3.03m Equilibrium: sum of resolved forces = 0

A thin elastic string, of modulus \(\lambda\) N and natural length 20 cm, passes round two small, smooth pegs \(A\) and \(B\) on the same horizontal level to form a closed loop. \(AB = 10\) cm. The ends of the string are attached to a weight \(P\) of mass 0.7 kg. When \(P\) rests in equilibrium, \(APB\) forms an equilateral triangle. \includegraphics{figure_2}

1. Find the value of \(\lambda\). [6 marks]
2. State one assumption that you have made about the weight \(P\), explaining how you have used this assumption in your solution. [1 mark]

A particle \(P\), of mass \(m\) kg, is attached to two light elastic strings, each of natural length \(l\) m and modulus of elasticity \(3mg\) N. The other ends of the strings are attached to the fixed points \(A\) and \(B\), where \(AB\) is horizontal and \(AB = 2l\) m. \includegraphics{figure_4} If \(P\) rests in equilibrium vertically below the mid-point of \(AB\), with each string making an angle \(\theta\) with the vertical, show that $$\cot \theta - \cos \theta = \frac{1}{6}.$$ [8 marks]

1. Show that the centre of mass of a uniform solid hemisphere of radius \(r\) is at a distance \(\frac{3r}{8}\) from the centre \(O\) of the plane face. [7 marks]

The figure shows the vertical cross-section of a rough solid hemisphere at rest on a rough inclined plane inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{10}\). \includegraphics{figure_7} \begin{enumerate}[label=(\alph*)] \setcounter{enumi}{1} \item Indicate on a copy of the figure the three forces acting on the hemisphere, clearly stating what they are, and paying special attention to their lines of action. [3 marks] \item Given that the plane face containing the diameter \(AB\) makes an angle \(\alpha\) with the vertical, show that \(\cos \alpha = \frac{4}{5}\). [6 marks] \end{enumerate]

A particle \(P\) of mass 0.2 kg is suspended by two identical light inelastic strings, with one end of each string attached to \(P\) and the other ends fixed to points \(O\) and \(X\) on the same horizontal level. Both strings are inclined at 30° to the horizontal.

1. Find the tension in the strings when \(P\) is at rest. [2 marks]

The string \(XP\) is suddenly cut, so that \(P\) begins to move in a vertical circle with centre \(O\).

1. Find the tension in the string \(OP\) when it makes an angle of 60° with the horizontal. [6 marks]

A light inelastic string of length \(l\) m passes through a small smooth ring which is fixed at a point \(O\) and is free to rotate about a vertical axis through \(O\). Particles \(P\) and \(Q\), of masses 0.06 kg and 0.04 kg respectively, are attached to the ends of the string.

1. \(Q\) describes a horizontal circle with centre \(P\), while \(P\) hangs at rest at a depth \(d\) m below \(O\). Show that \(d = \frac{2l}{5}\). [6 marks]
2. \(P\) and \(Q\) now both move in horizontal circles with the same angular velocity \(\omega\) rad s\(^{-1}\) about a vertical axis through \(O\). Show that \(OQ = \frac{3l}{5}\) m. [7 marks]

\includegraphics{figure_5}

\includegraphics{figure_2} Two uniform rods \(AB\) and \(BC\), each of length \(2L\), are freely jointed at \(B\), and \(AB\) is freely jointed to a fixed point at \(A\). The rods are held in equilibrium in a vertical plane by a light horizontal string attached at \(C\). The rods \(AB\) and \(BC\) make angles \(\alpha\) and \(\beta\) to the horizontal respectively. The weight of rod \(BC\) is \(75\) N, and the tension in the string is \(50\) N (see diagram).

1. Show that \(\tan \beta = \frac{1}{3}\). [3]
2. Given that \(\tan \alpha = \frac{12}{5}\), find the weight of \(AB\). [5]

\includegraphics{figure_6} Two uniform rods \(AB\) and \(AC\) are freely jointed at \(A\). Rod \(AB\) is of length \(2l\) and weight \(W\); rod \(AC\) is of length \(4l\) and weight \(2W\). The rods rest in equilibrium in a vertical plane on two rough horizontal steps, so that \(AB\) makes an angle of \(\theta\) with the horizontal, where \(\sin \theta = \frac{3}{5}\), and \(AC\) makes an angle of \(\varphi\) with the horizontal, where \(\sin \varphi = \frac{1}{5}\) (see diagram). The force of the step acting on \(AB\) at \(B\) has vertical component \(R\) and horizontal component \(F\).

1. By taking moments about \(A\) for the rod \(AB\), find an equation relating \(W\), \(R\) and \(F\). [3]
2. Show that \(R = \frac{75}{68}W\), and find the vertical component of the force acting on \(AC\) at \(C\). [6]
3. The coefficient of friction at \(B\) is equal to that at \(C\). Given that one of the rods is on the point of slipping, explain which rod this must be, and find the coefficient of friction. [4]

\includegraphics{figure_10} A block \(D\) of weight 50 N lies at rest in equilibrium on a fixed rough horizontal surface. A force of magnitude 15 N is applied to \(D\) at an angle \(\theta\) to the horizontal (see diagram).

