6.04d Integration: for centre of mass of laminas/solids

6. \begin{figure}[h] \end{figure} The shaded region shown in Figure 4 is bounded by the \(x\)-axis, the line with equation \(x = 9\) and the line with equation \(y = \frac { 1 } { 3 } x\). This shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution. This solid of revolution is used to model a solid right circular cone of height 9 cm and base radius 3 cm . The cone is non-uniform and the mass per unit volume of the cone at the point ( \(x , y , z\) ) is \(\lambda x \mathrm {~kg} \mathrm {~cm} ^ { - 3 }\), where \(0 \leqslant x \leqslant 9\) and \(\lambda\) is constant.

1. Find the distance of the centre of mass of the cone from its vertex. A toy is made by joining the circular plane face of the cone to the circular plane face of a uniform solid hemisphere of radius 3 cm , so that the centres of the two plane surfaces coincide. The weight of the cone is \(W\) newtons and the weight of the hemisphere is \(k W\) newtons.
   When the toy is placed on a smooth horizontal plane with any point of the curved surface of the hemisphere in contact with the plane, the toy will remain at rest.
2. Find the value of \(k\)

1. \hspace{0pt} [In this question you may quote, without proof, the formula for the distance of the centre of mass of a uniform circular arc from its centre.]

\begin{figure}[h] \end{figure} Five pieces of a uniform wire are joined together to form the rigid framework \(O A B C O\) shown in Figure 1, where

• \(O A , O B\) and \(B C\) are straight, with \(O A = O B = B C = r\)
• arc \(A B\) is one quarter of a circle with centre \(O\) and radius \(r\)
• arc \(O C\) is one quarter of a circle of radius \(r\)
• all five pieces of wire lie in the same plane
  1. Show that the centre of mass of arc \(A B\) is a distance \(\frac { 2 r } { \pi }\) from \(O A\).

Given that the distance of the centre of mass of the framework from \(O A\) is \(d\),

show that \(\mathrm { d } = \frac { 7 r } { 2 ( 3 + ) }\) The framework is freely pivoted at \(A\).
The framework is held in equilibrium, with \(A O\) vertical, by a horizontal force of magnitude \(F\) which is applied to the framework at \(C\). Given that the weight of the framework is \(W\)

find \(F\) in terms of \(W\)

5. \begin{figure}[h] \end{figure} A uniform lamina \(O A B\) is modelled by the finite region bounded by the \(x\)-axis, the \(y\)-axis and the curve with equation \(y = 9 - x ^ { 2 }\), for \(x \geqslant 0\), as shown shaded in Figure 3. The unit of length on both axes is 1 m . The area of the lamina is \(18 \mathrm {~m} ^ { 2 }\)

1. Show that the centre of mass of the lamina is 3.6 m from \(\boldsymbol { O B }\).
   [0pt] [ Solutions relying on calculator technology are not acceptable.] A light string has one end attached to the lamina at \(O\) and the other end attached to the ceiling. A second light string has one end attached to the lamina at \(A\) and the other end attached to the ceiling.
   The lamina hangs in equilibrium with the strings vertical and \(O A\) horizontal.
   The weight of the lamina is \(W\) The tension in the string attached to the lamina at \(A\) is \(\lambda W\)
2. Find the value of \(\lambda\)

7. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} The shaded region shown in Figure 5 is bounded by the line with equation \(x = a\) and the curve with equation \(x ^ { 2 } + y ^ { 2 } = 4 a ^ { 2 }\) This shaded region is rotated through \(180 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution. This solid is used to model a dome with height \(a\) metres and base radius \(\sqrt { 3 } a\) metres.
The dome is modelled as being non-uniform with the mass per unit volume of the dome at the point \(( x , y , z )\) equal to \(\frac { \lambda } { x ^ { 2 } } \mathrm {~kg} \mathrm {~m} ^ { - 3 }\), where \(a \leqslant x \leqslant 2 a\) and \(\lambda\) is a constant.

