3.03m Equilibrium: sum of resolved forces = 0

Two light elastic strings each have natural length \(0.75\) m and modulus of elasticity \(49\) N. A particle \(P\) of mass \(2\) kg is attached to one end of each string. The other ends of the strings are attached to fixed points \(A\) and \(B\), where \(AB\) is horizontal and \(AB = 1.5\) m. \includegraphics{figure_2} The particle is held at the mid-point of \(AB\). The particle is released from rest, as shown in Figure 2.

1. Find the speed of \(P\) when it has fallen a distance of \(1\) m. [6]

Given instead that \(P\) hangs in equilibrium vertically below the mid-point of \(AB\), with \(\angle APB = 2\alpha\),

1. show that \(\tan \alpha + 5 \sin \alpha = 5\). [6]

A small ball \(B\), of mass 0.8 kg, is suspended from a horizontal ceiling by two light inextensible strings. \(B\) is in equilibrium under gravity with both strings inclined at 30° to the horizontal, as shown. \includegraphics{figure_2}

1. Find the tension, in N, in either string. [3 marks]
2. Calculate the magnitude of the least horizontal force that must be applied to \(B\) in this position to cause one string to become slack. [4 marks]

\includegraphics{figure_1} A light inextensible string has its ends attached to two fixed points \(A\) and \(B\). The point \(A\) is vertically above \(B\). A smooth ring \(R\) of mass \(m\) kg is threaded on the string and is pulled by a force of magnitude \(1.6\) N acting upwards at \(45°\) to the horizontal. The section \(AR\) of the string makes an angle of \(30°\) with the downward vertical and the section \(BR\) is horizontal (see diagram). The ring is in equilibrium with the string taut.

1. Give a reason why the tension in the part \(AR\) of the string is the same as that in the part \(BR\). [1]
2. Show that the tension in the string is \(0.754\) N, correct to 3 significant figures. [3]
3. Find the value of \(m\). [3]

\includegraphics{figure_5} Two small rings \(A\) and \(B\) are attached to opposite ends of a light inextensible string. The rings are threaded on a rough wire which is fixed vertically. \(A\) is above \(B\). A horizontal force is applied to a point \(P\) of the string. Both parts \(AP\) and \(BP\) of the string are taut. The system is in equilibrium with angle \(BAP = \alpha\) and angle \(ABP = \beta\) (see diagram). The weight of \(A\) is \(2\) N and the tensions in the parts \(AP\) and \(BP\) of the string are \(7\) N and \(T\) N respectively. It is given that \(\cos \alpha = 0.28\) and \(\sin \alpha = 0.96\), and that \(A\) is in limiting equilibrium.

1. Find the coefficient of friction between the wire and the ring \(A\). [7]
2. By considering the forces acting at \(P\), show that \(T \cos \beta = 1.96\). [2]
3. Given that there is no frictional force acting on \(B\), find the mass of \(B\). [3]

\includegraphics{figure_6} A smooth ring \(R\) of weight \(W\) N is threaded on a light inextensible string. The ends of the string are attached to fixed points \(A\) and \(B\), where \(A\) is vertically above \(B\). A horizontal force of magnitude \(P\) N acts on \(R\). The system is in equilibrium with the string taut; \(AR\) makes an angle \(\alpha\) with the downward vertical and \(BR\) makes an angle \(\beta\) with the upward vertical (see Fig. 1).

1. By considering the vertical components of the forces acting on \(R\), show that \(\alpha < \beta\). [3]

1. \includegraphics{figure_6ii} It is given that when \(P = 14\), \(AR = 0.4\) m, \(BR = 0.3\) m and the distance of \(R\) from the vertical line \(AB\) is \(0.24\) m (see Fig. 2). Find
   1. the tension in the string, [3]
   2. the value of \(W\). [3]
2. For the case when \(P = 0\),
   1. describe the position of \(R\), [1]
   2. state the tension in the string. [1]

A block of mass \(m\) kg is at rest on a horizontal plane. The coefficient of friction between the block and the plane is \(0.2\).

