3.03n Equilibrium in 2D: particle under forces

A smooth bead \(B\) of mass 0.3 kg is threaded on a light inextensible string of length 0.9 m. One end of the string is attached to a fixed point \(A\), and the other end of the string is attached to a fixed point \(C\) which is vertically below \(A\). The tension in the string is 7 N, and the bead rotates with angular speed \(ω\) rad s\(^{-1}\) in a horizontal circle about the vertical axis through \(A\) and \(C\).

1. Given that \(B\) moves in a circle with centre \(C\) and radius 0.2 m, calculate \(ω\), and hence find the kinetic energy of \(B\). [5]
2. Given instead that angle \(ABC = 90°\), and that \(AB\) makes an angle \(\tan^{-1}(\frac{4}{3})\) with the vertical, calculate \(T\) and \(ω\). [6]

\includegraphics{figure_4} A particle \(P\) is moving inside a smooth hollow cone which has its vertex downwards and its axis vertical, and whose semi-vertical angle is \(45°\). A light inextensible string parallel to the surface of the cone connects \(P\) to the vertex. \(P\) moves with constant angular speed in a horizontal circle of radius \(0.67\) m (see diagram). The tension in the string is equal to the weight of \(P\). Calculate the angular speed of \(P\). [6]

A particle \(P\) of mass \(0.2\,\text{kg}\) is attached to one end of a light elastic string of natural length \(0.75\,\text{m}\) and modulus of elasticity \(21\,\text{N}\). The other end of the string is attached to a fixed point \(A\) which is \(0.8\,\text{m}\) vertically above a smooth horizontal surface. \(P\) rests in equilibrium on the surface.

1. Find the magnitude of the force exerted on \(P\) by the surface. [2]

\(P\) is now projected horizontally along the surface with speed \(3\,\text{m s}^{-1}\).

1. Calculate the extension of the string at the instant when \(P\) leaves the surface. [3]
2. Hence find the speed of \(P\) at the instant when it leaves the surface. [3]

\includegraphics{figure_3} Particles \(A\) and \(B\), of masses \(3m\) and \(m\) respectively, are connected by a light inextensible string of length \(a\) that passes through a fixed smooth ring \(R\). Particle \(B\) hangs in equilibrium vertically below the ring. Particle \(A\) moves in horizontal circles on a smooth horizontal surface with speed \(\frac{2}{3}\sqrt{ga}\). The angle between \(AR\) and \(BR\) is \(\theta\) (see diagram). The normal reaction between \(A\) and the surface is \(\frac{15}{2}mg\).

1. Find \(\cos \theta\). [3]

\includegraphics{figure_1} A particle of weight 10 N is attached to one end of a light elastic string. The other end of the string is attached to a fixed point \(A\) on a horizontal ceiling. A horizontal force of 7.5 N acts on the particle. In the equilibrium position, the string makes an angle \(\theta\) with the ceiling (see diagram). The string has natural length 0.8 m and modulus of elasticity 50 N.

1. Find the tension in the string. [2]
2. Find the vertical distance between the particle and the ceiling. [3]

\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_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_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]

\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 has mass \(2\) kg. It is attached at \(B\) to the ends of two light inextensible strings \(AB\) and \(BC\). When the particle hangs in equilibrium, \(AB\) makes an angle of \(30°\) with the vertical, as shown in Fig. 1. The magnitude of the tension in \(BC\) is twice the magnitude of the tension in \(AB\).

1. Find, in degrees to one decimal place, the size of the angle that \(BC\) makes with the vertical. [4]
2. Hence find, to 3 significant figures, the magnitude of the tension in \(AB\). [4]

\includegraphics{figure_1} A tennis ball \(P\) is attached to one end of a light inextensible string, the other end of the string being attached to a the top of a fixed vertical pole. A girl applies a horizontal force of magnitude 50 N to \(P\), and \(P\) is in equilibrium under gravity with the string making an angle of \(40°\) with the pole, as shown in Fig. 1. By modelling the ball as a particle find, to 3 significant figures,

1. the tension in the string, [3]
2. the weight of \(P\). [4]

\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

A car of mass 1000 kg is moving along a straight horizontal road with a constant acceleration of \(j\) m s\(^{-2}\). The resistance to motion is modelled as a constant force of magnitude 1200 N. When the car is travelling at 12 m s\(^{-1}\), the power generated by the engine of the car is 24 kW.

1. Calculate the value of \(j\). [4]

When the car is travelling at 14 m s\(^{-1}\), the engine is switched off and the car comes to rest, without braking, in a distance of \(d\) metres. Assuming the same model for resistance,

1. use the work-energy principle to calculate the value of \(d\). [3]

1. Give a reason why the model used for the resistance to motion may not be realistic. [1]

A uniform ladder \(AB\), of mass \(m\) and length \(2a\), has one end \(A\) on rough horizontal ground. The other end \(B\) rests against a smooth vertical wall. The ladder is in a vertical plane perpendicular to the wall. The ladder makes an angle \(α\) with the horizontal, where \(\tan α = \frac{4}{3}\). A child of mass \(2m\) stands on the ladder at \(C\) where \(AC = \frac{1}{4}a\), as shown in Fig. 1. The ladder and the child are in equilibrium. By modelling the ladder as a rod and the child as a particle, calculate the least possible value of the coefficient of friction between the ladder and the ground. [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 \(BD\) 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_1} A uniform ladder \(AB\), of mass \(m\) and length \(2a\), has one end \(A\) on rough horizontal ground. The other end \(B\) rests against a smooth vertical wall. The ladder is in a vertical plane perpendicular to the wall. The ladder makes an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \frac{4}{3}\). A child of mass \(2m\) stands on the ladder at \(C\) where \(AC = \frac{1}{2}a\), as shown in Fig. 1. The ladder and the child are in equilibrium. By modelling the ladder as a rod and the child as a particle, calculate the least possible value of the coefficient of friction between the ladder and the ground. [9]

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]

\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]

\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]