3D force systems: reduction to single force

6. The vertices of a tetrahedron \(P Q R S\) have position vectors \(\mathbf { p } , \mathbf { q } , \mathbf { r }\) and \(\mathbf { s }\) respectively, where $$\mathbf { p } = - 3 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } , \quad \mathbf { q } = 4 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } , \quad \mathbf { r } = \mathbf { i } - 2 \mathbf { j } + \mathbf { k } , \quad \mathbf { s } = 4 \mathbf { i } + \mathbf { k }$$ Forces of magnitude 20 N and \(2 \sqrt { } 13 \mathrm {~N}\) act along \(S Q\) and \(S R\) respectively. A third force acts at \(P\).
Given that the system of three forces reduces to a couple \(\mathbf { G }\), find

1. the third force,
2. the magnitude of \(\mathbf { G }\).
   (6)
   (Total 12 marks)

1. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a rigid body. \(\mathbf { F } _ { 1 } = ( 12 \mathbf { i } - 4 \mathbf { j } + 6 \mathbf { k } ) \mathrm { N }\) and acts at the point with position vector \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } , \mathbf { F } _ { 2 } = ( - 3 \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }\) and acts at the point with position vector \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\). The force \(\mathbf { F } _ { 3 }\) acts at the point with position vector \(( 2 \mathbf { i } - \mathbf { k } ) \mathrm { m }\).

Given that this set of forces is equivalent to a couple, find

1. \(\mathbf { F } _ { 3 }\),
2. the magnitude of the couple.

3. A system of forces acting on a rigid body consists of two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) acting at a point \(A\) of the body, together with a couple of moment \(\mathbf { G } . \mathbf { F } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( - 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) N\). The position vector of the point \(A\) is \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\) and \(\mathbf { G } = ( 7 \mathbf { i } - 3 \mathbf { j } + 8 \mathbf { k } ) \mathrm { Nm }\). Given that the system is equivalent to a single force \(\mathbf { R }\),

1. find \(\mathbf { R }\),
2. find a vector equation for the line of action of \(\mathbf { R }\).
   (Total 9 marks) \section*{4.} \section*{Figure 1}
   \includegraphics[max width=\textwidth, alt={}]{43ce237f-c8e4-428a-b8cd-04673e62abb9-3_896_515_276_772}
   A thin uniform rod \(P Q\) has mass \(m\) and length \(3 a\). A thin uniform circular disc, of mass \(m\) and radius \(a\), is attached to the rod at \(Q\) in such a way that the rod and the diameter \(Q R\) of the disc are in a straight line with \(P R = 5 a\). The rod together with the disc form a composite body, as shown in Figure 1. The body is free to rotate about a fixed smooth horizontal axis \(L\) through \(P\), perpendicular to \(P Q\) and in the plane of the disc.

1. Two forces \(\mathbf { F } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( 3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }\) act on a rigid body.

The force \(\mathbf { F } _ { 1 }\) acts through the point with position vector ( \(2 \mathbf { i } + \mathbf { k }\) ) m and the force \(\mathbf { F } _ { 2 }\) acts through the point with position vector \(( \mathbf { j } + 2 \mathbf { k } ) \mathrm { m }\).

1. If the two forces are equivalent to a single force \(\mathbf { R }\), find
   1. \(\mathbf { R }\),
   2. a vector equation of the line of action of \(\mathbf { R }\), in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\).
2. If the two forces are equivalent to a single force acting through the point with position vector \(( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) \mathrm { m }\) together with a couple of moment \(\mathbf { G }\), find the magnitude of \(\mathbf { G }\).

2. Three forces \(\mathbf { F } _ { 1 } = ( a \mathbf { i } + b \mathbf { j } - 2 \mathbf { k } ) \mathrm { N } , \mathbf { F } _ { 2 } = ( - \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 3 } = ( - \mathbf { i } - 3 \mathbf { j } + \mathbf { k } ) \mathrm { N }\), where \(a\) and \(b\) are constants, act on a rigid body. The force \(\mathbf { F } _ { 1 }\) acts through the point with position vector \(\mathbf { k } \mathrm { m }\), the force \(\mathbf { F } _ { 2 }\) acts through the point with position vector \(( 3 \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\) and the force \(\mathbf { F } _ { 3 }\) acts through the point with position vector \(( \mathbf { j } + 2 \mathbf { k } ) \mathrm { m }\). The system of three forces is equivalent to a single force \(\mathbf { R }\) acting through the origin together with a couple of moment \(\mathbf { G }\). The direction of \(\mathbf { R }\) is parallel to the direction of \(\mathbf { G }\). Find the value of \(a\) and the value of \(b\).

4. Two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act on a rigid body, where \(\mathbf { F } _ { 1 } = ( 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( \mathbf { i } + 4 \mathbf { k } ) \mathrm { N }\). The force \(\mathbf { F } _ { 1 }\) acts through the point with position vector \(( \mathbf { i } + \mathbf { k } ) \mathrm { m }\) relative to a fixed origin \(O\). The force \(\mathbf { F } _ { 2 }\) acts through the point with position vector ( \(2 \mathbf { j }\) ) m . The two forces are equivalent to a single force \(\mathbf { F }\).

