3.03e Resolve forces: two dimensions

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]

\includegraphics{figure_2} A particle \(P\) of mass 2 kg is pushed up a line of greatest slope of a rough plane by a horizontal force of magnitude \(X\) newtons, as shown in Figure 2. The force acts in the vertical plane which contains \(P\) and a line of greatest slope of the plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan\alpha = \frac{3}{4}\). The coefficient of friction between \(P\) and the plane is 0.5 Given that the acceleration of \(P\) is 1.45 m s\(^{-2}\), find the value of \(X\). [10]

\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 2 kg is attached to one end of a light inextensible string. A particle \(Q\) of mass 5 kg is attached to the other end of the string. The string passes over a small smooth light pulley. The pulley is fixed at a point on the intersection of a rough horizontal table and a fixed smooth inclined plane. The string lies along the table and also lies in a vertical plane which contains a line of greatest slope of the inclined plane. This plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\). Particle \(P\) is at rest on the table, a distance \(d\) metres from the pulley. Particle \(Q\) is on the inclined plane with the string taut, as shown in Figure 3. The coefficient of friction between \(P\) and the table is \(\frac{1}{4}\). The system is released from rest and \(P\) slides along the table towards the pulley. Assuming that \(P\) has not reached the pulley and that \(Q\) remains on the inclined plane,

1. write down an equation of motion for \(P\), [2]
2. write down an equation of motion for \(Q\), [2]
3. 1. find the acceleration of \(P\),
   2. find the tension in the string. [5]

When \(P\) has moved a distance 0.5 m from its initial position, the string breaks. Given that \(P\) comes to rest just as it reaches the pulley,

1. find the value of \(d\). [7]

Two forces, \(\mathbf{F}_1\) and \(\mathbf{F}_2\), act on a particle \(A\). \(\mathbf{F}_1 = (2i - 3j)\) N and \(\mathbf{F}_2 = (pi + qj)\) N, where \(p\) and \(q\) are constants. Given that the resultant of \(\mathbf{F}_1\) and \(\mathbf{F}_2\) is parallel to \((\mathbf{i} + 2\mathbf{j})\),

1. show that \(2p - q + 7 = 0\) [5] Given that \(q = 11\) and that the mass of \(A\) is 2 kg, and that \(\mathbf{F}_1\) and \(\mathbf{F}_2\) are the only forces acting on \(A\),
2. find the magnitude of the acceleration of \(A\). [5]

A suitcase of mass 40 kg is being dragged in a straight line along a rough horizontal floor at constant speed using a thin strap. The strap is inclined at \(20°\) above the horizontal. The coefficient of friction between the suitcase and the floor is \(\frac{3}{4}\). The strap is modelled as a light inextensible string and the suitcase is modelled as a particle. Find the tension in the strap. [7]

Two forces \(\mathbf{F_1}\) and \(\mathbf{F_2}\) act on a particle. The force \(\mathbf{F_1}\) has magnitude 8 N and acts due east. The resultant of \(\mathbf{F_1}\) and \(\mathbf{F_2}\) is a force of magnitude 14 N acting in a direction whose bearing is \(120°\). Find

1. the magnitude of \(\mathbf{F_2}\), [4]
2. the direction of \(\mathbf{F_2}\), giving your answer as a bearing to the nearest degree. [5]

\includegraphics{figure_2} Figure 2 shows two particles \(A\) and \(B\), of masses \(3m\) and \(4m\) respectively, attached to the ends of a light inextensible string. Initially \(A\) is held at rest on the surface of a fixed rough inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac{3}{4}\). The coefficient of friction between \(A\) and the plane is \(\frac{1}{4}\). The string passes over a small smooth light pulley \(P\) which is fixed at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. The particle \(B\) hangs freely and is vertically below \(P\). The system is released from rest with the string taut and with \(B\) at a height of 1.75 m above the ground. In the subsequent motion, \(A\) does not hit the pulley. For the period before \(B\) hits the ground,

1. write down an equation of motion for each particle. [4]
2. Hence show that the acceleration of \(B\) is \(\frac{8}{35}g\). [5]
3. Explain how you have used the fact that the string is inextensible in your calculation. [1]

When \(B\) hits the ground, \(B\) does not rebound and comes immediately to rest.

1. Find the distance travelled by \(A\) from the instant when the system is released to the instant when \(A\) first comes to rest. [7]

\includegraphics{figure_2} A rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\) A particle \(P\) of mass 2 kg is held in equilibrium on the plane by a horizontal force of magnitude \(X\) newtons, as shown in Figure 2. The force acts in a vertical plane which contains a line of greatest slope of the inclined plane.

