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

[The centre of mass of a semi-circular lamina of radius \(r\) is \(\frac{4r}{3\pi}\) from the centre] \includegraphics{figure_3} A template \(T\) consists of a uniform plane lamina \(PQRQS\), as shown in Figure 3. The lamina is bounded by two semicircles, with diameters \(SQ\) and \(QR\), and by the sides \(SP\), \(PQ\) and \(QR\) of the rectangle \(PQRS\). The point \(O\) is the mid-point of \(SR\), \(PQ = 12\) cm and \(QR = 2\) cm.

1. Show that the centre of mass of \(T\) is a distance \(\frac{4|2x^2 - 3|}{8x + 3\pi}\) cm from \(SR\). [7]

The template \(T\) is freely suspended from the point \(P\) and hangs in equilibrium. Given that \(x = 2\) and that \(\theta\) is the angle that \(PQ\) makes with the horizontal,

1. show that \(\tan \theta = \frac{48 + 9\pi}{22 + 6\pi}\). [4]

\includegraphics{figure_2} A uniform triangular lamina \(ABC\) of mass \(M\) is such that \(AB = AC\), \(BC = 2a\) and the distance of \(A\) from \(BC\) is \(h\). A line, parallel to \(BC\) and at a distance \(\frac{2h}{3}\) from \(A\), cuts \(AB\) at \(D\) and cuts \(AC\) at \(E\), as shown in Figure 2. It is given that the mass of the trapezium \(BCED\) is \(\frac{5M}{9}\).

1. Show that the centre of mass of the trapezium \(BCED\) is \(\frac{7h}{45}\) from \(BC\). [5]

\includegraphics{figure_3} The portion \(ADE\) of the lamina is folded through 180° about \(DE\) to form the folded lamina shown in Figure 3.

1. Find the distance of the centre of mass of the folded lamina from \(BC\). [4]

The folded lamina is freely suspended from \(D\) and hangs in equilibrium. The angle between \(DE\) and the downward vertical is \(\alpha\).

1. Find tan \(\alpha\) in terms of \(a\) and \(h\). [4]

1. Use algebraic integration to show that the centre of mass of a uniform solid right circular cone of height \(h\) is at a distance \(\frac{3}{4}h\) from the vertex of the cone. [You may assume that the volume of a cone of height \(h\) and base radius \(r\) is \(\frac{1}{3}\pi r^2 h\)] [5]

\includegraphics{figure_2} A uniform solid \(S\) consists of a right circular cone, of radius \(r\) and height \(5r\), fixed to a hemisphere of radius \(r\). The centre of the plane face of the hemisphere is \(O\) and this plane face coincides with the base of the cone, as shown in Figure 2.

1. Find the distance of the centre of mass of \(S\) from \(O\). [5]

The point \(A\) lies on the circumference of the base of the cone. The solid is suspended by a string attached at \(A\) and hangs freely in equilibrium.

1. Find the size of the angle between \(OA\) and the vertical. [3]

The mass of the hemisphere is \(M\). A particle of mass \(kM\) is fixed to the surface of the hemisphere on the axis of symmetry of \(S\). The solid is again suspended by the string attached at \(A\) and hangs freely in equilibrium. The axis of symmetry of \(S\) is now horizontal.

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

\includegraphics{figure_3} A container is formed by removing a right circular solid cone of height \(4l\) from a uniform solid right circular cylinder of height \(6l\). The centre \(O\) of the plane face of the cone coincides with the centre of a plane face of the cylinder and the axis of the cone coincides with the axis of the cylinder, as shown in Figure 3. The cylinder has radius \(2l\) and the base of the cone has radius \(l\).

1. Find the distance of the centre of mass of the container from \(O\). [6]

\includegraphics{figure_4} The container is placed on a plane which is inclined at an angle \(\theta°\) to the horizontal. The open face is uppermost, as shown in Figure 4. The plane is sufficiently rough to prevent the container from sliding. The container is on the point of toppling.

1. Find the value of \(\theta\). [4]

\includegraphics{figure_2} Figure 2 shows the region \(R\) bounded by the curve with equation \(y^2 = rx\), where \(r\) is a positive constant, the \(x\)-axis and the line \(x = r\). A uniform solid of revolution \(S\) is formed by rotating \(R\) through one complete revolution about the \(x\)-axis.

1. Show that the distance of the centre of mass of \(S\) from \(O\) is \(\frac{4}{5}r\). [6]

The solid is placed with its plane face on a plane which is inclined at an angle \(\alpha\) to the horizontal. The plane is sufficiently rough to prevent \(S\) from sliding. Given that \(S\) does not topple,

1. find, to the nearest degree, the maximum value of \(\alpha\). [4]

\includegraphics{figure_2} A uniform lamina occupies the region \(R\) bounded by the \(x\)-axis and the curve $$y = \sin x, \quad 0 \leq x \leq \pi,$$ as shown in Figure 2.

1. Show, by integration, that the \(y\)-coordinate of the centre of mass of the lamina is \(\frac{\pi}{8}\). [6]

\includegraphics{figure_3} A uniform prism \(S\) has cross-section \(R\). The prism is placed with its rectangular face on a table which is inclined at an angle \(\theta^{\circ}\) to the horizontal. The cross-section \(R\) lies in a vertical plane as shown in Figure 3. The table is sufficiently rough to prevent \(S\) sliding. Given that \(S\) does not topple,

1. find the largest possible value of \(\theta\). [3]

\includegraphics{figure_2} The region \(R\) is bounded by the curve with equation \(y = e^x\), the line \(x = 1\), the line \(x = 2\) and the \(x\)-axis as shown in Figure 2. A uniform solid \(S\) is formed by rotating \(R\) through \(2\pi\) about the \(x\)-axis.