1. Show that the distance of the centre of mass of the dome from the centre of its plane face is \(\left( 4 \ln 2 - \frac { 5 } { 2 } \right) a\) metres. A solid uniform right circular cone has base radius \(\sqrt { 3 } a\) metres and perpendicular height \(4 a\) metres. A toy is formed by attaching the plane surface of the dome to the plane surface of the cone, as shown in Figure 6. The weight of the cone is \(k W\) and the weight of the dome is \(2 W\) The centre of mass of the toy is a distance \(d\) metres from the plane face of the dome.
2. Show that \(d = \frac { | k + 5 - 8 \ln 2 | } { 2 + k } a\) The toy is suspended from a point on the circumference of the plane face of the dome and hangs freely in equilibrium with the plane face of the dome at an angle \(\alpha\) to the downward vertical.
   Given that \(\tan \alpha = \frac { 1 } { 2 \sqrt { 3 } }\)
3. find the exact value of \(k\).

2. \begin{figure}[h] \end{figure} A uniform rod of length \(28 a\) is cut into seven identical rods each of length \(4 a\). These rods are joined together to form the rigid framework \(A B C D E A\) shown in Figure 1. All seven rods lie in the same plane.
The distance of the centre of mass of the framework from \(E D\) is \(d\).

1. Show that \(d = \frac { 8 \sqrt { 3 } } { 7 } a\) The weight of the framework is \(W\).
   The framework is freely pivoted about a horizontal axis through \(C\).
   The framework is held in equilibrium in a vertical plane, with \(A C\) vertical and \(A\) below \(C\), by a horizontal force that is applied to the framework at \(A\). The force acts in the same vertical plane as the framework and has magnitude \(F\).
2. Find \(F\) in terms of \(W\).

4. \begin{figure}[h] \end{figure} A uniform lamina \(O A B\) is in the shape of the region \(R\).
Region \(R\) lies in the first quadrant and is bounded by the curve with equation \(\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 36 } = 1\), the \(x\)-axis, and the \(y\)-axis, as shown shaded in Figure 3. The point \(A\) is the point of intersection of the curve and the \(x\)-axis.
The point \(B\) is the point of intersection of the curve and the \(y\)-axis.
One unit on each axis represents 1 m .
The area of \(R\) is \(6 \pi\) The centre of mass of \(R\) lies at the point with coordinates \(( \bar { x } , \bar { y } )\)

1. Use algebraic integration to show that \(\bar { x } = \frac { 16 } { 3 \pi }\)
2. Use algebraic integration to find the exact value of \(\bar { y }\) The lamina is freely suspended from \(A\) and hangs in equilibrium with \(O A\) at angle \(\theta ^ { \circ }\) to the downward vertical.
3. Find the value of \(\theta\)

6. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} The shaded region, shown in Figure 4, is bounded by the \(x\)-axis, the line with equation \(x = 6\), the line with equation \(y = 2\) and the \(y\)-axis. This region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { x }\)-axis to form a solid of revolution. This solid is used to model a non-uniform cylinder of height 6 cm and radius 2 cm . The mass per unit volume of the cylinder at the point \(( x , y , z )\) is \(\lambda ( x + 2 ) \mathrm { kg } \mathrm { cm } ^ { - 3 }\), where \(0 \leqslant x \leqslant 6\) and \(\lambda\) is a constant.

1. Show that the mass of the cylinder is \(120 \lambda \pi \mathrm {~kg}\).
2. Show that the centre of mass of the cylinder is 3.6 cm from \(O\). The point \(O\) is the centre of one end of the cylinder. The point \(A\) is the centre of the other end of the cylinder. A uniform solid hemisphere of radius 3 cm has density \(\lambda \mathrm { kg } \mathrm { cm } ^ { - 3 }\). The hemisphere is attached to the cylinder with the centre of its circular face in contact with the point \(A\) on the cylinder to form the model shown in Figure 5. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c14975b7-6afa-44ce-beab-1cba2e82b249-20_309_673_1713_696} \captionsetup{labelformat=empty} \caption{Figure 5}
   \end{figure} The model is placed with the end containing \(O\) on a rough inclined plane which is inclined at angle \(\alpha ^ { \circ }\) to the horizontal. The plane is sufficiently rough to prevent the model from sliding. The model is on the point of toppling.
3. Find the value of \(\alpha\).

1. A flag pole is 15 m long.

The flag pole is non-uniform so that, at a distance \(x\) metres from its base, the mass per unit length of the flag pole, \(m \mathrm {~kg} \mathrm {~m} ^ { - 1 }\) is given by the formula \(m = 10 \left( 1 - \frac { x } { 25 } \right)\). The flag pole is modelled as a rod.

1. Show that the mass of the flag pole is 105 kg .
2. Find the distance of the centre of mass of the flag pole from its base.

