6.04c Composite bodies: centre of mass

6 Nm
8 Nm
10 Nm
14 Nm 3 A uniform disc has mass 6 kg and diameter 8 cm A uniform rectangular lamina, \(A B C D\), has mass 4 kg , width 8 cm and length 20 cm
The disc is fixed to the lamina to form a composite body as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-03_448_881_587_577} The sides \(A B , A D\) and \(C D\) are tangents to the disc.
Calculate the distance of the centre of mass of the composite body from \(A D\) Circle your answer.
4 cm
5.6 cm
6.4 cm
8.8 cm 4 A car of mass 1400 kg drives around a horizontal circular bend of radius 60 metres.
The car has a constant speed of \(12 \mathrm {~ms} ^ { - 1 }\) on the bend.
Calculate the magnitude of the resultant force acting on the car.
[0pt] [2 marks] \(5 \quad\) A region bounded by the curve with equation \(y = 4 - x ^ { 2 }\), the \(x\)-axis and the \(y\)-axis is shown below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-04_641_380_408_831} The region is rotated through \(360 ^ { \circ }\) around the \(x\)-axis to create a uniform solid.
5

1. Show that the distance of the centre of mass of the solid from the circular face is \(\frac { 5 } { 8 }\) [0pt] [5 marks]
   5
2. The solid is suspended in equilibrium from a point on the edge of the circular face.
   Find the angle between the circular face and the horizontal, giving your answer to the nearest degree.
   6 In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A sphere of mass 0.8 kg is attached to one end of a string of length 2 metres.
   The other end of the string is attached to a fixed point \(O\) The sphere is released from rest with the string taut and at an angle of \(30 ^ { \circ }\) to the vertical, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-06_464_218_676_909} 6
   1. Find the speed of the sphere when it is directly below \(O\) 6
   2. State one assumption that you made about the string.
      6
   3. As the sphere moves, the string makes an angle \(\theta\) with the downward vertical. By finding an expression for the tension in the string in terms of \(\theta\), show that the tension is a maximum when the sphere is directly below \(O\) 6
   4. A physics student conducts an experiment and uses a device to measure the tension in the string when the sphere is directly below \(O\) They find that the tension is 9.5 newtons.
      Explain why this result is reasonable, showing any calculations that you make.

14 Nm 3 A uniform disc has mass 6 kg and diameter 8 cm A uniform rectangular lamina, \(A B C D\), has mass 4 kg , width 8 cm and length 20 cm
The disc is fixed to the lamina to form a composite body as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-03_448_881_587_577} The sides \(A B , A D\) and \(C D\) are tangents to the disc.
Calculate the distance of the centre of mass of the composite body from \(A D\) Circle your answer.
4 cm
5.6 cm
6.4 cm
8.8 cm 4 A car of mass 1400 kg drives around a horizontal circular bend of radius 60 metres.
The car has a constant speed of \(12 \mathrm {~ms} ^ { - 1 }\) on the bend.
Calculate the magnitude of the resultant force acting on the car.
[0pt] [2 marks] \(5 \quad\) A region bounded by the curve with equation \(y = 4 - x ^ { 2 }\), the \(x\)-axis and the \(y\)-axis is shown below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-04_641_380_408_831} The region is rotated through \(360 ^ { \circ }\) around the \(x\)-axis to create a uniform solid.
5

