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

1. Prove that the centre of mass of a uniform solid hemisphere of radius \(r\) is at a distance \(\frac{3r}{8}\) from its plane face. [7 marks]

A solid cylinder of radius \(\frac{3r}{4}\) and height \(kr\), where \(k < 1\), is welded to a uniform hemisphere of radius \(r\) made of the same material, so that their axes of symmetry coincide. The figure shows the cross section of the resulting solid. If the centre of mass of this solid is at \(O\), the centre of the plane face of the hemisphere, \includegraphics{figure_5}

1. find the value of \(k\). [4 marks]

1. Prove that the centre of mass of a uniform solid right circular cone of height \(h\) and base radius \(r\) is at a distance \(\frac{3h}{4}\) from the vertex. [7 marks]

An item of confectionery consists of a thin wafer in the form of a hollow right circular cone of height \(h\) and mass \(m\), filled with solid chocolate, also of mass \(m\), to a depth of \(kh\) as shown. The centre of mass of the item is at \(O\), the centre of the horizontal plane face of the chocolate. \includegraphics{figure_3}

1. Show that \(k = \frac{8h}{15}\). [3 marks]

In the packaging process, the cone has to move on a conveyor belt inclined at an angle \(\alpha\) to the horizontal as shown. If the belt is rough enough to prevent sliding, and the maximum value of \(\alpha\) for which the cone does not topple is \(45°\), \includegraphics{figure_4}

1. find the radius of the base of the cone in terms of \(h\). [4 marks]

An ornamental tower is made from a solid right circular cylinder of mass \(M\) and height \(h\) by removing three identical cylindrical sections, each of height \(\frac{h}{8}\), equally spaced above a base of height \(\frac{h}{4}\), as shown. The tower is held in position by light, thin vertical strips \(AB\) and \(CD\). \includegraphics{figure_2} Find the distance of the centre of mass of the tower from its horizontal base. [7 marks]

A uniform lamina is in the shape of the region enclosed by the coordinate axes and the curve with equation \(y = 1 + \cos x\), as shown. \includegraphics{figure_4}

1. Show by integration that the centre of mass of the lamina is at a distance \(\frac{\pi^2 - 4}{2\pi}\) from the \(y\)-axis. [9 marks]

Given that the centre of mass is at a distance 0·75 units from the \(x\)-axis, and that \(P\) is the point \((0, 2)\) and \(O\) is the origin \((0, 0)\),

1. find, to the nearest degree, the angle between the line \(OP\) and the vertical when the lamina is freely suspended from \(P\). [3 marks]

\includegraphics{figure_2} Two uniform rods \(AB\) and \(AC\), each of mass \(2m\) and length \(2L\), are freely jointed at \(A\). The mid-points of the rods are \(D\) and \(E\) respectively. A light inextensible string of length \(s\) is fixed to \(E\) and passes round small, smooth light pulleys at \(D\) and \(A\). A particle \(P\) of mass \(m\) is attached to the other end of the string and hangs vertically. The points \(A\), \(B\) and \(C\) lie in the same vertical plane with \(B\) and \(C\) on a smooth horizontal surface. The angles \(PAB\) and \(PAC\) are each equal to \(\theta\) (\(\theta > 0\)), as shown in Fig. 2.

1. Find the length of \(AP\) in terms of \(s\), \(L\) and \(\theta\). [2]
2. Show that the potential energy \(V\) of the system is given by $$V = 2mgL(3\cos\theta + \sin\theta) + \text{constant}.$$ [4]
3. Hence find the value of \(\theta\) for which the system is in equilibrium. [4]
4. Determine whether this position of equilibrium is stable or unstable. [4]

\includegraphics{figure_1} A smooth wire with ends \(A\) and \(B\) is in the shape of a semi-circle of radius \(a\). The mid-point of \(AB\) is \(O\) and is fixed in a vertical plane and hangs below \(AB\) which is horizontal. A small ring \(R\), of mass \(m\sqrt{2}\), is threaded on the wire and is attached to two light inextensible strings. The other end of each string is attached to a particle of mass \(\frac{3m}{2}\). The particles hang vertically under gravity, as shown in Figure 1.

