3D force systems: equilibrium conditions

Given forces acting on a rigid body in 3D, find unknown force/couple components required for equilibrium, including finding resultant force and couple vectors.

5 questions · Challenging +1.3

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Edexcel M5 2009 June Q5
16 marks Challenging +1.2
  1. Two forces \(\mathbf { F } _ { 1 } = ( 2 \mathbf { i } + \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( - 2 \mathbf { j } - \mathbf { k } ) \mathrm { N }\) act on a rigid body. The force \(\mathbf { F } _ { 1 }\) acts at the point with position vector \(\mathbf { r } _ { 1 } = ( 3 \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\) and the force \(\mathbf { F } _ { 2 }\) acts at the point with position vector \(\mathbf { r } _ { 2 } = ( \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }\). A third force \(\mathbf { F } _ { 3 }\) acts on the body such that \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) are in equilibrium.
    1. Find the magnitude of \(\mathbf { F } _ { 3 }\).
    2. Find a vector equation of the line of action of \(\mathbf { F } _ { 3 }\).
    The force \(\mathbf { F } _ { 3 }\) is replaced by a fourth force \(\mathbf { F } _ { 4 }\), acting through the origin \(O\), such that \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 4 }\) are equivalent to a couple.
  2. Find the magnitude of this couple.
Edexcel M5 2013 June Q2
11 marks Challenging +1.2
2. Three forces \(\mathbf { F } _ { 1 } = ( 3 \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { N } , \mathbf { F } _ { 2 } = ( 2 \mathbf { i } - \mathbf { k } ) \mathrm { N }\), and \(\mathbf { F } _ { 3 }\) act on a rigid body. The force \(\mathbf { F } _ { 1 }\) acts through the point with position vector \(( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) \mathrm { m }\), the force \(\mathbf { F } _ { 2 }\) acts through the point with position vector \(( \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }\) and the force \(\mathbf { F } _ { 3 }\) acts through the point with position vector \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\). Given that the system \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) reduces to a couple \(\mathbf { G }\),
  1. find \(\mathbf { G }\). The line of action of \(\mathbf { F } _ { 3 }\) is changed so that the system \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) now reduces to a couple \(( 6 \mathbf { i } + 8 \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }\) m.
  2. Find an equation of the new line of action of \(\mathbf { F } _ { 3 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors.
Edexcel M5 2013 June Q4
15 marks Challenging +1.2
  1. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a rigid body. The forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act through the points with position vectors \(\mathbf { r } _ { 1 }\) and \(\mathbf { r } _ { 2 }\) respectively. \(\mathbf { r } _ { 1 } = ( - 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m }\), \(\mathbf { F } _ { 1 } = ( 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } ) \mathrm { N }\) \(\mathbf { r } _ { 2 } = ( 3 \mathbf { i } + 2 \mathbf { k } ) \mathrm { m }\), \(\mathbf { F } _ { 2 } = ( - 2 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \mathrm { N }\) Given that the system \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) is in equilibrium,
    1. find \(\mathbf { F } _ { 3 }\),
    2. find a vector equation of the line of action of \(\mathbf { F } _ { 3 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\).
    The force \(\mathbf { F } _ { 3 }\) is replaced by a force \(\mathbf { F } _ { 4 }\) acting through the point with position vector \(( \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }\). The system \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 4 }\) is equivalent to a single force ( \(3 \mathbf { i } + \mathbf { j } + \mathbf { k }\) ) N acting through the point with position vector \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\) together with a couple.
  2. Find the magnitude of this couple.
Edexcel M5 2015 June Q3
12 marks Challenging +1.2
  1. A rigid body is in equilibrium under the action of three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 } \mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act at the points with position vectors \(\mathbf { r } _ { 1 }\) and \(\mathbf { r } _ { 2 }\) respectively, where \(\mathbf { F } _ { 1 } = ( 2 \mathbf { j } + \mathbf { k } ) \mathrm { N } \quad \mathbf { r } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } ) \mathrm { m } \mathbf { F } _ { 2 } = ( - 2 \mathbf { i } - \mathbf { j } ) \mathrm { N } \quad \mathbf { r } _ { 2 } = ( - \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\)
    1. Find the magnitude of \(\mathbf { F } _ { 3 }\)
    2. Find a vector equation of the line of action of \(\mathbf { F } _ { 3 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(t\) is a scalar parameter.
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Edexcel M5 2017 June Q3
15 marks Challenging +1.8
  1. The position vectors of the points \(P\) and \(Q\) on a rigid body are \(( \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }\) and \(( \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\) respectively, relative to a fixed origin \(O\). A force \(\mathbf { F } _ { 1 }\) of magnitude 6 N acts at \(P\) in the direction \(( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } )\). A force \(\mathbf { F } _ { 2 }\) of magnitude 14 N acts at \(Q\) in the direction \(( 3 \mathbf { i } - 6 \mathbf { j } + 2 \mathbf { k } )\). When a force \(\mathbf { F } _ { 3 }\) acts at \(O\), which is also a point on the rigid body, the system of three forces is equivalent to a couple of moment \(\mathbf { G }\)
    1. Find \(\mathbf { F } _ { 3 }\)
    2. Find G
    When an additional force \(\mathbf { F } _ { 4 } = ( \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } ) \mathrm { N }\) also acts at \(O\), the system of four forces is equivalent to a single force \(\mathbf { R }\).
  2. Write down \(\mathbf { R }\).
  3. Find an equation of the line of action of \(\mathbf { R }\) in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(t\) is a parameter.