3.03e Resolve forces: two dimensions

\includegraphics{figure_2} Three horizontal forces of magnitudes \(15\) N, \(11\) N and \(13\) N act on a particle \(P\) in the directions shown in the diagram. The angles \(\alpha\) and \(\beta\) are such that \(\sin \alpha = 0.28\), \(\cos \alpha = 0.96\), \(\sin \beta = 0.8\) and \(\cos \beta = 0.6\).

1. Show that the component, in the \(y\)-direction, of the resultant of the three forces is zero. [4]
2. Find the magnitude of the resultant of the three forces. [3]
3. State the direction of the resultant of the three forces. [1]

\includegraphics{figure_1} Two horizontal forces \(\mathbf{P}\) and \(\mathbf{Q}\) act at the origin \(O\) of rectangular coordinates \(Oxy\) (see diagram). The components of \(\mathbf{P}\) in the \(x\)- and \(y\)-directions are \(14\) N and \(5\) N respectively. The components of \(\mathbf{Q}\) in the \(x\)- and \(y\)-directions are \(-9\) N and \(7\) N respectively.

1. Write down the components, in the \(x\)- and \(y\)-directions, of the resultant of \(\mathbf{P}\) and \(\mathbf{Q}\). [2]
2. Hence find the magnitude of this resultant, and the angle the resultant makes with the positive \(x\)-axis. [4]

\includegraphics{figure_2} Three horizontal forces of magnitudes 15 N, 11 N and 13 N act on a particle \(P\) in the directions shown in the diagram. The angles \(\alpha\) and \(\beta\) are such that \(\sin \alpha = 0.28\), \(\cos \alpha = 0.96\), \(\sin \beta = 0.8\) and \(\cos \beta = 0.6\).

1. Show that the component, in the \(y\)-direction, of the resultant of the three forces is zero. [4]
2. Find the magnitude of the resultant of the three forces. [3]
3. State the direction of the resultant of the three forces. [1]

\includegraphics{figure_1} Two perpendicular forces have magnitudes \(x\) N and \(3x\) N (see diagram). Their resultant has magnitude \(6\) N.

1. Calculate \(x\). [3]
2. Find the angle the resultant makes with the smaller force. [3]

A block \(B\) of weight \(10\) N is projected down a line of greatest slope of a plane inclined at an angle of \(20°\) to the horizontal. \(B\) travels down the plane at constant speed.

1. 1. Find the components perpendicular and parallel to the plane of the contact force between \(B\) and the plane. [2]
   2. Hence show that the coefficient of friction is \(0.364\), correct to \(3\) significant figures. [2]
2. \includegraphics{figure_6} \(B\) is in limiting equilibrium when acted on by a force of \(T\) N directed towards the plane at an angle of \(45°\) to a line of greatest slope (see diagram). Given that the frictional force on \(B\) acts down the plane, find \(T\). [7]

\includegraphics{figure_2} Figure 2 shows a large block of mass 50 kg being pulled on rough horizontal ground by means of a rope attached to the block. The tension in the rope is 200 N and it makes an angle of 40° with the horizontal. Under these conditions, the block is on the point of moving. Modelling the block as a particle,

1. show that the coefficient of friction between the block and the ground is 0.424 correct to 3 significant figures. [6 marks]

The angle with the horizontal at which the rope is being pulled is reduced to 30°. Ignoring air resistance and assuming that the tension in the rope and the coefficient of friction remain unchanged,

1. find the acceleration of the block. [6 marks]

\includegraphics{figure_2} A particle \(P\), of mass 2 kg, lies on a rough plane inclined at an angle of 30° to the horizontal. A force \(H\), whose line of action is parallel to the line of greatest slope of the plane, is applied to the particle as shown in Figure 2. The coefficient of friction between the particle and the plane is \(\frac{1}{\sqrt{3}}\). Given that the particle is on the point of moving up the plane,

1. draw a diagram showing all the forces acting on the particle, [2 marks]
2. show that the ratio of the magnitude of the frictional force to the magnitude of \(H\) is equal to \(1 : 2\) [7 marks]

The force \(H\) is now removed but \(P\) remains at rest.

1. Use the principle of friction to explain how this is possible. [2 marks]

A sledge of mass 4 kg rests in limiting equilibrium on a rough slope inclined at an angle 10° to the horizontal. By modelling the sledge as a particle,

1. show that the coefficient of friction, \(\mu\), between the sledge and the ground is 0.176 correct to 3 significant figures. [6 marks]

The sledge is placed on a steeper part of the slope which is inclined at an angle 30° to the horizontal. The value of \(\mu\) remains unchanged.

1. Find the minimum extra force required along the line of greatest slope to prevent the sledge from slipping down the hill. [5 marks]

\includegraphics{figure_8} A conical shell has radius 6 m and height 8 m. The shell, with its vertex \(V\) downwards, is rotating about its vertical axis. A particle, of mass 0.4 kg, is in contact with the rough inner surface of the shell. The particle is 4 m above the level of \(V\) (see diagram). The particle and shell rotate with the same constant angular speed. The coefficient of friction between the particle and the shell is \(\mu\).