1. Show that the distance of the centre of mass of the solid from the circular face is \(\frac { 5 } { 8 }\) [0pt] [5 marks]
   5
2. The solid is suspended in equilibrium from a point on the edge of the circular face.
   Find the angle between the circular face and the horizontal, giving your answer to the nearest degree.
   6 In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A sphere of mass 0.8 kg is attached to one end of a string of length 2 metres.
   The other end of the string is attached to a fixed point \(O\) The sphere is released from rest with the string taut and at an angle of \(30 ^ { \circ }\) to the vertical, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-06_464_218_676_909} 6
   1. Find the speed of the sphere when it is directly below \(O\) 6
   2. State one assumption that you made about the string.
      6
   3. As the sphere moves, the string makes an angle \(\theta\) with the downward vertical. By finding an expression for the tension in the string in terms of \(\theta\), show that the tension is a maximum when the sphere is directly below \(O\) 6
   4. A physics student conducts an experiment and uses a device to measure the tension in the string when the sphere is directly below \(O\) They find that the tension is 9.5 newtons.
      Explain why this result is reasonable, showing any calculations that you make.
      7 Two particles, \(A\) and \(B\), are moving on a smooth horizontal surface. A straight line has been marked on the surface and the particles are on opposite sides of the line. Particle \(A\) has mass 2 kg and moves with velocity \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the line. Particle \(B\) has mass 3 kg and moves with velocity \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(45 ^ { \circ }\) to the line. The particles and their velocities are shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-08_451_739_858_653} The particles collide when they reach the line and then move together as a single combined particle. 7
   5. Show that the magnitude of the impulse on particle \(A\) during the collision is 7.55 Ns correct to three significant figures.
      7
   6. State the magnitude of the impulse on \(B\) during the collision, giving a reason for your answer. 7
   7. Find the size of the angle between the straight line and the impulse acting on \(B\), giving your answer to the nearest degree. 7
   8. During the collision, one particle crosses the straight line.
      State which particle crosses the line, giving a reason for your answer.
      [0pt] [1 mark] 8 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A block has mass 10 kg and is at rest 1 metre from a fixed point \(O\) on a horizontal surface. One end of an elastic string is attached to the block and the other end of the elastic string is attached to the point \(O\) The elastic string has modulus of elasticity 40 newtons and natural length 2 metres.
      The coefficient of friction between the block and the surface is 0.6 A force is applied to the block so that it starts to move towards a vertical wall.
      The block moves on a line that is perpendicular to the wall.
      The force has magnitude 100 newtons and acts at an angle of \(30 ^ { \circ }\) to the horizontal.
      The situation is shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-10_239_1339_1176_354} 8
   9. Show that the distance that the block has moved, when the forces acting on it are in equilibrium, is 3.9 metres correct to two significant figures.
      [0pt] [4 marks]
      8
   10. State one limitation of the model that you have used. 8
   11. Find the maximum speed of the block.
       8
   12. The vertical wall is 8.7 metres from \(O\) Determine whether the block reaches the wall, showing any calculations that you make. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-13_2492_1721_217_150}

4. \begin{figure}[h] \end{figure} The uniform lamina \(A B C D\) shown in Figure 2 is in the shape of an isosceles trapezium.

• \(B C\) is parallel to \(A D\) and angle \(B A D\) is equal to angle \(A D C\)
• \(B C = 5 a\) and \(A D = 7 a\)
• the perpendicular distance between \(B C\) and \(A D\) is \(3 a\)
• the distance of the centre of mass of \(A B C D\) from \(A D\) is \(d\)
  1. Show that \(d = \frac { 17 } { 12 } a\)

The uniform lamina \(P Q R S\) is a rectangle with \(P Q = 5 a\) and \(Q R = 9 a\).
The lamina \(A B C D\) in Figure 2 is used to cut a hole in \(P Q R S\) to form the template shown shaded in Figure 3. \begin{figure}[h] \end{figure}

The template is freely suspended from \(P\) and hangs in equilibrium with \(P S\) at an angle of \(\theta ^ { \circ }\) to the downward vertical.

Find the value of \(\theta\)

\includegraphics{figure_4} Three identical uniform discs, \(A\), \(B\) and \(C\), each have mass \(m\) and radius \(a\). They are joined together by uniform rods, each of which has mass \(\frac{1}{4}m\) and length \(2a\). The discs lie in the same plane and their centres form the vertices of an equilateral triangle of side \(4a\). Each rod has one end rigidly attached to the circumference of a disc and the other end rigidly attached to the circumference of an adjacent disc, so that the rod lies along the line joining the centres of the two discs (see diagram).