1. Show that, when the radius \(OR\) makes an angle \(2\theta\) with the vertical, the potential energy, \(V\), of the system is given by $$V = \sqrt{2}mga(3 \cos \theta - \cos 2\theta) + \text{constant}.$$ [7]
2. Find the values of \(\theta\) for which the system is in equilibrium. [6]
3. Determine the stability of the position of equilibrium for which \(\theta > 0\). [4]

A straight rod \(AB\) of length \(a\) has variable density. At a distance \(x\) from \(A\) its mass per unit length is \(k(a + 2x)\), where \(k\) is a positive constant. Find the distance from \(A\) of the centre of mass of the rod. [5]

The region bounded by the \(x\)-axis, the lines \(x = 1\) and \(x = 2\) and the curve \(y = \frac{1}{x^2}\) for \(1 \leq x \leq 2\), is occupied by a uniform lamina of mass 24 kg. The unit of length is the metre. Find the moment of inertia of this lamina about the \(x\)-axis. [8]

The region bounded by the curve \(y = 2e^{\frac{1}{2}x}\) for \(0 \leq x \leq 2\), the \(x\)-axis, the \(y\)-axis and the line \(x = 2\), is occupied by a uniform lamina.

1. Find the exact value of the \(y\)-coordinate of the centre of mass of the lamina. [6]

As shown in the diagram below, a uniform lamina occupies the closed region bounded by the \(x\)-axis, the \(y\)-axis and the curve \(y = f(x)\) where $$f(x) = \begin{cases} 2e^{\frac{1}{2}x} & 0 \leq x \leq 2, \\ \frac{2}{3}(5-x)e & 2 \leq x \leq 5. \end{cases}$$ \includegraphics{figure_4}

1. Find the exact value of the \(x\)-coordinate of the centre of mass of the lamina. [7]

\includegraphics{figure_4} **Figure 1** A uniform lamina of mass \(M\) is in the shape of a right-angled triangle \(OAB\). The angle \(OAB\) is \(90°\), \(OA = a\) and \(AB = 2a\), as shown in Figure 1.

1. Prove, using integration, that the moment of inertia of the lamina \(OAB\) about the edge \(OA\) is \(\frac{8}{3}Ma^2\). (You may assume without proof that the moment of inertia of a uniform rod of mass \(m\) and length \(2l\) about an axis through one end and perpendicular to the rod is \(\frac{4}{3}ml^2\).) [6]

The lamina \(OAB\) is free to rotate about a fixed smooth horizontal axis along the edge \(OA\) and hangs at rest with \(B\) vertically below \(A\). The lamina is then given a horizontal impulse of magnitude \(J\). The impulse is applied to the lamina at the point \(B\), in a direction which is perpendicular to the plane of the lamina. Given that the lamina first comes to instantaneous rest after rotating through an angle of \(120°\),

1. find an expression for \(J\), in terms of \(M\), \(a\) and \(g\). [7]

A uniform lamina \(ABC\) of mass \(m\) is in the shape of an isosceles triangle with \(AB = AC = 5a\) and \(BC = 8a\).

1. Show, using integration, that the moment of inertia of the lamina about an axis through \(A\), parallel to \(BC\), is \(\frac{9}{2}ma^2\). [6]

The foot of the perpendicular from \(A\) to \(BC\) is \(D\). The lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis which passes through \(D\) and is perpendicular to the plane of the lamina. The lamina is released from rest when \(DA\) makes an angle \(\alpha\) with the downward vertical. It is given that the moment of inertia of the lamina about an axis through \(D\), perpendicular to \(BC\) and in the plane of the lamina, is \(\frac{8}{3}ma^2\).

1. Find the angular acceleration of the lamina when \(DA\) makes an angle \(\theta\) with the downward vertical. [8]

Given that \(\alpha\) is small,

1. find an approximate value for the period of oscillation of the lamina about the vertical. [2]

Prove, using integration, that the moment of inertia of a uniform solid right circular cone, of mass \(M\) and base radius \(a\), about its axis is \(\frac{3}{10}Ma^2\). [You may assume, without proof, that the moment of inertia of a uniform circular disc, of mass \(m\) and radius \(r\), about an axis through its centre and perpendicular to its plane is \(\frac{1}{2}mr^2\).] [10]

A uniform rod \(PQ\), of mass \(m\) and length \(3a\), is free to rotate about a fixed smooth horizontal axis \(L\), which passes through the end \(P\) of the rod and is perpendicular to the rod. The rod hangs at rest in equilibrium with \(Q\) vertically below \(P\). One end of a light inextensible string of length \(2a\) is attached to the rod at \(P\) and the other end is attached to a particle of mass \(3m\). The particle is held with the string taut, and horizontal and perpendicular to \(L\), and is then released. After colliding, the particle sticks to the rod forming a body \(B\).