1. Find the magnitude of \(\mathbf { F }\).
2. Find, in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\), a vector equation of the line of action of \(\mathbf { F }\).

Two forces \(\mathbf{F}_1 = (i + 2j + 3k)\) N and \(\mathbf{F}_2 = (3i + j + 2k)\) N act on a rigid body. The force \(\mathbf{F}_1\) acts through the point with position vector \((2i + k)\) m and the force \(\mathbf{F}_2\) acts through the point with position vector \((j + 2k)\) m.

1. If the two forces are equivalent to a single force \(\mathbf{R}\), find
   1. \(\mathbf{R}\), [2]
   2. a vector equation of the line of action of \(\mathbf{R}\), in the form \(\mathbf{r} = \mathbf{a} + \lambda \mathbf{b}\). [6]

1. If the two forces are equivalent to a single force acting through the point with position vector \((i + 2j + k)\) m together with a couple of moment \(\mathbf{G}\), find the magnitude of \(\mathbf{G}\). [5]

The points \(P\) and \(Q\) have position vectors \(4i - 6j - 12k\) and \(2i + 4j + 4k\) respectively, relative to a fixed origin \(O\). Three forces, \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\), act along \(\overrightarrow{OP}\), \(\overrightarrow{OQ}\) and \(\overrightarrow{QP}\) respectively, and have magnitudes \(7\) N, \(3\) N and \(3\sqrt{10}\) N respectively.

1. Express \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) in vector form. [3]

1. Show that the resultant of \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) is \((2i - 10j - 16k)\) N. [2]

1. Find a vector equation of the line of action of this resultant, giving your answer in the form \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are constant vectors and \(\lambda\) is a parameter. [5]

Three forces \(\mathbf{F}_1 = (3i - j + k)\) N, \(\mathbf{F}_2 = (2i - k)\) N, and \(\mathbf{F}_3\) act on a rigid body. The force \(\mathbf{F}_1\) acts through the point with position vector \((i + 2j + k)\) m, the force \(\mathbf{F}_2\) acts through the point with position vector \((i - 2j)\) m and the force \(\mathbf{F}_3\) acts through the point with position vector \((i + j + k)\) m. Given that the system \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) reduces to a couple \(\mathbf{G}\),

1. find \(\mathbf{G}\). [6]

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 \((6i + 8j + 2k)\) N m.

1. 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. [5]

A force system consists of three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) acting on a rigid body. \(\mathbf{F}_1 = (\mathbf{i} + 2\mathbf{j})\) N and acts at the point with position vector \((-\mathbf{i} + 4\mathbf{j})\) m. \(\mathbf{F}_2 = (-\mathbf{j} + \mathbf{k})\) N and acts at the point with position vector \((2\mathbf{i} + \mathbf{j} + \mathbf{k})\) m. \(\mathbf{F}_3 = (3\mathbf{i} - \mathbf{j} + \mathbf{k})\) N and acts at the point with position vector \((\mathbf{i} - \mathbf{j} + 2\mathbf{k})\) m. It is given that this system can be reduced to a single force \(\mathbf{R}\).

1. Find \(\mathbf{R}\). [2]
2. Find a vector equation of the line of action of \(\mathbf{R}\), giving your answer in the form \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are constant vectors and \(\lambda\) is a parameter. [10]

The points \(P\) and \(Q\) have position vectors \(4\mathbf{i} - 6\mathbf{j} - 12\mathbf{k}\) and \(2\mathbf{i} + 4\mathbf{j} + 4\mathbf{k}\) respectively, relative to a fixed origin \(O\). Three forces, \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\), act along \(\overrightarrow{OP}\), \(\overrightarrow{OQ}\) and \(\overrightarrow{QP}\) respectively, and have magnitudes 7 N, 3 N and \(3\sqrt{10}\) N respectively.

1. Express \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) in vector form. [3]
2. Show that the resultant of \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) is \((2\mathbf{i} - 10\mathbf{j} - 16\mathbf{k})\) N. [2]
3. Find a vector equation of the line of action of this resultant, giving your answer in the form \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are constant vectors and \(\lambda\) is a parameter. [5]

Two forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\) and a couple \(\mathbf{G}\) act on a rigid body. The force \(\mathbf{F}_1 = (3\mathbf{i} + 4\mathbf{j})\) N acts through the point with position vector \(2\mathbf{i}\) m relative to a fixed origin \(O\). The force \(\mathbf{F}_2 = (2\mathbf{i} - \mathbf{j} + \mathbf{k})\) N acts through the point with position vector \((\mathbf{i} + \mathbf{j})\) m relative to \(O\). The forces and couple are equivalent to a single force \(\mathbf{F}\) acting through \(O\).

1. Find \(\mathbf{F}\). [2]
2. Find \(\mathbf{G}\). [5]