1. Find the moment of inertia of this object about an axis \(l\), which is perpendicular to the plane of the object and through the centre of disc \(A\). [6]

The object is free to rotate about the horizontal axis \(l\). It is released from rest in the position shown, with the centre of disc \(B\) vertically above the centre of disc \(A\).

1. Write down the change in the vertical position of the centre of mass of the object when the centre of disc \(B\) is vertically below the centre of disc \(A\). Hence find the angular velocity of the object when the centre of disc \(B\) is vertically below the centre of disc \(A\). [4]

\includegraphics{figure_4} Three identical uniform discs, \(A\), \(B\) and \(C\), each have mass \(m\) and radius \(a\). They are joined together by uniform rods, each of which has mass \(\frac{1}{3}m\) and length \(2a\). The discs lie in the same plane and their centres form the vertices of an equilateral triangle of side \(4a\). Each rod has one end rigidly attached to the circumference of a disc and the other end rigidly attached to the circumference of an adjacent disc, so that the rod lies along the line joining the centres of the two discs (see diagram).

1. Find the moment of inertia of this object about an axis \(l\), which is perpendicular to the plane of the object and through the centre of disc \(A\). [6]
2. The object is free to rotate about the horizontal axis \(l\). It is released from rest in the position shown, with the centre of disc \(B\) vertically above the centre of disc \(A\). Write down the change in the vertical position of the centre of mass of the object when the centre of disc \(B\) is vertically below the centre of disc \(A\). Hence find the angular velocity of the object when the centre of disc \(B\) is vertically below the centre of disc \(A\). [4]

A uniform circular disc has diameter \(AB\), mass \(2m\) and radius \(a\). A particle of mass \(m\) is attached to the disc at \(B\). The disc is able to rotate about a smooth fixed horizontal axis through \(A\). The axis is tangential to the disc. Show that the moment of inertia of the system about the axis is \(\frac{5}{2}ma^2\). [4] The disc is held with \(AB\) horizontal and released. Find the angular speed of the system when \(B\) is directly below \(A\). [5] The disc is slightly displaced from the position of equilibrium in which \(B\) is below \(A\). At time \(t\) the angle between \(AB\) and the vertical is \(\theta\). Write down the equation of motion, and find the approximate period of small oscillations about the equilibrium position. [5]

Answer only one of the following two alternatives. **EITHER** \includegraphics{figure_11a} A uniform plane object consists of three identical circular rings, \(X\), \(Y\) and \(Z\), enclosed in a larger circular ring \(W\). Each of the inner rings has mass \(m\) and radius \(r\). The outer ring has mass \(3m\) and radius \(R\). The centres of the inner rings lie at the vertices of an equilateral triangle of side \(2r\). The outer ring touches each of the inner rings and the rings are rigidly joined together. The fixed axis \(AB\) is the diameter of \(W\) that passes through the centre of \(X\) and the point of contact of \(Y\) and \(Z\) (see diagram). It is given that \(R = \left(1 + \frac{2}{3}\sqrt{3}\right)r\).

1. Show that the moment of inertia of the object about \(AB\) is \(\left(7 + 2\sqrt{3}\right)mr^2\). [8]

The line \(CD\) is the diameter of \(W\) that is perpendicular to \(AB\). A particle of mass \(9m\) is attached to \(D\). The object is now held with its plane horizontal. It is released from rest and rotates freely about the fixed horizontal axis \(AB\).

1. Find, in terms of \(g\) and \(r\), the angular speed of the object when it has rotated through \(60°\). [6]

**OR** Fish of a certain species live in two separate lakes, \(A\) and \(B\). A zoologist claims that the mean length of fish in \(A\) is greater than the mean length of fish in \(B\). To test his claim, he catches a random sample of 8 fish from \(A\) and a random sample of 6 fish from \(B\). The lengths of the 8 fish from \(A\), in appropriate units, are as follows. $$15.3 \quad 12.0 \quad 15.1 \quad 11.2 \quad 14.4 \quad 13.8 \quad 12.4 \quad 11.8$$ Assuming a normal distribution, find a 95% confidence interval for the mean length of fish in \(A\). [5] The lengths of the 6 fish from \(B\), in the same units, are as follows. $$15.0 \quad 10.7 \quad 13.6 \quad 12.4 \quad 11.6 \quad 12.6$$ Stating any assumptions that you make, test at the 5% significance level whether the mean length of fish in \(A\) is greater than the mean length of fish in \(B\). [7] Calculate a 95% confidence interval for the difference in the mean lengths of fish from \(A\) and from \(B\). [2]