1. Show that the moment of inertia of \(B\) about \(L\) is \(15ma^2\). [2]

1. Show that \(B\) first comes to instantaneous rest after it has turned through an angle \(\arccos\left(\frac{9}{25}\right)\). [10]

1. A uniform lamina of mass \(m\) is in the shape of a triangle \(ABC\). The perpendicular distance of \(C\) from the line \(AB\) is \(h\). Prove, using integration, that the moment of inertia of the lamina about \(AB\) is \(\frac{1}{6}mh^2\). [7]

1. Deduce the radius of gyration of a uniform square lamina of side \(2a\), about a diagonal. [3]

The points \(X\) and \(Y\) are the mid-points of the sides \(RQ\) and \(RS\) respectively of a square \(PQRS\) of side \(2a\). A uniform lamina of mass \(M\) is in the shape of \(PQXYS\).

1. Show that the moment of inertia of this lamina about \(XY\) is \(\frac{79}{84}Ma^2\). [6]

A pendulum consists of a uniform rod \(PQ\), of mass \(3m\) and length \(2a\), which is rigidly fixed at its end \(Q\) to the centre of a uniform circular disc of mass \(m\) and radius \(a\). The rod is perpendicular to the plane of the disc. The pendulum is free to rotate about a fixed smooth horizontal axis \(L\) which passes through the end \(P\) of the rod and is perpendicular to the rod.

1. Show that the moment of inertia of the pendulum about \(L\) is \(\frac{33}{4}ma^2\). [5]

The pendulum is released from rest in the position where \(PQ\) makes an angle \(\alpha\) with the downward vertical. At time \(t\), \(PQ\) makes an angle \(\theta\) with the downward vertical.

1. Show that the angular speed, \(\dot{\theta}\), of the pendulum satisfies $$\dot{\theta}^2 = \frac{40g(\cos\theta - \cos\alpha)}{33a}$$ [4]
2. Hence, or otherwise, find the angular acceleration of the pendulum. [3]

Given that \(\alpha = \frac{\pi}{20}\) and that \(PQ\) has length \(\frac{8}{33}\) m,

1. find, to 3 significant figures, an approximate value for the angular speed of the pendulum \(0.2\) s after it has been released from rest. [5]

1. A uniform lamina of mass \(m\) is in the shape of a triangle \(ABC\). The perpendicular distance of \(C\) from the line \(AB\) is \(h\). Prove, using integration, that the moment of inertia of the lamina about \(AB\) is \(\frac{1}{6}mh^2\). [7]
2. Deduce the radius of gyration of a uniform square lamina of side \(2a\), about a diagonal. [3]

The points \(X\) and \(Y\) are the mid-points of the sides \(RQ\) and \(RS\) respectively of a square \(PQRS\) of side \(2a\). A uniform lamina of mass \(M\) is in the shape of \(PQXYS\).

1. Show that the moment of inertia of this lamina about \(XY\) is \(\frac{79}{84}Ma^2\). [6]

A uniform rod \(AB\), of mass \(m\) and length \(2a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\). The axis \(L\) is perpendicular to the rod and passes through the point \(P\) of the rod, where \(AP = \frac{2}{3}a\).

1. Find the moment of inertia of the rod about \(L\). [3]

The rod is held at rest with \(B\) vertically above \(P\) and is slightly displaced.

1. Find the angular speed of the rod when \(PB\) makes an angle \(\theta\) with the upward vertical. [4]
2. Find the magnitude of the angular acceleration of the rod when \(PB\) makes an angle \(\theta\) with the upward vertical. [3]
3. Find, in terms of \(g\) and \(a\) only, the angular speed of the rod when the force acting on the rod at \(P\) is perpendicular to the rod. [5]

1. Prove, using integration, that the moment of inertia of a uniform circular disc, of mass \(m\) and radius \(a\), about an axis through the centre of the disc and perpendicular to the plane of the disc is \(\frac{1}{2}ma^2\). [5]

[You may assume without proof that the moment of inertia of a uniform hoop of mass \(m\) and radius \(r\) about an axis through its centre and perpendicular to its plane is \(mr^2\).] \includegraphics{figure_1} A uniform plane shape \(S\) of mass \(M\) is formed by removing a uniform circular disc with centre \(O\) and radius \(a\) from a uniform circular disc with centre \(O\) and radius \(2a\), as shown in Figure 1. The shape \(S\) is free to rotate about a fixed smooth axis \(L\), which passes through \(O\) and lies in the plane of the shape.