3. \begin{figure}[h] \end{figure} A uniform solid cylinder has radius \(2 a\) and height \(h ( h > a )\).
A solid hemisphere of radius \(a\) is removed from the cylinder to form the vessel \(V\).
The plane face of the hemisphere coincides with the upper plane face of the cylinder.
The centre \(O\) of the hemisphere is also the centre of the upper plane face of the cylinder, as shown in Figure 2.

1. Show that the centre of mass of \(V\) is \(\frac { 3 \left( 8 h ^ { 2 } - a ^ { 2 } \right) } { 8 ( 6 h - a ) }\) from \(O\). The vessel \(V\) is placed on a rough plane which is inclined at an angle \(\phi\) to the horizontal. The lower plane circular face of \(V\) is in contact with the inclined plane. Given that \(h = 5 a\), the plane is sufficiently rough to prevent \(V\) from slipping and \(V\) is on the point of toppling,
2. find, to three significant figures, the size of the angle \(\phi\).

5. \begin{figure}[h] \end{figure} A shop sign is modelled as a uniform rectangular lamina \(A B C D\) with a semicircular lamina removed. The semicircle has radius \(a , B C = 4 a\) and \(C D = 2 a\).
The centre of the semicircle is at the point \(E\) on \(A D\) such that \(A E = d\), as shown in Figure 3.

1. Show that the centre of mass of the sign is \(\frac { 44 a } { 3 ( 16 - \pi ) }\) from \(A D\). The sign is suspended using vertical ropes attached to the sign at \(A\) and at \(B\) and hangs in equilibrium with \(A B\) horizontal. The weight of the sign is \(W\) and the ropes are modelled as light inextensible strings.
2. Find, in terms of \(W\) and \(\pi\), the tension in the rope attached at \(B\). The rope attached at \(B\) breaks and the sign hangs freely in equilibrium suspended from \(A\), with \(A D\) at an angle \(\alpha\) to the downward vertical. Given that \(\tan \alpha = \frac { 11 } { 18 }\)
3. find \(d\) in terms of \(a\) and \(\pi\).

5 In this question you must show detailed reasoning. The region bounded by the \(x\)-axis, the \(y\)-axis, the line \(x = 4\) and the curve with equation \(\mathrm { y } = \frac { 15 } { \sqrt { \mathrm { x } ^ { 2 } + 9 } }\) is occupied by a uniform lamina. The centre of mass of the lamina is at the point \(G ( \bar { x } , \bar { y } )\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{857eca7f-c42d-49a9-ac39-a2fb5bcb9cd5-4_944_954_598_228}

1. Show that \(\bar { x } = \frac { 2 } { \ln 3 }\).
2. Determine the value of \(\bar { y }\). Give your answer correct to \(\mathbf { 3 }\) significant figures. \(P\) is the point on the curved edge of the lamina where \(x = 3\). The lamina is freely suspended from \(P\) and hangs in equilibrium in a vertical plane.
3. Determine the acute angle that the longest straight edge of the lamina makes with the vertical.

3 In this question you must show detailed reasoning. [In this question you may use the formula: Volume of cone \(= \frac { 1 } { 3 } \times\) base area × height.]
The region between the line \(\mathrm { y } = - 3 \mathrm { x } + 3 \mathrm { a }\), where \(a > 0\), the \(x\)-axis and the \(y\)-axis is rotated about the \(y\)-axis to form a uniform right circular cone C with base radius \(a\).

1. Show that the centre of mass of C is \(\frac { 3 } { 4 } a\) from its base. The cone C is fixed on top of a uniform cube, B , of edge length \(2 a\), so that there is no overlap. Fig. 3.1 shows a side view of C and B fixed together; Fig. 3.2 shows a view of C and B from above. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{37798594-8cb0-48aa-8401-090f09e25dff-3_570_323_785_246} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{37798594-8cb0-48aa-8401-090f09e25dff-3_309_319_982_753} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure} The centre of mass of the combined shape lies on the boundary of C and B .
   The density of \(B\) is not equal to the density of \(C\).
2. Determine the exact value of \(\frac { \text { density of } \mathrm { C } } { \text { density of } \mathrm { B } }\).
   [0pt] [3]

6 In this question you must show detailed reasoning. As shown in Fig. 6.1, the region R is bounded by the lines \(x = 1 , x = 2 , y = 0\) and the curve \(y = 2 x ^ { 2 }\) for \(1 \leq x \leq 2\). A uniform solid of revolution, S , is formed when R is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. \begin{figure}[h] \end{figure}