\includegraphics{figure_1} A frame consists of a uniform semicircular wire of radius 20 cm and mass 2 kg, and a uniform straight wire of length 40 cm and mass 0.9 kg. The ends of the semicircular wire are attached to the ends of the straight wire (see diagram). Find the distance of the centre of mass of the frame from the straight wire. [4]

\includegraphics{figure_7} The diagram shows the cross-section \(OABCDE\) through the centre of mass of a uniform prism on a rough inclined plane. The portion \(ADEO\) is a rectangle in which \(AD = OE = 0.6\) m and \(DE = AO = 0.8\) m; the portion \(BCD\) is an isosceles triangle in which angle \(BCD\) is a right angle, and \(A\) is the mid-point of \(BD\). The plane is inclined at \(45°\) to the horizontal, \(BC\) lies along a line of greatest slope of the plane and \(DE\) is horizontal.

1. Calculate the distance of the centre of mass of the prism from \(BD\). [3]

The weight of the prism is \(21\) N, and it is held in equilibrium by a horizontal force of magnitude \(P\) N acting along \(ED\).

1. 1. Find the smallest value of \(P\) for which the prism does not topple. [2]
   2. It is given that the prism is about to slip for this smallest value of \(P\). Calculate the coefficient of friction between the prism and the plane. [3]

The value of \(P\) is gradually increased until the prism ceases to be in equilibrium.

1. Show that the prism topples before it begins to slide, stating the value of \(P\) at which equilibrium is broken. [5]

\includegraphics{figure_2} A uniform wire has the shape of a semicircular arc, with diameter \(AB\) of length \(0.8 \text{ m}\). The wire is attached to a vertical wall by a smooth hinge at \(A\). The wire is held in equilibrium with \(AB\) inclined at \(70°\) to the upward vertical by a light string attached to \(B\). The other end of the string is attached to the point \(C\) on the wall \(0.8 \text{ m}\) vertically above \(A\). The tension in the string is \(15 \text{ N}\) (see diagram).

1. Show that the horizontal distance of the centre of mass of the wire from the wall is \(0.463 \text{ m}\), correct to 3 significant figures. [3]
2. Calculate the weight of the wire. [2]

\includegraphics{figure_4} A uniform solid cone has base radius \(0.4 \text{ m}\) and height \(4.4 \text{ m}\). A uniform solid cylinder has radius \(0.4 \text{ m}\) and weight equal to the weight of the cone. An object is formed by attaching the cylinder to the cone so that the base of the cone and a circular face of the cylinder are in contact and their circumferences coincide. The object rests in equilibrium with its circular base on a plane inclined at an angle of \(20°\) to the horizontal (see diagram).

1. Calculate the least possible value of the coefficient of friction between the plane and the object. [2]
2. Calculate the greatest possible height of the cylinder. [4]

\includegraphics{figure_4} A uniform solid cone has base radius \(0.4\) m and height \(4.4\) m. A uniform solid cylinder has radius \(0.4\) m and weight equal to the weight of the cone. An object is formed by attaching the cylinder to the cone so that the base of the cone and a circular face of the cylinder are in contact and their circumferences coincide. The object rests in equilibrium with its circular base on a plane inclined at an angle of \(20°\) to the horizontal (see diagram).

1. Calculate the least possible value of the coefficient of friction between the plane and the object. [2]
2. Calculate the greatest possible height of the cylinder. [4]

\includegraphics{figure_3} An object is made from a uniform solid hemisphere of radius \(0.56\) m and centre \(O\) by removing a hemisphere of radius \(0.28\) m and centre \(O\). The diagram shows a cross-section through \(O\) of the object.