1. Show that the moment of inertia of \(S\) about \(L\) is \(\frac{5}{4}Ma^2\). [4]

The shape \(S\) is at rest in a horizontal plane and is free to rotate about the axis \(L\). A particle of mass \(M\) falls vertically and strikes \(S\) at the point \(A\), where \(OA = \frac{3}{2}a\) and \(OA\) is perpendicular to \(L\). The particle adheres to \(S\) at \(A\). Immediately before the particle strikes \(S\) the speed of the particle is \(u\).

1. Find, in terms of \(M\) and \(u\), the loss in kinetic energy due to the impact. [8]

1. Prove, using integration, that the moment of inertia of a uniform rod, of mass \(m\) and length \(2a\), about an axis perpendicular to the rod through its centre is \(\frac{1}{3}ma^2\). [3]

A uniform wire of mass \(4m\) and length \(8a\) is bent into the shape of a square.

1. Find the moment of inertia of the square about the axis through the centre of the square perpendicular to its plane. [4]

A uniform circular disc, of mass \(2m\) and radius \(a\), is free to rotate in a vertical plane about a fixed, smooth horizontal axis through a point of its circumference. The axis is perpendicular to the plane of the disc. The disc hangs in equilibrium. A particle \(P\) of mass \(m\) is moving horizontally in the same plane as the disc with speed \(\sqrt{20ag}\). The particle strikes, and adheres to, the disc at one end of its horizontal diameter.

1. Find the angular speed of the disc immediately after \(P\) strikes it. [7]
2. Verify that the disc will turn through an angle of \(90°\) before first coming to instantaneous rest. [3]

A uniform square lamina \(ABCD\) of side \(a\) and mass \(m\) is free to rotate in vertical plane about a horizontal axis through \(A\). The axis is perpendicular to the plane of the lamina. The lamina is released from rest when \(t = 0\) and \(AC\) makes a small angle with the downward vertical through \(A\).

1. Show that the moment of inertia of the lamina about the axis is \(\frac{2}{3}ma^2\). [3]
2. Show that the motion of the lamina is approximately simple harmonic. [5]
3. Find the time \(t\) when \(AC\) is first vertical. [2]

A uniform rod \(AB\) of mass \(m\) and length \(4a\) is free to rotate in a vertical plane about a horizontal axis through the point \(O\) of the rod, where \(OA = a\). The rod is slightly disturbed from rest when \(B\) is vertically above \(A\).

1. Find the magnitude of the angular acceleration of the rod when it is horizontal. [4]
2. Find the angular speed of the rod when it is horizontal. [2]
3. Calculate the magnitude of the force acting on the rod at \(O\) when the rod is horizontal. [5]

A uniform lamina has the shape of the region enclosed by the curve \(y = x^2 + 1\) and the lines \(x = 0\), \(x = 4\) and \(y = 0\) The diagram below shows the lamina. \includegraphics{figure_5}

1. Find the coordinates of the centre of mass of the lamina, giving your answer in exact form. [4 marks]
2. The lamina is suspended from the point where the curve intersects the line \(x = 4\) and hangs in equilibrium. Find the angle between the vertical and the longest straight edge of the lamina, giving your answer correct to the nearest degree. [3 marks]

The finite region enclosed by the line \(y = kx\), the \(x\)-axis and the line \(x = 5\) is rotated through 360° around the \(x\) axis to form a solid cone.

1. 1. Use integration to show that the position of the centre of mass of the cone is independent of \(k\) [4 marks]
   2. State the distance between the base of the cone and its centre of mass. [1 mark]
2. State one assumption that you have made about the cone. [1 mark]
3. The plane face of the cone is placed on a rough inclined plane. The coefficient of friction between the cone and the plane is 0.8 The angle between the plane and the horizontal is gradually increased from 0° Find the range of values of \(k\) for which the cone slides before it topples. [4 marks]