1. Complete the diagram in the Printed Answer Booklet showing all the forces acting on \(D\). [1]

It is given that \(D\) remains at rest and the coefficient of friction between \(D\) and the surface is 0.2.

1. Show that $$15\cos\theta - 3\sin\theta \leqslant 10.$$ [5]

\includegraphics{figure_8} A child attempts to drag a sledge along horizontal ground by means of a rope attached to the sledge. The rope makes an angle of \(15°\) with the horizontal (see diagram). Given that the sledge remains at rest and that the frictional force acting on the sledge is 60 N, find the tension in the rope. [2]

\includegraphics{figure_10} A rectangular block \(B\) is at rest on a horizontal surface. A particle \(P\) of mass 2.5 kg is placed on the upper surface of \(B\). The particle \(P\) is attached to one end of a light inextensible string which passes over a smooth fixed pulley. A particle \(Q\) of mass 3 kg is attached to the other end of the string and hangs freely below the pulley. The part of the string between \(P\) and the pulley is horizontal (see diagram). The particles are released from rest with the string taut. It is given that \(B\) remains in equilibrium while \(P\) moves on the upper surface of \(B\). The tension in the string while \(P\) moves on \(B\) is 16.8 N.

1. Find the acceleration of \(Q\) while \(P\) and \(B\) are in contact. [2]
2. Determine the coefficient of friction between \(P\) and \(B\). [3]
3. Given that the coefficient of friction between \(B\) and the horizontal surface is \(\frac{5}{49}\), determine the least possible value for the mass of \(B\). [3]

\includegraphics{figure_9} A block \(B\) of weight \(10 \text{N}\) lies at rest in equilibrium on a rough plane inclined at \(\theta\) to the horizontal. A horizontal force of magnitude \(2 \text{N}\), acting above a line of greatest slope, is applied to \(B\) (see diagram).

1. Complete the diagram in the Printed Answer Booklet to show all the forces acting on \(B\). [1]

It is given that \(B\) remains at rest and the coefficient of friction between \(B\) and the plane is 0.8.

1. Determine the greatest possible value of \(\tan \theta\). [5]

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 three forces \(\mathbf{F_1}\), \(\mathbf{F_2}\) and \(\mathbf{F_3}\) are acting on a particle. \(\mathbf{F_1} = (25\mathbf{i} + 12\mathbf{j})\) N \(\mathbf{F_2} = (-7\mathbf{i} + 5\mathbf{j})\) N \(\mathbf{F_3} = (15\mathbf{i} - 28\mathbf{j})\) N The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertical respectively. The resultant of these three forces is \(\mathbf{F}\) newtons.

The fourth force, \(\mathbf{F_4}\), is applied to the particle so that the four forces are in equilibrium. Find \(\mathbf{F_4}\), giving your answer in terms of \(\mathbf{i}\) and \(\mathbf{j}\). [1 mark]

In this question use \(g = 9.8\) m s\(^{-2}\). The diagram shows a box, of mass 8.0 kg, being pulled by a string so that the box moves at a constant speed along a rough horizontal wooden board. The string is at an angle of 40° to the horizontal. The tension in the string is 50 newtons. \includegraphics{figure_16a} The coefficient of friction between the box and the board is \(\mu\) Model the box as a particle.

1. Show that \(\mu = 0.83\) [4 marks]
2. One end of the board is lifted up so that the board is now inclined at an angle of 5° to the horizontal. The box is pulled up the inclined board. The string remains at an angle of 40° to the board. The tension in the string is increased so that the box accelerates up the board at 3 m s\(^{-2}\) \includegraphics{figure_16b}
   1. Draw a diagram to show the forces acting on the box as it moves. [1 mark]
   2. Find the tension in the string as the box accelerates up the slope at 3 m s\(^{-2}\). [7 marks]

\includegraphics{figure_12} The ends of a light inextensible string are fixed to two points A and B in the same vertical line, with A above B. The string passes through a small smooth ring of mass \(m\). The ring is fastened to the string at a point P. When the string is taut the angle APB is a right angle, the angle BAP is \(\theta\) and the perpendicular distance of P from AB is \(r\). The ring moves in a horizontal circle with constant angular velocity \(\omega\) and the string taut as shown in Fig. 12.

1. By resolving horizontally and vertically, show that the tension in the part of the string BP is \(m(r\omega^2\cos\theta - g\sin\theta)\). [6]
2. Find a similar expression, in terms of \(r\), \(\omega\), \(m\), \(g\) and \(\theta\), for the tension in the part of the string AP. [2]

It is given that AB = 5a and AP = 4a.

1. Show that \(16a\omega^2 > 5g\). [3]

The ring is now free to move on the string but remains in the same position on the string as before. The string remains taut and the ring continues to move in a horizontal circle.

1. Find the period of the motion of the ring, giving your answer in terms of \(a\), \(g\) and \(\pi\). [5]

\includegraphics{figure_1} Three forces of magnitudes 4 N, 7 N and \(P\) N act at a point in the directions shown in the diagram. The forces are in equilibrium.