1. The frictional force on the particle is \(F\) N, and the normal force of the shell on the particle is \(R\) N. It is given that the speed of the particle is 4.5 ms\(^{-1}\), which is the smallest possible speed for the particle not to slip.
   1. By resolving vertically, show that \(4F + 3R = 19.6\). [2]
   2. By finding another equation connecting \(F\) and \(R\), find the values of \(F\) and \(R\) and show that \(\mu = 0.336\), correct to 3 significant figures. [6]
2. Find the largest possible angular speed of the shell for which the particle does not slip. [6]

One end of a light inextensible string of length \(l\) is attached to the vertex of a smooth cone of semi-vertical angle \(45°\). The cone is fixed to the ground with its axis vertical. The other end of the string is attached to a particle of mass \(m\) which rotates in a horizontal circle in contact with the outer surface of the cone. The angular speed of the particle is \(\omega\) (see diagram). The tension in the string is \(T\) and the contact force between the cone and the particle is \(R\).

1. By resolving horizontally and vertically, find two equations involving \(T\) and \(R\) and hence show that \(T = \frac{1}{2}ml(\sqrt{2}g + l\omega^2)\). [6]
2. When the string has length 0.8 m, calculate the greatest value of \(\omega\) for which the particle remains in contact with the cone. [4]

A smooth solid cone of semi-vertical angle \(60°\) is fixed to the ground with its axis vertical. A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point vertically above the vertex of the cone. \(P\) rotates in a horizontal circle on the surface of the cone with constant angular velocity \(\omega\). The string is inclined to the downward vertical at an angle of \(30°\) (see diagram).

1. Show that the magnitude of the contact force between the cone and the particle is \(\frac{1}{4}m(2\sqrt{3}g - 3a\omega^2)\). [6]
2. Given that \(a = 0.5\) m and \(m = 3.5\) kg, find, in either order, the greatest speed for which the particle remains in contact with the cone and the corresponding tension in the string. [3]

Aliya, whose mass is \(m\) kg, is playing rounders. She rounds the first base at a speed of \(v\) ms\(^{-1}\), making the turn on a horizontal circular path of radius \(r\) m.

1. Write down, in terms of \(m\), \(v\) and \(r\), the magnitude of the horizontal force acting on her. [1 mark]
2. Show that if she continues on the same circular path, the reaction force exerted on her by the ground must act at an angle \(\theta\) to the vertical, where \(\tan \theta = \frac{v^2}{gr}\). [6 marks]

\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]

\includegraphics{figure_11} A uniform rod \(AB\), of weight \(20 \text{N}\) and length \(2.8 \text{m}\), rests in equilibrium with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth wall inclined at \(55°\) to the horizontal. The rod, which rests in a vertical plane that is perpendicular to the wall, is inclined at \(30°\) to the horizontal (see diagram).

1. Show that the magnitude of the force acting on the rod at \(B\) is \(9.56 \text{N}\), correct to 3 significant figures. [3]
2. Determine the magnitude of the contact force between the rod and the ground. Give your answer correct to 3 significant figures. [5]

Lizzie is sat securely on a wooden sledge. The combined mass of Lizzie and the sledge is \(M\) kilograms. The sledge is being pulled forward in a straight line along a horizontal surface by means of a light inextensible rope, which is attached to the front of the sledge. This rope stays inclined at an acute angle \(\theta\) above the horizontal and remains taut as the sledge moves forward. \includegraphics{figure_17} The sledge remains in contact with the surface throughout. The coefficient of friction between the sledge and the surface is \(\mu\) and there are no other resistance forces. Lizzie and the sledge move forward with constant acceleration, \(a \text{ m s}^{-2}\) The tension in the rope is a constant \(T\) Newtons.

1. Show that $$T = \frac{M(a + \mu g)}{\cos \theta + \mu \sin \theta}$$ [7 marks]
2. It is known that when \(M = 30\), \(\theta = 30°\), and \(T = 40\), the sledge remains at rest. Lizzie uses these values with the relationship formed in part (a) to find the value for \(\mu\) Explain why her value for \(\mu\) may be incorrect. [2 marks]

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]

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} Fig. 12 shows a hemispherical bowl. The rim of this bowl is a circle with centre O and radius \(r\). The bowl is fixed with its rim horizontal and uppermost. A particle P, of mass \(m\), is connected by a light inextensible string of length \(l\) to the lowest point A on the bowl and describes a horizontal circle with constant angular speed \(\omega\) on the smooth inner surface of the bowl. The string is taut, and AP makes an angle \(\alpha\) with the vertical.

1. Show that the normal contact force between P and the bowl is of magnitude \(mg + 2mr\omega^2\cos^2\alpha\). [9]
2. Deduce that \(g < r\omega^2(k_1 + k_2\cos^2\alpha)\), stating the value of the constants \(k_1\) and \(k_2\). [3]

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]

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]

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]