3.03m Equilibrium: sum of resolved forces = 0

\includegraphics{figure_1} A boy is pulling a sledge of mass 8 kg in a straight line at a constant speed across rough horizontal ground by means of a rope. The rope is inclined at 30° to the ground, as shown in Figure 1. The coefficient of friction between the sledge and the ground is \(\frac{1}{5}\). By modelling the sledge as a particle and the rope as a light inextensible string, find the tension in the rope. [8]

\includegraphics{figure_3} A particle \(P\) of mass 4 kg is attached to one end of a light inextensible string. A particle \(Q\) of mass \(m\) kg is attached to the other end of the string. The string passes over a small smooth pulley which is fixed at a point on the intersection of two fixed inclined planes. The string lies in a vertical plane that contains a line of greatest slope of each of the two inclined planes. The first plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\) and the second plane is inclined to the horizontal at an angle \(\beta\), where \(\tan \beta = \frac{4}{3}\). Particle \(P\) is on the first plane and particle \(Q\) is on the second plane with the string taut, as shown in Figure 3. The first plane is rough and the coefficient of friction between \(P\) and the plane is \(\frac{1}{4}\). The second plane is smooth. The system is in limiting equilibrium. Given that \(P\) is on the point of slipping down the first plane,

1. find the value of \(m\), [10]
2. find the magnitude of the force exerted on the pulley by the string, [4]
3. find the direction of the force exerted on the pulley by the string. [1]

\includegraphics{figure_2} A particle of mass 0.4 kg is held at rest on a fixed rough plane by a horizontal force of magnitude \(P\) newtons. The force acts in the vertical plane containing the line of greatest slope of the inclined plane which passes through the particle. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\), as shown in Figure 2. The coefficient of friction between the particle and the plane is \(\frac{1}{3}\). Given that the particle is on the point of sliding up the plane, find

1. the magnitude of the normal reaction between the particle and the plane, [5]
2. the value of \(P\). [5]

\includegraphics{figure_2} A ring of mass 0.3 kg is threaded on a fixed, rough horizontal curtain pole. A light inextensible string is attached to the ring. The string and the pole lie in the same vertical plane. The ring is pulled downwards by the string which makes an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac{3}{4}\) as shown in Fig. 2. The tension in the string is 2.5 N. Given that, in this position, the ring is in limiting equilibrium,

1. find the coefficient of friction between the ring and the pole. [8]

\includegraphics{figure_3} The direction of the string is now altered so that the ring is pulled upwards. The string lies in the same vertical plane as before and again makes an angle \(\alpha\) with the horizontal, as shown in Fig. 3. The tension in the string is again 2.5 N.

1. Find the normal reaction exerted by the pole on the ring. [2]
2. State whether the ring is in equilibrium in the position shown in Fig. 3, giving a brief justification for your answer. You need make no further detailed calculation of the forces acting. [2]

\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]

\includegraphics{figure_1} A box of mass 1.5 kg is placed on a plane which is inclined at an angle of 30° to the horizontal. The coefficient of friction between the box and plane is \(\frac{1}{4}\). The box is kept in equilibrium by a light string which lies in a vertical plane containing a line of greatest slope of the plane. The string makes an angle of 20° with the plane, as shown in Fig. 2. The box is in limiting equilibrium and is about to move up the plane. The tension in the string is \(T\) newtons. The box is modelled as a particle. Find the value of \(T\). [10]

\includegraphics{figure_2} Two small rings, \(A\) and \(B\), each of mass \(2m\), are threaded on a rough horizontal pole. The coefficient of friction between each ring and the pole is \(\mu\). The rings are attached to the ends of a light inextensible string. A smooth ring \(C\), of mass \(3m\), is threaded on the string and hangs in equilibrium below the pole. The rings \(A\) and \(B\) are in limiting equilibrium on the pole, with \(\angle BAC = \angle ABC = \theta\), where \(\tan \theta = \frac{3}{4}\), as shown in Fig. 2.

