Three-dimensional force systems: finding resultant and couple

2. Three forces, \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a rigid body. \(\mathbf { F } _ { 1 } = ( 2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) \mathrm { N } , \mathbf { F } _ { 2 } = ( \mathbf { i } + \mathbf { j } - 4 \mathbf { k } )\) N and \(\mathbf { F } _ { 3 } = ( p \mathbf { i } + q \mathbf { j } + r \mathbf { k } ) \mathrm { N }\), where \(p , q\) and \(r\) are constants. All three forces act through the point with position vector \(( 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } ) \mathrm { m }\), relative to a fixed origin. The three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) are equivalent to a single force ( \(5 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }\) ) N , acting at the origin, together with a couple \(\mathbf { G }\).

1. Find the values of \(p , q\) and \(r\).
2. Find \(\mathbf { G }\).

2. Three forces, \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a rigid body. \(\mathbf { F } _ { 1 } = ( 2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) \mathrm { N } , \mathbf { F } _ { 2 } = ( \mathbf { i } + \mathbf { j } - 4 \mathbf { k } )\) N and \(\mathbf { F } _ { 3 } = ( p \mathbf { i } + q \mathbf { j } + r \mathbf { k } ) \mathrm { N }\), where \(p , q\) and \(r\) are constants. All three forces act through the point with position vector ( \(3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\) ) m, relative to a fixed origin. The three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) are equivalent to a single force ( \(5 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }\) ) N, acting at the origin, together with a couple \(\mathbf { G }\).

1. Find the values of \(p , q\) and \(r\).
2. Find \(\mathbf { G }\).

3. \section*{Figure 1} Figure 1 shows a box in the shape of a cuboid \(P Q R S T U V W\) where \(\overrightarrow { P Q } = 3 \mathbf { i }\) metres, \(\overrightarrow { P S } = 4 \mathbf { j }\) metres and \(\overrightarrow { P T } = 3 \mathbf { k }\) metres. A force \(( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { N }\) acts at \(Q\), a force \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) acts at \(R\), a force \(( - 2 \mathbf { j } + \mathbf { k } ) \mathrm { N }\) acts at \(T\), and a force \(( 2 \mathbf { j } + \mathbf { k } ) \mathrm { N }\) acts at \(W\). Given that these are the only forces acting on the box, find

1. the resultant force acting on the box,
2. the resultant vector moment about \(P\) of the four forces acting on the box. When an additional force \(\mathbf { F }\) acts on the box at a point \(X\) on the edge \(P S\), the box is in equilibrium.
3. Find \(\mathbf { F }\).
4. Find the length of \(P X\).

5. Two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act on a rigid body, where \(\mathbf { F } _ { 1 } = ( 3 \mathbf { i } + 4 \mathbf { j } - 6 \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( 5 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }\).
The force \(\mathbf { F } _ { 1 }\) acts at the point with position vector \(( \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }\), and the force \(\mathbf { F } _ { 2 }\) acts at the point with position vector ( \(3 \mathbf { i } - \mathbf { k }\) ) m. The two forces are equivalent to a single force \(\mathbf { F }\) acting at the point with position vector \(( \mathbf { i } - \mathbf { k } ) \mathrm { m }\), together with a couple \(\mathbf { G }\).

1. Find \(\mathbf { F }\).
2. Find the magnitude of \(\mathbf { G }\).
   (8)

3. A system of forces consists of two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) acting on a rigid body. \(\mathbf { F } _ { 1 } = ( - 2 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \mathrm { N }\) and acts at the point with position vector \(\mathbf { r } _ { 1 } = ( \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\). \(\mathbf { F } _ { 2 } = ( 3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }\) and acts at the point with position vector \(\mathbf { r } _ { 2 } = ( 4 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ) \mathrm { m }\).
Given that the system is equivalent to a single force \(\mathbf { R } \mathrm { N }\), acting at the point with position vector \(( 5 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \mathrm { m }\), together with a couple \(\mathbf { G N m }\), find

1. \(\mathbf { R }\),
2. the magnitude of \(\mathbf { G }\).
   (9)

6. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a rigid body at the points with position vectors, \(\mathbf { r } _ { 1 } , \mathbf { r } _ { 2 }\) and \(\mathbf { r } _ { 3 }\) respectively, where \(\mathbf { F } _ { 1 } = ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { N }\) \(\mathbf { F } _ { 2 } = ( 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) \mathrm { N }\) \(\mathbf { F } _ { 3 } = ( - \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }\) \(\mathbf { r } _ { 1 } = ( \mathbf { i } - \mathbf { k } ) \mathrm { m }\) \(\mathbf { r } _ { 2 } = ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\) \(\mathbf { r } _ { 3 } = ( \mathbf { i } + \mathbf { j } - \mathbf { k } ) \mathrm { m }\) The system of the three forces is equivalent to a single force \(\mathbf { R }\) acting at the point with position vector ( \(\mathbf { 3 i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\), together with a couple of moment \(\mathbf { G }\).