1. Show that the volume of S is \(\frac { 124 \pi } { 5 }\).
2. Show that the distance of the centre of mass of S from the centre of its smaller circular plane surface is \(\frac { 43 } { 62 }\). Fig. 6.2 shows S placed so that its smaller circular plane surface is in contact with a slope inclined at \(\alpha ^ { \circ }\) to the horizontal. S does not slip but is on the point of tipping. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a01b2e46-e213-4f20-bc2e-5852061d8b91-6_458_565_2014_694} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{figure}
3. Find the value of \(\alpha\), giving your answer in degrees correct to 3 significant figures.

2 The region bounded by the \(x\)-axis and the curve \(y = a x ( 2 - x )\), where \(a\) is a constant, is occupied by a uniform lamina \(L _ { 1 }\) (see Fig. 1). Units on the axes are metres. \begin{figure}[h] \end{figure}

1. Write down the value of the \(x\)-coordinate of the centre of mass of \(L _ { 1 }\).
2. Show that the \(y\)-coordinate of the centre of mass of \(L _ { 1 }\) is \(\frac { 2 } { 5 } a\). The mass of \(L _ { 1 }\) is \(M \mathrm {~kg}\). A uniform rectangular lamina of width 2 m and height \(a \mathrm {~m}\) is made from a different material from that of \(L _ { 1 }\) and has a mass of \(2 M \mathrm {~kg}\). A new lamina, \(L _ { 2 }\), is formed by joining the straight edge of \(L _ { 1 }\) to an edge of the rectangular lamina of length 2 m (see Fig. 2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a8c9d007-e67f-4637-9e74-630ba9a91442-2_547_273_1772_890} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} \(L _ { 2 }\) is freely suspended from one of its right-angled corners and hangs in equilibrium with its edge of length 2 m making an angle of \(20 ^ { \circ }\) with the horizontal.
3. Find the value of \(a\), giving your answer correct to 3 significant figures.

5 The triangular region shown below is rotated through \(360 ^ { \circ }\) around the \(x\)-axis, to form a solid cone. \includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-06_328_755_415_644} The coordinates of the vertices of the triangle are \(( 0,0 ) , ( 8,0 )\) and \(( 0,4 )\).
All units are in centimetres. 5

1. State an assumption that you should make about the cone in order to find the position of its centre of mass. 5
2. Using integration, prove that the centre of mass of the cone is 2 cm from its plane face.
   5
3. The cone is placed with its plane face on a rough board. One end of the board is lifted so that the angle between the board and the horizontal is gradually increased. Eventually the cone topples without sliding. 5 (c) (i) Find the angle between the board and the horizontal when the cone topples, giving your answer to the nearest degree. 5 (c) (ii) Find the range of possible values for the coefficient of friction between the cone and the board.

4 Fig. 4.1 shows a uniform lamina in the shape of a sector of a circle of radius \(r\) and angle \(2 \theta\) where \(\theta\) is in radians. The sector consists of a triangle \(O A B\) and a segment bounded by the chord \(A B\). \begin{figure}[h] \end{figure}

1. Explain why the centre of mass of the segment lies on the radius through the midpoint of \(A B\).
2. Show that the distance of the centre of mass of the segment from \(O\) is \(\frac { 2 r \sin ^ { 3 } \theta } { 3 ( \theta - \sin \theta \cos \theta ) }\). A uniform circular lamina of radius 5 units is placed with its centre at the origin, \(O\), of an \(x - y\) coordinate system. A component for a machine is made by removing and discarding a segment from the lamina. The radius of the circle from which the segment is formed is 3 units and the centre of this circle is \(O\). The centre of the straight edge of the segment has coordinates \(( 0,2 )\) and this edge is perpendicular to the \(y\)-axis (see Fig. 4.2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8859baf3-f8e8-4fbf-b54f-34f550b02c26-03_748_743_1594_251} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure}
3. Find the \(y\)-coordinate of the centre of mass of the component, giving your answer correct to 3 significant figures. \(C\) is the point on the component with coordinates \(( 0,5 )\). The component is now placed horizontally and supported only at \(O\). A particle of mass \(m \mathrm {~kg}\) is placed on the component at \(C\) and the component and particle are in equilibrium.
4. Find the mass of the component in terms of \(m\). Total Marks for Question Set 3: 40