1. When a horizontal force of magnitude \(5\) N acts on the block, the block is on the point of slipping. Find the value of \(m\). [3]

1. \includegraphics{figure_5ii} When a force of magnitude \(P\) N acts downwards on the block at an angle \(\alpha\) to the horizontal, as shown in the diagram, the frictional force on the block has magnitude \(6\) N and the block is again on the point of slipping. Find
   1. the value of \(\alpha\) in degrees,
   2. the value of \(P\).
   [8]

A block \(B\) of weight \(10\) N is projected down a line of greatest slope of a plane inclined at an angle of \(20°\) to the horizontal. \(B\) travels down the plane at constant speed.

1. 1. Find the components perpendicular and parallel to the plane of the contact force between \(B\) and the plane. [2]
   2. Hence show that the coefficient of friction is \(0.364\), correct to \(3\) significant figures. [2]
2. \includegraphics{figure_6} \(B\) is in limiting equilibrium when acted on by a force of \(T\) N directed towards the plane at an angle of \(45°\) to a line of greatest slope (see diagram). Given that the frictional force on \(B\) acts down the plane, find \(T\). [7]

Two forces each of magnitude \(4\text{ N}\) have a resultant of magnitude \(6\text{ N}\).

1. Calculate the angle between the two \(4\text{ N}\) forces. [4]

The two given forces of magnitude \(4\text{ N}\) act on a particle of mass \(m\text{ kg}\) which remains at rest on a smooth horizontal surface. The surface exerts a force of magnitude \(3\text{ N}\) on the particle.

1. Find \(m\), and give the acute angle between the surface and one of the \(4\text{ N}\) forces. [3]

A sledge of mass 25 kg is on a plane inclined at \(30°\) to the horizontal. The coefficient of friction between the sledge and the plane is 0.2.

1. \includegraphics{figure_6} The sledge is pulled up the plane, with constant acceleration, by means of a light cable which is parallel to a line of greatest slope (see Fig. 1). The sledge starts from rest and acquires a speed of \(0.8 \text{ m s}^{-1}\) after being pulled for 10 s. Ignoring air resistance, find the tension in the cable. [6]
2. \includegraphics{figure_7} On a subsequent occasion the cable is not in use and two people of total mass 150 kg are seated in the sledge. The sledge is held at rest by a horizontal force of magnitude \(P\) newtons, as shown in Fig. 2. Find the least value of \(P\) which will prevent the sledge from sliding down the plane. [7]

\includegraphics{figure_3} Fig. 3 shows a block of mass 15 kg on a rough, horizontal plane. A light string is fixed to the block at A, passes over a smooth, fixed pulley B and is attached at C to a sphere. The section of the string between the block and the pulley is inclined at 40° to the horizontal and the section between the pulley and the sphere is vertical. The system is in equilibrium and the tension in the string is 58.8 N.

1. The sphere has a mass of \(m\) kg. Calculate the value of \(m\). [2]
2. Calculate the frictional force acting on the block. [3]
3. Calculate the normal reaction of the plane on the block. [3]

Force \(\mathbf{F}\) is \(\begin{pmatrix} 4 \\ 1 \\ 2 \end{pmatrix}\) N and force \(\mathbf{G}\) is \(\begin{pmatrix} -6 \\ 2 \\ 4 \end{pmatrix}\) N.

1. Find the resultant of \(\mathbf{F}\) and \(\mathbf{G}\) and calculate its magnitude. [4]
2. Forces \(\mathbf{F}\), \(2\mathbf{G}\) and \(\mathbf{H}\) act on a particle which is in equilibrium. Find \(\mathbf{H}\). [3]

\includegraphics{figure_3} Figure 3 shows a block of mass 25 kg held in equilibrium on a plane inclined at an angle of 35° to the horizontal by means of a string which is at an angle of 15° to the line of greatest slope of the plane. In an initial model of the situation, the plane is assumed to be smooth. Giving your answers correct to 3 significant figures,

1. show that the tension in the string is 145 N. [3 marks]
2. find the magnitude of the reaction between the plane and the block. [4 marks]

In a more refined model, the plane is assumed to be rough. Given that the tension in the string can be increased to 200 N before the block begins to move up the slope,

1. find, correct to 3 significant figures, the magnitude of the frictional force and state the direction in which it acts. [4 marks]
2. Without performing any further calculations, state whether the reaction calculated in part (b) will increase, decrease or remain the same in the refined model. Give a reason for your answer. [3 marks]

\includegraphics{figure_2} Figure 2 shows a cable car \(C\) of mass 1 tonne which has broken down. The cable car is suspended in equilibrium by two perpendicular cables \(AC\) and \(BC\) which are attached to fixed points \(A\) and \(B\), at the same horizontal level on either side of a valley. The cable \(AC\) is inclined at an angle \(\alpha\) to the horizontal where \(\tan \alpha = \frac{3}{4}\).

