3.03m Equilibrium: sum of resolved forces = 0

\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} A particle of mass 0.3 kg is held at rest by two light inextensible strings. One string is attached at an angle of \(60°\) to a horizontal ceiling. The other string is attached at an angle \(α°\) to a vertical wall (see diagram). The tension in the string attached to the ceiling is 4 N. Find the tension in the string which is attached to the wall and find the value of \(α\). [6]

\includegraphics{figure_2} The diagram shows a smooth ring \(R\), of mass \(m\) kg, threaded on a light inextensible string. A horizontal force of magnitude 2 N acts on \(R\). The ends of the string are attached to fixed points \(A\) and \(B\) on a vertical wall. The part \(AR\) of the string makes an angle of 30° with the vertical, the part \(BR\) makes an angle of 40° with the vertical and the string is taut. The ring is in equilibrium. Find the tension in the string and find the value of \(m\). [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_2} Coplanar forces of magnitudes 16 N, 12 N, 24 N and 8 N act at a point in the directions shown in the diagram. Find the magnitude and direction of the single additional force acting at the same point which will produce equilibrium. [6]

\includegraphics{figure_4} A block of mass 8 kg is at rest on a plane inclined at 20° to the horizontal. The block is connected to a vertical wall at the top of the plane by a string. The string is taut and parallel to a line of greatest slope of the plane (see diagram).

1. Given that the tension in the string is 13 N, find the frictional and normal components of the force exerted on the block by the plane. [4]

The string is cut; the block remains at rest, but is on the point of slipping down the plane.

1. Find the coefficient of friction between the block and the plane. [2]

\includegraphics{figure_3} A small ring of mass 0.8 kg is threaded on a rough rod which is fixed horizontally. The ring is in equilibrium, acted on by a force of magnitude 7 N pulling upwards at 45° to the horizontal (see diagram).

1. Show that the normal component of the contact force acting on the ring has magnitude 3.05 N, correct to 3 significant figures. [2]
2. The ring is in limiting equilibrium. Find the coefficient of friction between the ring and the rod. [3]

\includegraphics{figure_3} A small ring of mass \(0.8 \text{ kg}\) is threaded on a rough rod which is fixed horizontally. The ring is in equilibrium, acted on by a force of magnitude \(7 \text{ N}\) pulling upwards at \(45°\) to the horizontal (see diagram).

1. Show that the normal component of the contact force acting on the ring has magnitude \(3.05 \text{ N}\), correct to 3 significant figures. [2]
2. The ring is in limiting equilibrium. Find the coefficient of friction between the ring and the rod. [3]

\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_3} A particle \(P\) of mass \(8 \text{ kg}\) is on a smooth plane inclined at an angle of \(30°\) to the horizontal. A force of magnitude \(100 \text{ N}\), making an angle of \(\theta°\) with a line of greatest slope and lying in the vertical plane containing the line of greatest slope, acts on \(P\) (see diagram).

1. Given that \(P\) is in equilibrium, show that \(\theta = 66.4\), correct to \(1\) decimal place, and find the normal reaction between the plane and \(P\). [4]
2. Given instead that \(\theta = 30\), find the acceleration of \(P\). [2]

\includegraphics{figure_3} Coplanar forces of magnitudes 8 N, 12 N and 18 N act at a point in the directions shown in the diagram. Find the magnitude and direction of the single additional force acting at the same point which will produce equilibrium. [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_2} A particle \(P\) of mass \(1.6\) kg is suspended in equilibrium by two light inextensible strings attached to points \(A\) and \(B\). The strings make angles of \(20°\) and \(40°\) respectively with the horizontal (see diagram). Find the tensions in the two strings. [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 ring of mass 4 kg is threaded on a fixed rough vertical rod. A light string is attached to the ring, and is pulled with a force of magnitude \(T\) N acting at an angle of \(60°\) to the downward vertical (see diagram). The ring is in equilibrium.