1. Show that the tension in the string is \(\frac{5}{2}mg\). [3]
2. Find the value of \(\mu\). [7]

\includegraphics{figure_1} A particle of weight 24 N is held in equilibrium by two light inextensible strings. One string is horizontal. The other string is inclined at an angle of 30° to the horizontal, as shown in Figure 1. The tension in the horizontal string is \(Q\) newtons and the tension in the other string is \(P\) newtons. Find

1. the value of \(P\), [3]
2. the value of \(Q\). [3]

\includegraphics{figure_1} A particle of mass \(m\) kg is attached at \(C\) to two light inextensible strings \(AC\) and \(BC\). The other ends of the strings are attached to fixed points \(A\) and \(B\) on a horizontal ceiling. The particle hangs in equilibrium with \(AC\) and \(BC\) inclined to the horizontal at 30° and 60° respectively, as shown in Figure 1. Given that the tension in \(AC\) is 20 N, find

1. the tension in \(BC\), [4]
2. the value of \(m\). [4]

A particle \(P\) of mass 2 kg is attached to one end of a light string, the other end of which is attached to a fixed point \(O\). The particle is held in equilibrium, with \(OP\) at \(30°\) to the downward vertical, by a force of magnitude \(F\) newtons. The force acts in the same vertical plane as the string and acts at an angle of \(30°\) to the horizontal, as shown in Figure 3. \includegraphics{figure_3} Find

1. the value of \(F\),
2. the tension in the string. [8]

\includegraphics{figure_1} A particle of weight \(W\) newtons is attached at \(C\) to the ends of two light inextensible strings \(AC\) and \(BC\). The other ends of the strings are attached to two fixed points \(A\) and \(B\) on a horizontal ceiling. The particle hangs in equilibrium with \(AC\) and \(BC\) inclined to the horizontal at \(30°\) and \(60°\) respectively, as shown in Fig. 1. Given the tension in \(AC\) is 50 N, calculate

1. the tension in \(BC\), to 3 significant figures, [3]
2. the value of \(W\). [3]

\includegraphics{figure_1} A smooth bead \(B\) is threaded on a light inextensible string. The ends of the string are attached to two fixed points \(A\) and \(C\) on the same horizontal level. The bead is held in equilibrium by a horizontal force of magnitude 6 N acting parallel to \(AC\). The bead \(B\) is vertically below \(C\) and \(\angle BAC = \alpha\), as shown in Figure 1. Given that \(\tan \alpha = \frac{3}{4}\), find

1. the tension in the string, [3]
2. the weight of the bead. [4]

\includegraphics{figure_1} A particle of weight \(W\) newtons is held in equilibrium on a rough inclined plane by a horizontal force of magnitude 4 N. The force acts in a vertical plane containing a line of greatest slope of the inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\), as shown in Figure 1. The coefficient of friction between the particle and the plane is \(\frac{1}{2}\). Given that the particle is on the point of sliding down the plane,

1. show that the magnitude of the normal reaction between the particle and the plane is 20 N,
2. find the value of \(W\). [9]

\includegraphics{figure_1} A particle of weight 8 N is attached at \(C\) to the ends of two light inextensible strings \(AC\) and \(BC\). The other ends, \(A\) and \(B\), are attached to a fixed horizontal ceiling. The particle hangs at rest in equilibrium, with the strings in a vertical plane. The string \(AC\) is inclined at 35° to the horizontal and the string \(BC\) is inclined at 25° to the horizontal, as shown in Figure 1. Find

1. the tension in the string \(AC\),
2. the tension in the string \(BC\).

