3.03n Equilibrium in 2D: particle under forces

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]

\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 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]

In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the directions east and north respectively. A particle \(R\) of mass \(2\) kg is moving on a smooth horizontal surface under the action of a single horizontal force \(\mathbf{F}\) N. At time \(t\) seconds, the velocity \(\mathbf{v} \text{ m s}^{-1}\) of \(R\), relative to a fixed origin \(O\), is given by \(\mathbf{v} = (pt^2 - 3t)\mathbf{i} + (8t + q)\mathbf{j}\), where \(p\) and \(q\) are constants and \(p < 0\).

1. Given that when \(t = 0.5\) the magnitude of \(\mathbf{F}\) is \(20\), find the value of \(p\). [6]

When \(t = 0\), \(R\) is at the point with position vector \((2\mathbf{i} - 3\mathbf{j})\) m.

1. Find, in terms of \(q\), an expression for the displacement vector of \(R\) at time \(t\). [4]

When \(t = 1\), \(R\) is at a point on the line \(L\), where \(L\) passes through \(O\) and the point with position vector \(2\mathbf{i} - 8\mathbf{j}\).

1. Find the value of \(q\). [3]

Two heavy boxes, \(M\) and \(N\), are connected securely by a length of rope. The mass of \(M\) is 50 kilograms. The mass of \(N\) is 80 kilograms. \(M\) is placed near the bottom of a rough slope. The slope is inclined at 60° above the horizontal. The rope is passed over a smooth pulley at the top end of the slope so that \(N\) hangs with the rope vertical. The boxes are initially held in this position, with the rope taut and running parallel to the line of greatest slope, as shown in the diagram below. \includegraphics{figure_21} When the boxes are released, \(M\) moves up the slope as \(N\) descends, with acceleration \(a\) m s\(^{-2}\) The tension in the rope is \(T\) newtons.

1. Explain why the equation of motion for \(N\) is $$80g - T = 80a$$ [1 mark]
2. Show that the normal reaction force between \(M\) and the slope is \(25g\) newtons. [1 mark]
3. The coefficient of friction, \(\mu\), between the slope and \(M\) is such that \(0 \leq \mu \leq 1\) Show that $$a \geq \frac{(11 - 5\sqrt{3})g}{26}$$ [6 marks]
4. State one modelling assumption you have made throughout this question. [1 mark]

The fixed points E and F are on the same horizontal level with EF = 1.6 m. A light string has natural length 0.7 m and modulus of elasticity 29.4 N. One end of the string is attached to E and the other end is attached to a particle of mass \(M\) kg. A second string, identical to the first, has one end attached to F and the other end attached to the particle. The system is in equilibrium in a vertical plane with each string stretched to a length of 1 m, as shown in Fig. 3. \includegraphics{figure_3}

1. Find the tension in each string. [2]
2. Find \(M\). [3]

Fig. 5 shows a light inextensible string of length 3.3 m passing through a small smooth ring R. The ends of the string are attached to fixed points A and B, where A is vertically above B. The ring R has mass 0.27 kg and is moving with constant speed in a horizontal circle of radius 1.2 m. The distances AR and BR are 2 m and 1.3 m respectively. \includegraphics{figure_5}

1. Show that the tension in the string is 6.37 N. [4]
2. Find the speed of R. [4]

A particle \(P\) of mass 0.5 kg is in equilibrium under the action of three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\). $$\mathbf{F}_1 = (9\mathbf{i} + 6\mathbf{j} - 12\mathbf{k})\text{N} \quad \text{and} \quad \mathbf{F}_2 = (6\mathbf{i} - 7\mathbf{j} + 3\mathbf{k})\text{N}.$$

1. Find the force \(\mathbf{F}_3\). [2]
2. Forces \(\mathbf{F}_2\) and \(\mathbf{F}_3\) are removed so that \(P\) moves in a straight line \(AB\) under the action of the single force \(\mathbf{F}_1\). The points \(A\) and \(B\) have position vectors \((2\mathbf{i} - 9\mathbf{j} + 7\mathbf{k})\) m and \((8\mathbf{i} - 5\mathbf{j} - \mathbf{k})\) m respectively. The particle \(P\) is initially at rest at \(A\).
   1. Verify that \(\mathbf{F}_1\) acts parallel to the vector \(\overrightarrow{AB}\).
   2. Find the work done by the force \(\mathbf{F}_1\) as the particle moves from \(A\) to \(B\).
   3. By using the work-energy principle, find the speed of \(P\) as it reaches \(B\). [7]

\includegraphics{figure_6} A particle of mass 5 kg is attached to a string \(AB\) and a rod \(BC\) at the point \(B\). The string \(AB\) is light and elastic with modulus \(\lambda\) N and natural length 2 m. The rod \(BC\) is light and of length 2 m. The end \(A\) of the string is attached to a fixed point and the end \(C\) of the rod is attached to another fixed point such that \(A\) is vertically above \(C\) with \(AC = 2\) m. When the particle rests in equilibrium, \(AB\) makes an angle of 50° with the downward vertical.

1. Determine, in terms of \(\lambda\), the tension in the string \(AB\). [3]
2. Calculate, in terms of \(\lambda\), the energy stored in the string \(AB\). [2]
3. Find, in terms of \(\lambda\), the thrust in the rod \(BC\). [4]

A uniform ladder \(AB\) of mass 35 kg and length 7 m rests with its end \(A\) on rough horizontal ground and its end \(B\) against a rough vertical wall. The ladder is inclined at an angle of \(45°\) to the horizontal. A man of mass 70 kg is standing on the ladder at a point \(C\), which is \(x\) metres from \(A\). The coefficient of friction between the ladder and the wall is \(\frac{1}{4}\) and the coefficient of friction between the ladder and the ground is \(\frac{1}{2}\). The system is in limiting equilibrium. Find \(x\). [8]

A parcel \(P\) of weight 50 N is being held in equilibrium by two light, inextensible strings \(AP\) and \(BP\). The string \(AP\) is attached to a wall at \(A\), and string \(BP\) passes over a smooth pulley which is at the same height as \(A\), as shown in the diagram. When the tension in \(BP\) is 40 N, the strings are at right angles to each other. \includegraphics{figure_10}

1. Find the tension in string \(AP\). [4]
2. Explain why the parcel can never be in equilibrium with both strings horizontal. [1]

A particle \(P\) of mass \(1.5\) kg is placed on a smooth horizontal table. The particle is initially at the origin of a \(2\)-dimensional vector system defined by perpendicular unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) in the plane of the table. The particle is subject to three forces of magnitudes \(10\) N, \(12\) N and \(F\) N, acting in the directions of the vectors \(3\mathbf{i} + 4\mathbf{j}\), \(-\mathbf{j}\) and \(-\cos \theta \mathbf{i} + \sin \theta \mathbf{j}\) respectively, and no others.

1. Given that the system is in equilibrium, determine \(F\) and \(\theta\). [6]

The force of magnitude \(12\) N is replaced by one of magnitude \(4\) N, but in the opposite direction. The particle is initially at rest.

1. Find the position vector of the particle \(3\) seconds later. [5]