1. The normal and frictional components of the contact force exerted on the ring by the rod are \(R\) N and \(F\) N respectively. Find \(R\) and \(F\) in terms of \(T\). [4]
2. The coefficient of friction between the rod and the ring is 0.7. Find the value of \(T\) for which the ring is about to slip. [3]

\includegraphics{figure_2} A block of mass 15 kg hangs in equilibrium below a horizontal ceiling attached to two strings as shown in the diagram. One of the strings is inclined at \(45°\) to the horizontal and the tension in this string is 120 N. The other string is inclined at \(θ°\) to the horizontal and the tension in this string is \(T\) N. Find the values of \(T\) and \(θ\). [6]

A smooth ring \(R\) of mass \(m\) kg is threaded on a light inextensible string \(ARB\). The ends of the string are attached to fixed points \(A\) and \(B\) with \(A\) vertically above \(B\). The string is taut and angle \(ARB = 90°\). The angle between the part \(AR\) of the string and the vertical is \(45°\). The ring is held in equilibrium in this position by a force of magnitude \(2.5\) N, acting on the ring in the direction \(BR\) (see diagram). Calculate the tension in the string and the mass of the ring. [4] \includegraphics{figure_1}

\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]

\includegraphics{figure_3} A particle \(P\) of mass 0.3 kg is held in equilibrium above a horizontal plane by a force of magnitude 5 N, acting vertically upwards. The particle is attached to two strings \(PA\) and \(PB\) of lengths 0.9 m and 1.2 m respectively. The points \(A\) and \(B\) lie on the plane and angle \(APB = 90°\) (see diagram). Find the tension in each of the strings. [5]

\includegraphics{figure_1} One end of a light inextensible string is attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(m\) kg which hangs vertically below \(A\). The particle is also attached to one end of a light elastic string of natural length \(0.25\) m. The other end of this string is attached to a point \(B\) which is \(0.6\) m from \(P\) and on the same horizontal level as \(P\). Equilibrium is maintained by a horizontal force of magnitude \(7\) N applied to \(P\) (see Fig. 1).

1. Calculate the modulus of elasticity of the elastic string. [2]
2. Find the value of \(m\). [4]

\includegraphics{figure_2} A light inextensible string of length \(a\) is threaded through a fixed smooth ring \(R\). One end of the string is attached to a particle \(A\) of mass \(3m\). The other end of the string is attached to a particle \(B\) of mass \(m\). The particle \(A\) hangs in equilibrium at a distance \(x\) vertically below the ring. The angle between \(AR\) and \(BR\) is \(\theta\) (see diagram). The particle \(B\) moves in a horizontal circle with constant angular speed \(2\sqrt{\frac{g}{a}}\). Show that \(\cos \theta = \frac{1}{3}\) and find \(x\) in terms of \(a\). [5]

\includegraphics{figure_4} A ring of weight \(W\), with radius \(a\) and centre \(O\), is at rest on a rough surface that is inclined to the horizontal at an angle \(\alpha\) where \(\tan\alpha = \frac{1}{3}\). The plane of the ring is perpendicular to the inclined surface and parallel to a line of greatest slope of the surface. The point \(P\) on the circumference of the ring is such that \(OP\) is parallel to the surface. A light inextensible string is attached to \(P\) and to the point \(Q\), which is on the surface, such that \(PQ\) is horizontal (see diagram). The points \(O\), \(P\) and \(Q\) are in the same vertical plane. The system is in limiting equilibrium and the coefficient of friction between the ring and the surface is \(\mu\).

1. Find, in terms of \(W\), the tension in the string \(PQ\). [4]
2. Find the value of \(\mu\). [3]

\includegraphics{figure_1} A block of mass 50 kg lies on a rough plane which is inclined to the horizontal at an angle \(\alpha\), where \(\tan\alpha = \frac{7}{24}\). The block is held at rest by a vertical rope, as shown in Figure 1, and is on the point of sliding down the plane. The block is modelled as a particle and the rope is modelled as a light inextensible string. Given that the friction force acting on the block has magnitude 65.8 N, find

1. the tension in the rope, [4]
2. the coefficient of friction between the block and the plane. [4]