2 The cover of a children's book is modelled as being a uniform lamina \(L . L\) occupies the region bounded by the \(x\)-axis, the curve \(y = 6 + \sin x\) and the lines \(x = 0\) and \(x = 5\) (see Fig. 2.1). The centre of mass of \(L\) is at the point \(( \bar { x } , \bar { y } )\). \begin{figure}[h] \end{figure}

1. Show that \(\bar { x } = 2.36\), correct to 3 significant figures.
2. Find \(\bar { y }\), giving your answer correct to 3 significant figures. The side of \(L\) along the \(y\)-axis is attached to the rest of the book and the book is placed on a rough horizontal plane. The attachment of the cover to the book is modelled as a hinge. The cover is held in equilibrium at an angle of \(\frac { 1 } { 3 } \pi\) radians to the horizontal by a force of magnitude \(P \mathrm {~N}\) acting at \(B\) perpendicular to the cover (see Fig. 2.2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d6bf2fa5-2f29-4632-b27d-ed8c5a0379cf-03_412_213_402_525} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
3. State two additional modelling assumptions, one about the attachment of the cover and one about the badge, which are necessary to allow the value of \(P\) to be determined.
4. Using the modelling assumptions, determine the value of \(P\) giving your answer correct to 3 significant figures.

Answer only one of the following two alternatives. **EITHER** \includegraphics{figure_11a} A uniform disc, of mass \(4m\) and radius \(a\), and a uniform ring, of mass \(m\) and radius \(2a\), each have centre \(O\). A wheel is made by fixing three uniform rods, \(OA\), \(OB\) and \(OC\), each of mass \(m\) and length \(2a\), to the disc and the ring, as shown in the diagram. Show that the moment of inertia of the wheel about an axis through \(A\), perpendicular to the plane of the wheel, is \(42ma^2\). [5] The axis through \(A\) is horizontal, and the wheel can rotate freely about this axis. The wheel is released from rest with \(O\) above the level of \(A\) and \(AO\) making an angle of \(30°\) with the horizontal. Find the angular speed of the wheel when \(AO\) is horizontal. [3] When \(AO\) is horizontal the disc becomes detached from the wheel. Find the angle that \(AO\) makes with the horizontal when the wheel first comes to instantaneous rest. [6] **OR** The continuous random variable \(T\) has probability density function given by $$f(t) = \begin{cases} 0 & t < 2, \\ \frac{2}{(t-1)^3} & t \geqslant 2. \end{cases}$$

1. Find the distribution function of \(T\), and find also P\((T > 5)\). [3]
2. Consecutive independent observations of \(T\) are made until the first observation that exceeds \(5\) is obtained. The random variable \(N\) is the total number of observations that have been made up to and including the observation exceeding \(5\). Find P\((N > E(N))\). [3]
3. Find the probability density function of \(Y\), where \(Y = \frac{1}{T-1}\). [8]

\(AB\) is a diameter of a uniform circular disc \(D\) of mass \(9m\), radius \(3a\) and centre \(O\). A lamina is formed by removing a circular disc, with centre \(O\) and radius \(a\), from \(D\). Show that the moment of inertia of the lamina, about a fixed horizontal axis \(l\) through \(A\) and perpendicular to the plane of the lamina, is \(112ma^2\). [5] A particle of mass \(3m\) is now attached to the lamina at \(B\). The system is free to rotate about the axis \(l\). The system is held with \(B\) vertically above \(A\) and is then slightly displaced and released from rest. The greatest speed of \(B\) in the subsequent motion is \(k\sqrt{(ga)}\). Find the value of \(k\), correct to 3 significant figures. [7]

\(AB\) is a diameter of a uniform circular disc \(D\) of mass \(9m\), radius \(3a\) and centre \(O\). A lamina is formed by removing a circular disc, with centre \(O\) and radius \(a\), from \(D\). Show that the moment of inertia of the lamina, about a fixed horizontal axis \(l\) through \(A\) and perpendicular to the plane of the lamina, is \(112ma^2\). [5] A particle of mass \(3m\) is now attached to the lamina at \(B\). The system is free to rotate about the axis \(l\). The system is held with \(B\) vertically above \(A\) and is then slightly displaced and released from rest. The greatest speed of \(B\) in the subsequent motion is \(k\sqrt{(ga)}\). Find the value of \(k\), correct to 3 significant figures. [7]

\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]