[8]

\includegraphics{figure_1} A box of mass 2 kg is held in equilibrium on a fixed rough inclined plane by a rope. The rope lies in a vertical plane containing a line of greatest slope of the inclined plane. The rope is inclined to the plane at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\), and the plane is at an angle of \(30°\) to the horizontal, as shown in Figure 1. The coefficient of friction between the box and the inclined plane is \(\frac{1}{2}\) and the box is on the point of slipping up the plane. By modelling the box as a particle and the rope as a light inextensible string, find the tension in the rope. [8]

\includegraphics{figure_4} A particle of mass \(m\) rests on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{3}{4}\). The particle is attached to one end of a light inextensible string which lies in a line of greatest slope of the plane and passes over a small light smooth pulley \(P\) fixed at the top of the plane. The other end of the string is attached to a particle \(B\) of mass \(3m\), and \(B\) hangs freely below \(P\), as shown in Fig. 4. The particles are released from rest with the string taut. The particle \(B\) moves down with acceleration of magnitude \(\frac{1}{3}g\). Find

1. the tension in the string, [4]
2. the coefficient of friction between \(A\) and the plane. [9]

\includegraphics{figure_2} Figure 2 shows a horizontal uniform pole \(AB\), of weight \(W\) and length \(2a\). The end \(A\) of the pole rests against a rough vertical wall. One end of a light inextensible string \(BD\) is attached to the pole at \(B\) and the other end is attached to the wall at \(D\). A particle of weight \(2W\) is attached to the pole at \(C\), where \(BC = x\). The pole is in equilibrium in a vertical plane perpendicular to the wall. The string is inclined at an angle \(θ\) to the horizontal, where \(\tan θ = \frac{5}{3}\). The pole is modelled as a uniform rod.

1. Show that the tension in \(BD\) is \(\frac{5(5a - 2x)}{6a}W\). [5]

The vertical component of the force exerted by the wall on the pole is \(\frac{1}{2}W\). Find

1. x in terms of \(a\), [3]

1. the horizontal component, in terms of \(W\), of the force exerted by the wall on the pole. [4]

\includegraphics{figure_3} A straight log \(AB\) has weight \(W\) and length \(2a\). A cable is attached to one end \(B\) of the log. The cable lifts the end \(B\) off the ground. The end \(A\) remains in contact with the ground, which is rough and horizontal. The log is in limiting equilibrium. The log makes an angle \(α\) to the horizontal, where \(\tan α = \frac{4}{3}\). The cable makes an angle \(β\) to the horizontal, as shown in Fig. 3. The coefficient of friction between the log and the ground is \(\frac{1}{3}\). The log is modelled as a uniform rod and the cable as light.

1. Show that the normal reaction on the log at \(A\) is \(\frac{3}{4}W\). [6]

1. Find the value of \(β\). [6]

The tension in the cable is \(kW\).

1. Find the value of \(k\). [2]

END

\includegraphics{figure_2} Figure 2 shows a horizontal uniform pole \(AB\), of weight \(W\) and length \(2a\). The end \(A\) of the pole rests against a rough vertical wall. One end of a light inextensible string \(BD\) is attached to the pole at \(B\) and the other end is attached to the wall at \(D\). A particle of weight \(2W\) is attached to the pole at \(C\), where \(BC = x\). The pole is in equilibrium in a vertical plane perpendicular to the wall. The string \(BD\) is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). The pole is modelled as a uniform rod.

1. Show that the tension in \(BD\) is \(\frac{5(5a - 2x)}{6a}W\). [5]

The vertical component of the force exerted by the wall on the pole is \(\frac{7}{4}W\). Find

1. \(x\) in terms of \(a\), [3]
2. the horizontal component, in terms of \(W\), of the force exerted by the wall on the pole. [4]

\includegraphics{figure_2} A ladder \(AB\), of weight \(W\) and length \(4a\), has one end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is \(\mu\). The other end \(B\) rests against a smooth vertical wall. The ladder makes an angle \(\theta\) with the horizontal, where \(\tan \theta = 2\). A load of weight \(4W\) is placed at the point \(C\) on the ladder, where \(AC = 3a\), as shown in Figure 2. The ladder is modelled as a uniform rod which is in a vertical plane perpendicular to the wall. The load is modelled as a particle. Given that the system is in limiting equilibrium,

1. show that \(\mu = 0.35\). [6]

A second load of weight \(kW\) is now placed on the ladder at \(A\). The load of weight \(4W\) is removed from \(C\) and placed on the ladder at \(B\). The ladder is modelled as a uniform rod which is in a vertical plane perpendicular to the wall. The loads are modelled as particles. Given that the ladder and the loads are in equilibrium,