1. Calculate the distance of the centre of mass of the object from \(O\). [4] [The volume of a hemisphere is \(\frac{2}{3}\pi r^3\).]
2. The object has weight \(24\) N. A uniform hemisphere \(H\) of radius \(0.28\) m is placed in the hollow part of the object to create a new uniform hemisphere with centre \(O\). The centre of mass of the non-uniform hemisphere is \(0.15\) m from \(O\). Calculate the weight of \(H\). [3]

\includegraphics{figure_2} An object is made from a uniform solid hemisphere of radius \(0.56\) m and centre \(O\) by removing a hemisphere of radius \(0.28\) m and centre \(O\). The diagram shows a cross-section through \(O\) of the object.

1. Calculate the distance of the centre of mass of the object from \(O\). [4] [The volume of a hemisphere is \(\frac{2}{3}\pi r^3\).]
2. Calculate the weight of \(H\). [3]

The object has weight \(24\) N. A uniform hemisphere \(H\) of radius \(0.28\) m is placed in the hollow part of the object to create a non-uniform hemisphere with centre \(O\). The centre of mass of the non-uniform hemisphere is \(0.15\) m from \(O\).

\includegraphics{figure_5} A uniform object is made by joining a solid cone of height 0.8 m and base radius 0.6 m and a cylinder. The cylinder has length 0.4 m and radius 0.5 m. The cylinder has a cylindrical hole of length 0.4 m and radius \(x\) m drilled through it along the axis of symmetry. A plane face of the cylinder is attached to the base of the cone so that the object has an axis of symmetry perpendicular to its base and passing through the vertex of the cone. The object is placed with points on the base of the cone and the base of the cylinder in contact with a horizontal surface (see diagram). The object is on the point of toppling.

1. Show that the centre of mass of the object is 0.15 m from the base of the cone. [3]
2. Find \(x\). [4]

[The volume of a cone is \(\frac{1}{3}\pi r^2 h\).]

\includegraphics{figure_2} A bow consists of a uniform curved portion \(AB\) of mass \(1.4 \text{ kg}\), and a uniform taut string of mass \(m \text{ kg}\) which joins \(A\) and \(B\). The curved portion \(AB\) is an arc of a circle centre \(O\) and radius \(0.8 \text{ m}\). Angle \(AOB\) is \(\frac{2}{3}\pi\) radians (see diagram). The centre of mass of the bow (including the string) is \(0.65 \text{ m}\) from \(O\). Calculate \(m\). [6]

A uniform solid cylinder has radius 0.7 m and height \(h\) m. A uniform solid cone has base radius 0.7 m and height 2.4 m. The cylinder and the cone both rest in equilibrium each with a circular face in contact with a horizontal plane. The plane is now tilted so that its inclination to the horizontal, \(θ°\), is increased gradually until the cone is about to topple.

1. Find the value of \(θ\) at which the cone is about to topple. [2]
2. Given that the cylinder does not topple, find the greatest possible value of \(h\). [2]

The plane is returned to a horizontal position, and the cone is fixed to one end of the cylinder so that the plane faces coincide. It is given that the weight of the cylinder is three times the weight of the cone. The curved surface of the cone is placed on the horizontal plane (see diagram). \includegraphics{figure_4}

1. Given that the solid immediately topples, find the least possible value of \(h\). [5]

\includegraphics{figure_6} A uniform lamina \(OABCD\) consists of a semicircle \(BCD\) with centre \(O\) and radius \(0.6\) m and an isosceles triangle \(OAB\), joined along \(OB\) (see diagram). The triangle has area \(0.36\) m\(^2\) and \(AB = AO\).

1. Show that the centre of mass of the lamina lies on \(OB\). [4]
2. Calculate the distance of the centre of mass of the lamina from \(O\). [4]

\includegraphics{figure_4} The diagram shows the cross-section of a uniform solid consisting of a cylinder of radius \(0.4\) m and height \(1.5\) m with a hemisphere of radius \(0.4\) m on top.