1. Find \(\mathbf { R }\).
2. Find \(\mathbf { G }\).

1. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a rigid body at the points with position vectors \(\mathbf { r } _ { 1 } , \mathbf { r } _ { 2 }\) and \(\mathbf { r } _ { 3 }\) respectively, where \(\mathbf { F } _ { 1 } = ( 2 \mathbf { j } - \mathbf { k } ) \mathrm { N }\) \(\mathbf { F } _ { 3 } = ( \mathbf { i } + \mathbf { j } ) \mathrm { N }\) \(\mathbf { r } _ { 1 } = ( 4 \mathbf { j } - \mathbf { k } ) \mathrm { m }\) \(\mathbf { r } _ { 3 } = ( 3 \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\) \(\mathbf { F } _ { 1 } = ( 2 \mathbf { j } - \mathbf { k } ) \mathrm { N }\) \(\mathbf { r } _ { 1 } = ( 4 \mathbf { j } - \mathbf { k } ) \mathrm { m }\)

$$\begin{aligned} & \mathbf { F } _ { 2 } = ( \mathbf { i } + \mathbf { k } ) \mathrm { N } \\ & \mathbf { r } _ { 2 } = ( 2 \mathbf { i } + \mathbf { k } ) \mathrm { m } \end{aligned}$$ j The system of the three forces is equivalent to a single force \(\mathbf { R }\) acting through the point with position vector \(( \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\), together with a couple of moment \(\mathbf { G }\).

1. Find \(\mathbf { R }\).
2. Find \(\mathbf { G }\). respectively, where The

10 JANUARY 2006 Afternoon
1 hour 30 minutes
Additional materials:
8 page answer booklet
Graph paper
List of Formulae (MF1) TIME
1 hour 30 minutes

• Write your name, centre number and candidate number in the spaces provided on the answer booklet.
• Answer all the questions.
• Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy is specified in the question or is clearly appropriate.
• The acceleration due to gravity is denoted by \(\mathrm { g } \mathrm { m } \mathrm { s } ^ { - 2 }\). Unless otherwise instructed, when a numerical value is needed, use \(g = 9.8\).
• You are permitted to use a graphical calculator in this paper.

• The number of marks is given in brackets [ ] at the end of each question or part question.
• The total number of marks for this paper is 72.
• Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper.
• You are reminded of the need for clear presentation in your answers.

2 The force \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { N }\) acts at the point with coordinates \(( 0,2 )\) The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed along the \(x\)-axis and the \(y\)-axis respectively.
Calculate the magnitude of the moment of this force about the origin.
Circle your answer.
[0pt] [1 mark]
6 Nm
8 Nm
10 Nm
14 Nm

Two forces \(\mathbf{F}_1 = (3i + k)\) N and \(\mathbf{F}_2 = (4i + j - k)\) N act on a rigid body. The force \(\mathbf{F}_1\) acts at the point with position vector \((2i - j + 3k)\) m and the force \(\mathbf{F}_2\) acts at the point with position vector \((-3i + 2k)\) m. The two forces are equivalent to a single force \(\mathbf{R}\) acting at the point with position vector \((i + 2j + k)\) m together with a couple of moment \(\mathbf{G}\). Find,

1. \(\mathbf{R}\), [2]
2. \(\mathbf{G}\). [4]

A third force \(\mathbf{F}_3\) is now added to the system. The force \(\mathbf{F}_3\) acts at the point with position vector \((2i - k)\) m and the three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) are equivalent to a couple.

1. Find the magnitude of the couple. [6]

Two forces \(\mathbf{F}_1 = (3\mathbf{i} + \mathbf{k})\) N and \(\mathbf{F}_2 = (4\mathbf{i} + \mathbf{j} - \mathbf{k})\) N act on a rigid body. The force \(\mathbf{F}_1\) acts at the point with position vector \((2\mathbf{i} - \mathbf{j} + 3\mathbf{k})\) m and the force \(\mathbf{F}_2\) acts at the point with position vector \((-3\mathbf{i} + 2\mathbf{k})\) m. The two forces are equivalent to a single force \(\mathbf{R}\) acting at the point with position vector \((\mathbf{i} + 2\mathbf{j} + \mathbf{k})\) m together with a couple of moment \(\mathbf{G}\). Find,

1. \(\mathbf{R}\), [2]
2. \(\mathbf{G}\). [4]

A third force \(\mathbf{F}_3\) is now added to the system. The force \(\mathbf{F}_3\) acts at the point with position vector \((2\mathbf{i} - \mathbf{k})\) m and the three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) are equivalent to a couple.

1. Find the magnitude of the couple. [6]

Three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) act on a rigid body at the points with position vectors \(\mathbf{r}_1\), \(\mathbf{r}_2\) and \(\mathbf{r}_3\) respectively. \(\mathbf{F}_1 = (2\mathbf{i} + 3\mathbf{j} - \mathbf{k})\) N and \(\mathbf{r}_1 = (\mathbf{i} + \mathbf{j} - 2\mathbf{k})\) m, \(\mathbf{F}_2 = (\mathbf{i} - 4\mathbf{j} - 2\mathbf{k})\) N and \(\mathbf{r}_2 = (3\mathbf{i} - \mathbf{j} - \mathbf{k})\) m, \(\mathbf{F}_3 = (-3\mathbf{i} + \mathbf{j} + 3\mathbf{k})\) N and \(\mathbf{r}_3 = (\mathbf{i} - 2\mathbf{j} + \mathbf{k})\) m. Show that the system is equivalent to a couple and find the magnitude of the vector moment of this couple. [9]

A force of magnitude 7 N acts at each end of a rod of length 20 cm, forming a couple. The forces act at right angles to the rod, as shown in the diagram below. \includegraphics{figure_2} Find the magnitude of the resultant moment of the couple. Circle your answer. [1 mark] 1.4 N m 2.8 N m 140 N m 280 N m

Three forces represented by the vectors \(-4\mathbf{i} + \mathbf{j} + 2\mathbf{j}\) and \(k\mathbf{i} - 2\mathbf{j}\) act at the points with coordinates \((0, 0)\), \((3, 0)\) and \((0, 4)\) respectively.

1. Given that the three forces form a couple, find the value of \(k\). [2]
2. Find the magnitude and direction of the couple. [3]