1. Show that the tension in the cable \(AC\) is 5880 N and find the tension in the cable \(BC\). [7 marks] A gust of wind then blows along the valley.
2. Explain the effect that this will have on the tension in the two cables. [2 marks]

A non-uniform ladder \(AB\), of length \(3a\), has its centre of mass at \(G\), where \(AG = 2a\). The ladder rests in limiting equilibrium with the end \(B\) against a smooth vertical wall and the end \(A\) resting on rough horizontal ground. The angle between \(AB\) and the horizontal in this position is \(\alpha\), where \(\tan \alpha = \frac{14}{9}\). \includegraphics{figure_3} Calculate the coefficient of friction between the ladder and the ground. [7 marks]

A stick of mass \(0.75\) kg is at rest with one end \(X\) on a rough horizontal floor and the other end \(Y\) leaning against a smooth vertical wall. The coefficient of friction between the stick and the floor is \(0.6\). Modelling the stick as a uniform rod, find the smallest angle that the stick can make with the floor before it starts to slip. \includegraphics{figure_2} [6 marks]

\includegraphics{figure_7} A uniform ladder \(AB\), of mass \(m\) kg and length \(2a\) m, rests with its upper end \(A\) in contact with a smooth vertical wall and its lower end \(B\) in contact with a fixed peg on horizontal ground. The ladder makes an angle \(\alpha\) with the ground, where \(\tan \alpha = \frac{3}{4}\).

1. Show that the magnitude of the resultant force acting on the ladder at \(B\) is \(\frac{\sqrt{13}}{3}mg\). [7 marks]
2. Find, to the nearest degree, the direction of this resultant force at \(B\). [3 marks]

The peg will break when the horizontal force acting on it exceeds \(2mg\) N. A painter of mass \(6m\) kg starts to climb the ladder from \(B\).

1. Find, in terms of \(a\), the greatest distance up the ladder that the painter can safely climb. [7 marks]

A uniform plank of wood \(XY\), of mass 1.4 kg, rests with its upper end \(X\) against a rough vertical wall and its lower end \(Y\) on rough horizontal ground. The coefficient of friction between the plank and both the wall and the ground is \(\mu\). The plank is in limiting equilibrium at both ends and the vertical component of the force exerted on the plank by the ground has magnitude 12 N. Find the value of \(\mu\), to 2 decimal places. [8 marks]

\includegraphics{figure_5} A uniform rod \(AB\) of length 60 cm and weight 15 N is freely suspended from its end \(A\). The end \(B\) of the rod is attached to a light inextensible string of length 80 cm whose other end is fixed to a point \(C\) which is at the same horizontal level as \(A\). The rod is in equilibrium with the string at right angles to the rod (see diagram).

1. Show that the tension in the string is 4.5 N. [4]
2. Find the magnitude and direction of the force acting on the rod at \(A\). [6]

\includegraphics{figure_5} A uniform rod \(AB\), of mass 3 kg and length 4 m, is in limiting equilibrium with \(A\) on rough horizontal ground. The rod is at an angle of 60° to the horizontal and is supported by a small smooth peg \(P\), such that the distance \(AP\) is 2.5 m (see diagram). Find

1. the force acting on the rod at \(P\), [3]
2. the coefficient of friction between the ground and the rod. [5]

A non-uniform beam \(AB\) of length 4 m and mass 5 kg has its centre of mass at the point \(G\) of the beam where \(AG = 2.5\) m. The beam is freely suspended from its end \(A\) and is held in a horizontal position by means of a wire attached to the end \(B\). The wire makes an angle of \(20°\) with the vertical and the tension is \(T\) N (see diagram).

1. Calculate \(T\). [3]
2. Calculate the magnitude and the direction of the force acting on the beam at \(A\). [7]

One end of a light inextensible string of length \(l\) is attached to the vertex of a smooth cone of semi-vertical angle \(45°\). The cone is fixed to the ground with its axis vertical. The other end of the string is attached to a particle of mass \(m\) which rotates in a horizontal circle in contact with the outer surface of the cone. The angular speed of the particle is \(\omega\) (see diagram). The tension in the string is \(T\) and the contact force between the cone and the particle is \(R\).

