Forces in equilibrium (find unknowns)

2 \includegraphics[max width=\textwidth, alt={}, center]{fa0e0e0d-b0a6-44e0-8b4f-4923e235c6c6-2_465_478_479_836} Three coplanar forces of magnitudes \(F \mathrm {~N} , 12 \mathrm {~N}\) and 15 N are in equilibrium acting at a point \(P\) in the directions shown in the diagram. Find \(\alpha\) and \(F\).

2 \includegraphics[max width=\textwidth, alt={}, center]{f4f2996b-5382-4b0d-9804-b5f5945946b3-2_636_519_664_813} Three horizontal forces of magnitudes \(F \mathrm {~N} , 63 \mathrm {~N}\) and 25 N act at \(O\), the origin of the \(x\)-axis and \(y\)-axis. The forces are in equilibrium. The force of magnitude \(F \mathrm {~N}\) makes an angle \(\theta\) anticlockwise with the positive \(x\)-axis. The force of magnitude 63 N acts along the negative \(y\)-axis. The force of magnitude 25 N acts at \(\tan ^ { - 1 } 0.75\) clockwise from the negative \(x\)-axis (see diagram). Find the value of \(F\) and the value of \(\tan \theta\).

1 \includegraphics[max width=\textwidth, alt={}, center]{b9080e9f-2c23-43ce-b171-bd68648dc56b-2_711_398_269_877} Each of two identical light elastic strings has natural length 0.25 m and modulus of elasticity 4 N . A particle \(P\) of mass 0.6 kg is attached to one end of each of the strings. The other ends of the strings are attached to fixed points \(A\) and \(B\) which are 0.8 m apart on a smooth horizontal table. The particle is held at rest on the table, at a point 0.3 m from \(A B\) for which \(A P = B P\) (see diagram).

1. Find the tension in the strings.
2. The particle is released. Find its initial acceleration.

2 A small box has weight \(\mathbf { W } \mathrm { N }\) and is held in equilibrium by two strings with tensions \(\mathbf { T } _ { 1 } \mathrm {~N}\) and \(\mathbf { T } _ { 2 } \mathrm {~N}\). This situation is shown in Fig. 2 which also shows the standard unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) that are horizontal and vertically upwards, respectively. \begin{figure}[h] \end{figure} The tension \(\mathbf { T } _ { 1 }\) is \(10 \mathbf { i } + 24 \mathbf { j }\).

1. Calculate the magnitude of \(\mathbf { T } _ { 1 }\) and the angle between \(\mathbf { T } _ { 1 }\) and the vertical. The magnitude of the weight is \(w \mathrm {~N}\).
2. Write down the vector \(\mathbf { W }\) in terms of \(w\) and \(\mathbf { j }\). The tension \(\mathbf { T } _ { 2 }\) is \(k \mathbf { i } + 10 \mathbf { j }\), where \(k\) is a scalar.
3. Find the values of \(k\) and of \(w\).

2 The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) shown in Fig. 2 are in the horizontal and vertically upwards directions. \begin{figure}[h] \end{figure} Forces \(\mathbf { p }\) and \(\mathbf { q }\) are given, in newtons, by \(\mathbf { p } = 12 \mathbf { i } - 5 \mathbf { j }\) and \(\mathbf { q } = 16 \mathbf { i } + 1.5 \mathbf { j }\).

1. Write down the force \(\mathbf { p } + \mathbf { q }\) and show that it is parallel to \(8 \mathbf { i } - \mathbf { j }\).
2. Show that the force \(3 \mathbf { p } + 10 \mathbf { q }\) acts in the horizontal direction.
3. A particle is in equilibrium under forces \(k \mathbf { p } , 3 \mathbf { q }\) and its weight \(\mathbf { w }\). Show that the value of \(k\) must be - 4 and find the mass of the particle.

9 Three forces \(\binom { 7 } { - 6 } \mathrm {~N} , \binom { 2 } { 5 } \mathrm {~N}\) and \(\mathbf { F N }\) act on a particle.
Given that the particle is in equilibrium under the action of these three forces, calculate \(\mathbf { F }\).

8 A particle is in equilibrium under the action of the following three forces: \(( 2 p \mathbf { i } - 4 \mathbf { j } ) N , ( - 3 q \mathbf { i } + 5 p \mathbf { j } ) N\) and \(( - 13 \mathbf { i } - 6 \mathbf { j } ) N\).
Find the values of p and q .

3. A particle is in equilibrium under the action of three forces \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) acting in the same horizontal plane. \(P\) has magnitude 9 N and acts on a bearing of \(030 ^ { \circ } . Q\) has magnitude 12 N . and acts on a bearing of \(225 ^ { \circ }\).

1. Find the values of \(a\) and \(b\) such that \(\mathbf { R } = ( a \mathbf { i } + b \mathbf { j } ) \mathrm { N }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the directions due East and due North respectively.
2. Calculate the magnitude and direction of \(\mathbf { R }\)

4 In this question, \(\mathbf { i }\) is a horizontal unit vector and \(\mathbf { j }\) is a unit vector pointing vertically upwards.
A force \(\mathbf { F }\) is \(- \mathbf { i } + 5 \mathbf { j }\).

1. Calculate the magnitude of \(\mathbf { F }\). Calculate also the angle between \(\mathbf { F }\) and the upward vertical. Force \(\mathbf { G }\) is \(2 a \mathbf { i } + a \mathbf { j }\) and force \(\mathbf { H }\) is \(- 2 \mathbf { i } + 3 b \mathbf { j }\), where \(a\) and \(b\) are constants. The force \(\mathbf { H }\) is the resultant of forces \(4 \mathbf { F }\) and \(\mathbf { G }\).
2. Find \(\mathbf { G }\) and \(\mathbf { H }\).

A particle \(P\) is acted upon by three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) given by \(\mathbf{F}_1 = (6\mathbf{i} - 4\mathbf{j}) \text{ N}\), \(\mathbf{F}_2 = (-3\mathbf{i} + 9\mathbf{j}) \text{ N}\) and \(\mathbf{F}_3 = (a\mathbf{i} + b\mathbf{j}) \text{ N}\), where \(a\) and \(b\) are constants. Given that \(P\) is in equilibrium,

1. find the value of \(a\) and the value of \(b\). [2]

The force \(\mathbf{F}_3\) is now removed. The resultant of \(\mathbf{F}_1\) and \(\mathbf{F}_2\) is \(\mathbf{R}\).

1. Find the magnitude of \(\mathbf{R}\). [3]
2. Find the angle, to \(0.1°\), that \(\mathbf{R}\) makes with \(\mathbf{i}\). [2]