢 , where 𝐢 is fixed to a different point on the same vertical wall. If the tension in the wire equals 39 N, calculate the moment of the tension about point 𝐴 in newton-metres.

  • A 3 2 4 + 4 3 2 i k
  • B 5 4 + 2 4 i k
  • C 1 3 + 1 2 i k
  • D 2 1 6 + 7 2 i k
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