Equilibrium of particle under coplanar forces

8 Three forces act in a vertical plane on an object of mass 250 kg , as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{7ac7dfd0-4c3e-4eb7-920f-ce5b24ad1281-5_481_1139_408_447} The two forces \(P\) newtons and \(Q\) newtons each act at \(80 ^ { \circ }\) to the horizontal. The object accelerates horizontally at \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) under the action of these forces.

1. Show that $$P = 125 \left( \frac { a } { \cos 80 ^ { \circ } } + \frac { g } { \sin 80 ^ { \circ } } \right)$$
2. Find the value of \(a\) for which \(Q\) is zero.

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 \(\theta\).

3 A ship has a mass of 500 tonnes. Two tugs are used to pull the ship using cables that are horizontal. One tug exerts a force of 100000 N at an angle of \(25 ^ { \circ }\) to the centre line of the ship. The other tug exerts a force of \(T \mathrm {~N}\) at an angle of \(20 ^ { \circ }\) to the centre line of the ship. The diagram shows the ship and forces as viewed from above. \includegraphics[max width=\textwidth, alt={}, center]{01338c87-302c-420f-a473-39b0796ccaed-06_279_844_539_664} The ship accelerates in a straight line along its centre line.

1. \(\quad\) Find \(T\).
2. A resistance force of magnitude 20000 N directly opposes the motion of the ship. Find the acceleration of the ship.
   [0pt] [4 marks]
   \includegraphics[max width=\textwidth, alt={}]{01338c87-302c-420f-a473-39b0796ccaed-06_1419_1714_1288_153}

3. \begin{figure}[h] \end{figure} Figure 1 shows the forces acting on a particle, \(P\). These consist of a 20 N force to the South, a 6 N force to the East, an 18 N force \(30 ^ { \circ }\) West of North and two unknown forces \(X\) and \(Y\) which act to the North-East and North respectively. Given that \(P\) is in equilibrium,

1. show that \(X\) has magnitude \(3 \sqrt { } 2 \mathrm {~N}\),
2. find the exact value of \(Y\).

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]

11 \includegraphics[max width=\textwidth, alt={}, center]{35d24778-1203-4d5d-be4b-bb375344fe09-4_285_700_1043_721} Three forces are acting on a particle \(A\) as shown in the diagram. The forces act in the same plane and the particle is in equilibrium.

1. Evaluate \(P\) and \(\theta\). The 8 N force is removed.
2. State the direction of the instantaneous acceleration of \(A\).

\includegraphics{figure_2} Coplanar forces of magnitudes 20 N, \(P\) N, \(3P\) N and \(4P\) N act at a point in the directions shown in the diagram. The system is in equilibrium. Find \(P\) and \(\theta\). [6]

\includegraphics{figure_4} Three coplanar forces of magnitudes 20 N, 100 N and \(F\) N act at a point. The directions of these forces are shown in the diagram. Given that the three forces are in equilibrium, find \(F\) and \(\alpha\). [6]

\includegraphics{figure_5} Four coplanar forces act at a point. The magnitudes of the forces are \(F\) N, \(10\) N, \(50\) N and \(40\) N. The directions of the forces are as shown in the diagram.

1. Given that the forces are in equilibrium, find the value of \(F\) and the value of \(\theta\). [6]
2. Given instead that \(F = 10\sqrt{2}\) and \(\theta = 45\), find the direction and the exact magnitude the resultant force. [3]

\includegraphics{figure_3} Four coplanar forces of magnitude \(P\) N, 10 N, 16 N and 2 N act at a point in the directions shown in the diagram. It is given that the forces are in equilibrium. Find the values of \(\theta\) and \(P\). [6]

\includegraphics{figure_5} The diagram shows a block \(D\) of mass 100 kg supported by two sloping struts \(AD\) and \(BD\), each attached at an angle of \(45°\) to fixed points \(A\) and \(B\) respectively on a horizontal floor. The block is also held in place by a vertical rope \(CD\) attached to a fixed point \(C\) on a horizontal ceiling. The tension in the rope \(CD\) is 500 N and the block rests in equilibrium.

1. Find the magnitude of the force in each of the struts \(AD\) and \(BD\). [3] A horizontal force of magnitude \(F\) N is applied to the block in a direction parallel to \(AB\).
2. Find the value of \(F\) for which the magnitude of the force in the strut \(AD\) is zero. [3]

\includegraphics{figure_4} Four coplanar forces act at a point. The magnitudes of the forces are \(F\) N, \(2F\) N, \(3F\) N and \(30\) N. The directions of the forces are as shown in the diagram. Given that the forces are in equilibrium, find the value of \(F\) and the value of \(\theta\). [6]

\includegraphics{figure_3} Coplanar forces of magnitudes 8 N, 12 N, 10 N and \(P\) N act at a point in the directions shown in the diagram. The system is in equilibrium. Find \(P\) and \(\theta\). [6]

\includegraphics{figure_1} Coplanar forces of magnitudes \(P\) N, \(Q\) N, 16 N and 22 N act at a point in the directions shown in the diagram. The forces are in equilibrium. Find the values of \(P\) and \(Q\). [5]

\includegraphics{figure_3} Coplanar forces of magnitudes \(52\) N, \(39\) N and \(P\) N act at a point in the directions shown in the diagram. The system is in equilibrium. Find the values of \(P\) and \(\theta\). [4]

\includegraphics{figure_3} A particle is moving under the action of three forces as shown in the diagram. The particle is in equilibrium. Find the magnitudes of forces \(P\) and \(Q\). [6]

\includegraphics{figure_1} Coplanar forces of magnitudes 40 N, 32 N, \(P\) N and 17 N act at a point in the directions shown in the diagram. The system is in equilibrium. Find the values of \(P\) and \(\theta\). [6]

\includegraphics{figure_3} Four coplanar forces of magnitudes \(F\) N, \(5\) N, \(25\) N and \(15\) N are acting at a point \(P\) in the directions shown in the diagram. Given that the forces are in equilibrium, find the values of \(F\) and \(α\). [6]

\includegraphics{figure_3} A particle is in equilibrium on a smooth horizontal table when acted on by the three horizontal forces shown in the diagram.

1. Find the values of \(F\) and \(\theta\). [4]
2. The force of magnitude 7 N is now removed. State the magnitude and direction of the resultant of the remaining two forces. [2]

\includegraphics{figure_5} A small ring \(P\) is threaded on a fixed smooth horizontal rod \(AB\). Three horizontal forces of magnitudes 4.5 N, 7.5 N and \(F\) N act on \(P\) (see diagram).

1. Given that these three forces are in equilibrium, find the values of \(F\) and \(\theta\). [6]

1. It is given instead that the values of \(F\) and \(\theta\) are 9.5 and 30 respectively, and the acceleration of the ring is 1.5 m s\(^{-2}\). Find the mass of the ring. [2]

A smooth sphere of mass \(M\) and radius \(a\) rests in contact with a smooth vertical wall and a smooth inclined plane. The plane makes an angle \(\alpha\) with the horizontal.

1. Find the magnitude of each of the contact forces acting on the sphere.
2. Find the range of values of \(\alpha\) for which this equilibrium is possible.

[8]

\includegraphics{figure_1} In Fig. 1, \(\angle AOC = 90°\) and \(\angle BOC = \theta°\). A particle at \(O\) is in equilibrium under the action of three coplanar forces. The three forces have magnitude 8 N, 12 N and \(X\) N and act along \(OA\), \(OB\) and \(OC\) respectively. Calculate

1. the value, to one decimal place, of \(\theta\), [3]
2. the value, to 2 decimal places, of \(X\). [3]