1. Find the distance of the centre of mass above the base of the cylinder. [5]
2. The solid can just rest in equilibrium on a plane inclined at angle \(\alpha\) to the horizontal. Find \(\alpha\). [3]

\includegraphics{figure_4} \(ABCDEF\) is the cross-section through the centre of mass of a uniform solid prism. \(ABCF\) is a rectangle in which \(AB = CF = 1.6\) m, and \(BC = AF = 0.4\) m. \(CDE\) is a triangle in which \(CD = 1.8\) m, \(CE = 0.4\) m, and angle \(DCE = 90°\). The prism stands on a rough horizontal surface. A horizontal force of magnitude \(T\) N acts at \(B\) in the direction \(CB\) (see diagram). The prism is in equilibrium.

1. Show that the distance of the centre of mass of the prism from \(AB\) is \(0.488\) m. [4]
2. Given that the weight of the prism is \(100\) N, find the greatest and least possible values of \(T\). [3]

A uniform circular disc has centre \(O\) and radius \(1.2\text{ m}\). The centre of the disc is at the origin of \(x\)- and \(y\)-axes. Two circular holes with centres at \(A\) and \(B\) are made in the disc (see diagram). The point \(A\) is on the negative \(x\)-axis with \(OA = 0.5\text{ m}\). The point \(B\) is on the negative \(y\)-axis with \(OB = 0.7\text{ m}\). The hole with centre \(A\) has radius \(0.3\text{ m}\) and the hole with centre \(B\) has radius \(0.4\text{ m}\). Find the distance of the centre of mass of the object from

1. the \(x\)-axis, [4]
2. the \(y\)-axis. [3]

The object can rotate freely in a vertical plane about a horizontal axis through \(O\).

1. Calculate the angle which \(OA\) makes with the vertical when the object rests in equilibrium. [2]

\includegraphics{figure_6} A uniform circular disc has centre \(O\) and radius \(1.2\,\text{m}\). The centre of the disc is at the origin of \(x\)- and \(y\)-axes. Two circular holes with centres at \(A\) and \(B\) are made in the disc (see diagram). The point \(A\) is on the negative \(x\)-axis with \(OA = 0.5\,\text{m}\). The point \(B\) is on the negative \(y\)-axis with \(OB = 0.7\,\text{m}\). The hole with centre \(A\) has radius \(0.3\,\text{m}\) and the hole with centre \(B\) has radius \(0.4\,\text{m}\). Find the distance of the centre of mass of the object from

1. the \(x\)-axis, [4]
2. the \(y\)-axis. [3]

The object can rotate freely in a vertical plane about a horizontal axis through \(O\).

1. Calculate the angle which \(OA\) makes with the vertical when the object rests in equilibrium. [2]

\includegraphics{figure_2} A uniform wire is bent to form an object which has a semicircular arc with diameter \(AB\) of length 1.2 m, with a smaller semicircular arc with diameter \(BC\) of length 0.6 m. The end \(C\) of the smaller arc is at the centre of the larger arc (see diagram). The two semicircular arcs of the wire are in the same plane.

1. Show that the distance of the centre of mass of the object from the line \(ACB\) is 0.191 m, correct to 3 significant figures. [3]

The object is freely suspended at \(A\) and hangs in equilibrium.

1. Find the angle between \(ACB\) and the vertical. [4]

\includegraphics{figure_4} The diagram shows the cross-section \(ABCD\) through the centre of mass of a uniform solid prism. \(AB = 0.9\) m, \(BC = 2a\) m, \(AD = a\) m and angle \(ABC =\) angle \(BAD = 90°\).

1. Calculate the distance of the centre of mass of the prism from \(AD\). [2]
2. Express the distance of the centre of mass of the prism from \(AB\) in terms of \(a\). [2]

The prism has weight 18 N and rests in equilibrium on a rough horizontal surface, with \(AD\) in contact with the surface. A horizontal force of magnitude 6 N is applied to the prism. This force acts through the centre of mass in the direction \(BC\).

1. Given that the prism is on the point of toppling, calculate \(a\). [3]