Equilibrium of particle under coplanar forces

2 A particle is in equilibrium under the action of four horizontal forces of magnitudes 5 newtons, 8 newtons, \(P\) newtons and \(Q\) newtons, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{cfe0bdbc-35e3-485f-a922-b652a72f4c95-2_355_357_1146_852}

1. Show that \(P = 9\).
2. Find the value of \(Q\).

4 A particle, of mass \(m \mathrm {~kg}\), remains in equilibrium under the action of three forces, which act in a vertical plane, as shown in the diagram. The force with magnitude 60 N acts at \(48 ^ { \circ }\) above the horizontal and the force with magnitude 50 N acts at an angle \(\theta\) above the horizontal.
\includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-08_576_647_548_701}

1. By resolving horizontally, find \(\theta\).
2. Find \(m\).
   \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-09_2484_1709_223_153}
   \begin{center} \begin{tabular}{|l|l|} \hline & \begin{tabular}{l}

4 A particle, of weight \(W\) newtons, is held in equilibrium by two forces of magnitudes 10 newtons and 20 newtons. The 10 -newton force is horizontal and the 20 -newton force acts at an angle \(\theta\) above the horizontal, as shown in the diagram. All three forces act in the same vertical plane.
\includegraphics[max width=\textwidth, alt={}, center]{828e8db1-efcf-4878-8292-ba5bbd80115c-3_406_608_520_717}

1. \(\quad\) Find \(\theta\).
2. \(\quad\) Find \(W\).
3. Calculate the mass of the particle.

2 Three forces are in equilibrium in a vertical plane, as shown in the diagram. There is a vertical force of magnitude 40 N and a horizontal force of magnitude 60 N . The third force has magnitude \(F\) newtons and acts at an angle \(\theta\) above the horizontal.
\includegraphics[max width=\textwidth, alt={}, center]{788534a5-abbb-4d6a-87b2-c54e859a128a-04_490_894_456_571}

1. \(\quad\) Find \(F\).
2. \(\quad\) Find \(\theta\).

1 Forces of magnitude \(4 \mathrm {~N} , 3 \mathrm {~N} , 5 \mathrm {~N}\) and \(R \mathrm {~N}\) act on a particle in the directions shown in Fig. 1. \begin{figure}[h] \end{figure} The particle is in equilibrium. Find each of the following.

• The value of \(R\).
• The value of \(\theta\).

2 The vertices of a triangular lamina, which is in the \(x - y\) plane, are at the origin O and the points \(\mathrm { A } ( 4,0 )\) and \(\mathrm { B } ( 0,3 )\). Forces, of magnitude \(T _ { 1 } \mathrm {~N} , T _ { 2 } \mathrm {~N}\) and 10 N , whose lines of action are in the \(x - y\) plane, are applied to the lamina at \(\mathrm { O } , \mathrm { A }\) and B respectively, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-2_814_922_1135_246}

1. 1. Show that \(\sin \alpha = 0.6\).
   2. Write down the value of \(\cos \alpha\). The lamina is in equilibrium.
2. Determine the values of \(T _ { 1 } , T _ { 2 }\) and \(\theta\).

1. The diagram below shows four coplanar horizontal forces of magnitude \(F \mathrm {~N} , 12 \mathrm {~N} , 16 \mathrm {~N}\) and 20 N acting at a point \(P\) in the directions shown.
   \includegraphics[max width=\textwidth, alt={}, center]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-14_792_862_593_607}

Given that the forces are in equilibrium, calculate the value of \(F\) and the size of the angle \(\alpha\). [7]