1. Draw a closed figure to represent the three forces. [1]
2. Hence, or otherwise, find the following.
   1. The value of \(\theta\). [2]
   2. The value of \(P\). [2]

Three coplanar horizontal forces of magnitude \(21\) N, \(11\) N and \(8\) N act on a particle \(P\) in the directions shown in the diagram. \includegraphics{figure_7}

1. Given that \(\tan\alpha = \frac{3}{4}\), calculate the magnitude of the resultant force. [5]
2. Explain why the forces cannot be in equilibrium whatever the value of \(\alpha\). [1]

\includegraphics{figure_4} As shown in the diagram, \(AB\) is a long thin rod which is fixed vertically with \(A\) above \(B\). One end of a light inextensible string of length \(1\) m is attached to \(A\) and the other end is attached to a particle \(P\) of mass \(m_1\) kg. One end of another light inextensible string of length \(1\) m is also attached to \(P\). Its other end is attached to a small smooth ring \(R\), of mass \(m_2\) kg, which is free to move on \(AB\). Initially, \(P\) moves in a horizontal circle of radius \(0.6\) m with constant angular velocity \(\omega\) rad s\(^{-1}\). The magnitude of the tension in string \(AP\) is denoted by \(T_1\) N while that in string \(PR\) is denoted by \(T_2\) N.

1. By considering forces on \(R\), express \(T_2\) in terms of \(m_2\). [2]
2. Show that
   1. \(T_1 = \frac{4g}{5}(m_1 + m_2)\). [2]
   2. \(\omega^2 = \frac{4g(m_1 + 2m_2)}{4m_1}\). [3]
3. Deduce that, in the case where \(m_1\) is much bigger than \(m_2\), \(\omega \approx 3.5\). [2]

In a different case, where \(m_1 = 2.5\) and \(m_2 = 2.8\), \(P\) slows down. Eventually the system comes to rest with \(P\) and \(R\) hanging in equilibrium.

1. Find the total energy lost by \(P\) and \(R\) as the angular velocity of \(P\) changes from the initial value of \(\omega\) rad s\(^{-1}\) to zero. [5]

A cylindrical object with mass 8 kg rests on two cylindrical bars of equal radius. The lines connecting the centre of each of the bars to the centre of the object make an angle of \(40°\) to the vertical. \includegraphics{figure_2}

1. Draw a diagram showing all the forces acting on the object. Describe each of the forces using words. [2]
2. Calculate the magnitude of the force on each of the bars due to the cylindrical object. [7]

In this question use \(g = 10 \text{ m s}^{-2}\). A particle of mass 3 kg rests in limiting equilibrium on a rough plane inclined at \(30°\) to the horizontal.

1. Find the exact value of the coefficient of friction between the particle and the plane. [2]

A horizontal force of 36 N is now applied to the particle.

1. Find how far down the plane the particle travels after the force has been applied for 4 s. [6]

Three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) acting on a particle are given by $$\mathbf{F}_1 = (3\mathbf{i} - 2a\mathbf{j})\text{N}, \quad \mathbf{F}_2 = (2b\mathbf{i} + 3a\mathbf{j})\text{N} \quad \text{and} \quad \mathbf{F}_3 = (-2\mathbf{i} + b\mathbf{j})\text{N}.$$ The particle is in equilibrium under the action of these three forces. Find the value of \(a\) and the value of \(b\). [3]

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]

The points \(A\), \(B\) and \(C\) lie in a vertical plane and have position vectors \(4\mathbf{i}\), \(3\mathbf{j}\) and \(7\mathbf{i} + 4\mathbf{j}\), respectively. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertically upwards, respectively. The units of the components are metres.

1. Show that angle \(BAC\) is a right angle. [2]

\includegraphics{figure_10} Strings \(AB\) and \(AC\) are attached to \(B\) and \(C\), and joined at \(A\). A particle of weight 20 N is attached at \(A\) (see diagram). The particle is in equilibrium.

1. By resolving in the directions \(AB\) and \(AC\), determine the magnitude of the tension in each string. [3]
2. Express the tension in the string \(AB\) as a vector, in terms of \(\mathbf{i}\) and \(\mathbf{j}\). [3]

\includegraphics{figure_13} Particles \(A\) and \(B\) of masses \(2m\) and \(m\), respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley \(P\). The particle \(A\) rests in equilibrium on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\alpha \leqslant 45°\) and \(B\) is above the plane. The vertical plane defined by \(APB\) contains a line of greatest slope of the plane, and \(PA\) is inclined at angle \(2\alpha\) to the horizontal (see diagram).

1. Show that the normal reaction \(R\) between \(A\) and the plane is \(mg(2 \cos \alpha - \sin \alpha)\). [3]
2. Show that \(R \geqslant \frac{1}{2}mg\sqrt{2}\). [3]

The coefficient of friction between \(A\) and the plane is \(\mu\). The particle \(A\) is about to slip down the plane.

1. Show that \(0.5 < \tan \alpha \leqslant 1\). [3]
2. Express \(\mu\) as a function of \(\tan \alpha\) and deduce its maximum value as \(\alpha\) varies. [3]

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]