1. Show that when \(X = 14.7\) there is no frictional force acting on \(P\) [3] The coefficient of friction between \(P\) and the plane is 0.5
2. Find the smallest possible value of \(X\). [8]

[In this question, \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors.] A particle \(A\) of mass 0.5 kg is at rest on a smooth horizontal plane. At time \(t = 0\), two forces, \(\mathbf{F}_1 = (-3\mathbf{i} + 2\mathbf{j})\) N and \(\mathbf{F}_2 = (p\mathbf{i} + q\mathbf{j})\) N, where \(p\) and \(q\) are constants, are applied to \(A\). Given that \(A\) moves in the direction of the vector \((\mathbf{i} - 2\mathbf{j})\),

1. show that \(2p + q - 4 = 0\) [4] Given that \(p = 5\)
2. Find the speed of \(A\) at time \(t = 4\) seconds. [5]

\includegraphics{figure_1} A small box is pushed along a floor. The floor is modelled as a rough horizontal plane and the box is modelled as a particle. The coefficient of friction between the box and the floor is \(\frac{1}{2}\). The box is pushed by a force of magnitude 100 N which acts at an angle of 30° with the floor, as shown in Figure 1. Given that the box moves with constant speed, find the mass of the box. [7]

\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_2} A parcel of weight \(10\) N lies on a rough plane inclined at an angle of \(30°\) to the horizontal. A horizontal force of magnitude \(P\) newtons acts on the parcel, as shown in Figure 2. The parcel is in equilibrium and on the point of slipping up the plane. The normal reaction of the plane on the parcel is \(18\) N. The coefficient of friction between the parcel and the plane is \(\mu\). Find

1. the value of \(P\), [4]
2. the value of \(\mu\). [5]

The horizontal force is removed.

1. Determine whether or not the parcel moves. [5]

\includegraphics{figure_3} A box of mass 30 kg is being pulled along rough horizontal ground at a constant speed using a rope. The rope makes an angle of 20° with the ground, as shown in Figure 3. The coefficient of friction between the box and the ground is 0.4. The box is modelled as a particle and the rope as a light, inextensible string. The tension in the rope is \(P\) newtons.

1. Find the value of \(P\). [8]

The tension in the rope is now increased to 150 N.

1. Find the acceleration of the box. [6]

\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_5} Figure 5 shows two particles \(A\) and \(B\), of mass \(2m\) and \(4m\) respectively, connected by a light inextensible string. Initially \(A\) is held at rest on a rough inclined plane which is fixed to horizontal ground. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan\alpha = \frac{3}{4}\). The coefficient of friction between \(A\) and the plane is \(\frac{1}{4}\). The string passes over a small smooth pulley \(P\) which is fixed at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane and \(B\) hangs vertically below \(P\). The system is released from rest with the string taut, with \(A\) at the point \(X\) and with \(B\) at a height \(h\) above the ground. For the motion until \(B\) hits the ground,

1. give a reason why the magnitudes of the accelerations of the two particles are the same, [1]
2. write down an equation of motion for each particle, [4]
3. find the acceleration of each particle. [5]

Particle \(B\) does not rebound when it hits the ground and \(A\) continues moving up the plane towards \(P\). Given that \(A\) comes to rest at the point \(Y\), without reaching \(P\),

1. find the distance \(XY\) in terms of \(h\). [6]

\includegraphics{figure_2} A box of mass \(6 \text{ kg}\) lies on a rough plane inclined at an angle of \(30°\) to the horizontal. The box is held in equilibrium by means of a horizontal force of magnitude \(P\) newtons, as shown in Fig. 2. The line of action of the force is in the same vertical plane as a line of greatest slope of the plane. The coefficient of friction between the box and the plane is \(0.4\). The box is modelled as a particle. Given that the box is in limiting equilibrium and on the point of moving up the plane, find,

1. the normal reaction exerted on the box by the plane, [4]
2. the value of \(P\). [3]

The horizontal force is removed.

1. Show that the box will now start to move down the plane. [5]

\includegraphics{figure_3} Particles \(A\) and \(B\), of mass \(2m\) and \(m\) respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley fixed at the edge of a rough horizontal table. Particle \(A\) is held on the table, while \(B\) rests on a smooth plane inclined at \(30°\) to the horizontal, as shown in Fig. 3. The string is in the same vertical plane as a line of greatest slope of the inclined plane. The coefficient of friction between \(A\) and the table is \(\mu\). The particle \(A\) is released from rest and begins to move. By writing down an equation of motion for each particle,

1. show that, while both particles move with the string taut. Each particle has an acceleration of magnitude \(\frac{1}{5}(1 - 4\mu)g\). [7]

When each particle has moved a distance \(h\), the string breaks. The particle \(A\) comes to rest before reaching the pulley. Given that \(\mu = 0.2\),

1. find, in terms of \(h\), the total distance moved by \(A\). [6]

For the model described above,

1. state two physical factors, apart from air resistance, which could be taken into account to make the model more realistic. [2]

\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]