1. By resolving horizontally and vertically, find two equations involving \(T\) and \(R\) and hence show that \(T = \frac{1}{2}ml(\sqrt{2}g + l\omega^2)\). [6]
2. When the string has length 0.8 m, calculate the greatest value of \(\omega\) for which the particle remains in contact with the cone. [4]

\includegraphics{figure_1} A uniform lamina \(ABDC\) is bounded by two semicircular arcs \(AB\) and \(CD\), each with centre \(O\) and of radii \(3a\) and \(a\) respectively, and two straight edges, \(AC\) and \(DB\), which lie on the line \(AOB\) (see Fig. 1).

1. Show that the distance of the centre of mass of the lamina from \(O\) is \(\frac{13a}{3\pi}\). [5]

\includegraphics{figure_2} The lamina has mass 3 kg and is freely pivoted to a fixed point at \(A\). The lamina is held in equilibrium with \(AB\) vertical by means of a light string attached to \(B\). The string lies in the same plane as the lamina and is at an angle of \(40°\) below the horizontal (see Fig. 2).

1. Calculate the tension in the string. [3]
2. Find the direction of the force acting on the lamina at \(A\). [4]

A smooth solid cone of semi-vertical angle \(60°\) is fixed to the ground with its axis vertical. A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point vertically above the vertex of the cone. \(P\) rotates in a horizontal circle on the surface of the cone with constant angular velocity \(\omega\). The string is inclined to the downward vertical at an angle of \(30°\) (see diagram).

1. Show that the magnitude of the contact force between the cone and the particle is \(\frac{1}{4}m(2\sqrt{3}g - 3a\omega^2)\). [6]
2. Given that \(a = 0.5\) m and \(m = 3.5\) kg, find, in either order, the greatest speed for which the particle remains in contact with the cone and the corresponding tension in the string. [3]

A uniform ladder \(AB\), of weight \(W\) and length \(2a\), rests with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth vertical wall. The ladder is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{12}{13}\). A man of weight \(6W\) is standing on the ladder at a distance \(x\) from \(A\) and the system is in equilibrium.

1. Show that the magnitude of the frictional force exerted by the ground on the ladder is \(\frac{5W}{24}\left(1 + \frac{6x}{a}\right)\). [5]

The coefficient of friction between the ladder and the ground is \(\frac{1}{3}\).

1. Find, in terms of \(a\), the greatest value of \(x\) for which the system is in equilibrium. [3]

The bottom of the ladder \(A\) is moved closer to the wall so that the ladder is now inclined at an angle \(\alpha\) to the horizontal. The man of weight \(6W\) can now stand at the top of the ladder \(B\) without the ladder slipping.

1. Find the least possible value of \(\tan \alpha\). [3]

\includegraphics{figure_2} Fig. 2 shows a framework in a vertical plane made from the equal, light, rigid rods AB, BC, AD, BD, BE, CE and DE. [The triangles ABD, BDE and BCE are all equilateral.] The rods AB, BC and DE are horizontal. The rods are freely pin-jointed to each other at A, B, C, D and E. The pin-joint at A is also fixed to an inclined plane. The plane is smooth and parallel to the rod AD. The pin-joint at D rests on this plane. The following external forces act on the framework: a vertical load of \(LN\) at C; the normal reaction force \(RN\) of the plane on the framework at D; the horizontal and vertical forces \(XN\) and \(YN\), respectively, acting at A.

1. Write down equations for the horizontal and vertical equilibrium of the framework. [3]
2. By considering moments, find the relationship between \(R\) and \(L\). Hence show that \(X = \sqrt{3}L\) and \(Y = 0\). [4]
3. Draw a diagram showing all the forces acting on the pin-joints, including the forces internal to the rods. [2]
4. Show that the internal force in the rod AD is zero. [2]
5. Find the forces internal to AB, CE and BC in terms of \(L\) and state whether each is a tension or a thrust (compression). [You may leave your answers in surd form.] [7]
6. Without calculating their values in terms of \(L\), show that the forces internal to the rods BD and BE have equal magnitude but one is a tension and the other a thrust. [2]