1. Find the range of possible values of \(k\). [7]

\includegraphics{figure_2} A horizontal uniform rod \(AB\) has mass \(m\) and length \(4a\). The end \(A\) rests against a rough vertical wall. A particle of mass \(2m\) is attached to the rod at the point \(C\), where \(AC = 3a\). One end of a light inextensible string \(BD\) is attached to the rod at \(B\) and the other end is attached to the wall at a point \(D\), where \(D\) is vertically above \(A\). The rod is in equilibrium in a vertical plane perpendicular to the wall. The string is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac{3}{4}\), as shown in Figure 2.

1. Find the tension in the string. [5]
2. Show that the horizontal component of the force exerted by the wall on the rod has magnitude \(\frac{5}{8}mg\). [3]

The coefficient of friction between the wall and the rod is \(\mu\). Given that the rod is in limiting equilibrium,

1. find the value of \(\mu\). [4]

\includegraphics{figure_2} A ladder \(AB\), of mass \(m\) and length \(4a\), has one end \(A\) resting on rough horizontal ground. The other end \(B\) rests against a smooth vertical wall. A load of mass \(3m\) is fixed on the ladder at the point \(C\), where \(AC = a\). The ladder is modelled as a uniform rod in a vertical plane perpendicular to the wall and the load is modelled as a particle. The ladder rests in limiting equilibrium making an angle of 30° with the wall, as shown in Figure 2. Find the coefficient of friction between the ladder and the ground. [10]

A cyclist and her bicycle have a total mass of \(70\) kg. She cycles along a straight horizontal road with constant speed \(3.5 \text{ ms}^{-1}\). She is working at a constant rate of \(490\) W.

1. Find the magnitude of the resistance to motion. [4]

The cyclist now cycles down a straight road which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{14}\), at a constant speed \(U \text{ ms}^{-1}\). The magnitude of the non-gravitational resistance to motion is modelled as \(40U\) newtons. She is now working at a constant rate of \(24\) W.

1. Find the value of \(U\). [7]

\includegraphics{figure_5} A small ring \(R\) of mass \(m\) is free to slide on a smooth straight wire which is fixed at an angle of \(30°\) to the horizontal. The ring is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(\lambda\). The other end of the string is attached to a fixed point \(A\) of the wire, as shown in Fig. 5. The ring rests in equilibrium at the point \(B\), where \(AB = \frac{a}{2}\).

1. Show that \(\lambda = 4mg\). [3]

The ring is pulled down to the point \(C\), where \(BC = \frac{1}{4}a\), and released from rest. At time \(t\) after \(R\) is released the extension of the string is \((\frac{1}{4}a + x)\).

1. Obtain a differential equation for the motion of \(R\) while the string remains taut, and show that it represents simple harmonic motion with period \(\pi\sqrt{\left(\frac{a}{g}\right)}\). [6]
2. Find, in terms of \(g\), the greatest magnitude of the acceleration of \(R\) while the string remains taut. [2]
3. Find, in terms of \(a\) and \(g\), the time taken for \(R\) to move from the point at which it first reaches maximum speed to the point where the string becomes slack for the first time. [5]

\includegraphics{figure_1} A particle \(P\) of mass \(m\) is attached to the ends of two light inextensible strings \(AP\) and \(BP\) each of length \(l\). The ends \(A\) and \(B\) are attached to fixed points, with \(A\) vertically above \(B\) and \(AB = \frac{3}{4}l\), as shown in Fig. 1. The particle \(P\) moves in a horizontal circle with constant angular speed \(\omega\). The centre of the circle is the mid-point of \(AB\) and both strings remain taut.

1. Show that the tension \(AP\) is \(\frac{1}{6}m(3l\omega^2 + 4g)\). [7]
2. Find, in terms of \(m\), \(l\), \(\omega\) and \(g\), an expression for the tension in \(BP\). [2]
3. Deduce that \(\omega^2 \geq \frac{4g}{3l}\). [2]