1. Show that the volume of \(S\) is \(\frac{1}{2}\pi (e^4 - e^2)\). [4]
2. Find, to 3 significant figures, the \(x\)-coordinate of the centre of mass of \(S\). [6]

\includegraphics{figure_2} A model tree is made by joining a uniform solid cylinder to a uniform solid cone made of the same material. The centre \(O\) of the base of the cone is also the centre of one end of the cylinder, as shown in Fig. 2. The radius of the cylinder is \(r\) and the radius of the base of the cone is \(2r\). The height of the cone and the height of the cylinder are each \(h\). The centre of mass of the model is at the point \(G\).

1. Show that \(OG = \frac{1}{14}h\). [8]

The model stands on a desk top with its plane face in contact with the desk top. The desk top is tilted until it makes an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \frac{r}{7h}\). The desk top is rough enough to prevent slipping and the model is about to topple.

1. Find \(r\) in terms of \(h\). [4]

\includegraphics{figure_3} The shaded region \(R\) is bounded by part of the curve with equation \(y = \frac{1}{4}(x - 2)^2\), the \(x\)-axis and the \(y\)-axis, as shown in Fig. 3. The unit of length on both axes is 1 cm. A uniform solid \(S\) is made by rotating \(R\) through \(360°\) about the \(x\)-axis. Using integration,

1. calculate the volume of the solid \(S\), leaving your answer in terms of \(\pi\), [4]
2. show that the centre of mass of \(S\) is \(\frac{4}{5}\) cm from its plane face. [7]

\includegraphics{figure_4} A tool is modelled as having two components, a solid uniform cylinder \(C\) and the solid \(S\). The diameter of \(C\) is 4 cm and the length of \(C\) is 8 cm. One end of \(C\) coincides with the plane face of \(S\). The components are made of different materials. The weight of \(C\) is \(10W\) newtons and the weight of \(S\) is \(2W\) newtons. The tool lies in equilibrium with its axis of symmetry horizontal on two smooth supports \(A\) and \(B\), which are at the ends of the cylinder, as shown in Fig. 4.

1. Find the magnitude of the force of the support \(A\) on the tool. [5]

A uniform solid is formed by rotating the region enclosed between the curve with equation \(y = \sqrt{x}\), the \(x\)-axis and the line \(x = 4\), through one complete revolution about the \(x\)-axis. Find the distance of the centre of mass of the solid from the origin \(O\). [5]

A bowl consists of a uniform solid metal hemisphere, of radius \(a\) and centre \(O\), from which is removed the solid hemisphere of radius \(\frac{1}{4}a\) with the same centre \(O\).

1. Show that the distance of the centre of mass of the bowl from \(O\) is \(\frac{45}{112}a\). [5]

The bowl is fixed with its plane face uppermost and horizontal. It is now filled with liquid. The mass of the bowl is \(M\) and the mass of the liquid is \(kM\), where \(k\) is a constant. Given that the distance of the centre of mass of the bowl and liquid together from \(O\) is \(\frac{17}{48}a\),

1. Find the value of \(k\). [5]

The rudder on a ship is modelled as a uniform plane lamina having the same shape as the region \(R\) which is enclosed between the curve with equation \(y = 2x - x^2\) and the \(x\)-axis.

1. Show that the area of \(R\) is \(\frac{4}{3}\). [4]
2. Find the coordinates of the centre of mass of the lamina. [5]

An open container \(C\) is modelled as a thin uniform hollow cylinder of radius \(h\) and height \(h\) with a base but no lid. The centre of the base is \(O\).

1. Show that the distance of the centre of mass of \(C\) from \(O\) is \(\frac{1}{4}h\). [5]

The container is filled with uniform liquid. Given that the mass of the container is \(M\) and the mass of the liquid is \(M\),

1. find the distance of the centre of mass of the filled container from \(O\). [5]

The finite region bounded by the \(x\)-axis, the curve \(y = \frac{1}{x}\), the line \(x = \frac{1}{4}\) and the line \(x = 1\), is rotated through one complete revolution about the \(x\)-axis to form a uniform solid of revolution.

1. Show that the volume of the solid is \(21\pi\). [4]
2. Find the coordinates of the centre of mass of the solid. [5]

\includegraphics{figure_3} Figure 3 shows a uniform equilateral triangular lamina \(PRT\) with sides of length \(2a\).

1. Using calculus, prove that the centre of mass of \(PRT\) is at a distance \(\frac{2\sqrt{3}}{3}a\) from \(R\). [6]

\includegraphics{figure_4} The circular sector \(PQU\), of radius \(a\) and centre \(P\), and the circular sector \(TUS\), of radius \(a\) and centre \(T\), are removed from \(PRT\) to form the uniform lamina \(QRSU\) shown in Figure 4.

1. Show that the distance of the centre of mass of \(QRSU\) from \(U\) is \(\frac{2a}{3\sqrt{3} - \pi}\). [6]

\includegraphics{figure_3} A uniform solid right circular cylinder has height \(h\) and radius \(r\). The centre of one plane face is \(O\) and the centre of the other plane face is \(Y\). A cylindrical hole is made by removing a solid cylinder of radius \(\frac{1}{4}r\) and height \(\frac{1}{4}h\) from the end with centre \(O\). The axis of the cylinder removed is parallel to \(OY\) and meets the end with centre \(O\) at \(X\), where \(OX = \frac{1}{4}r\). One plane face of the cylinder removed coincides with the plane face through \(O\) of the original cylinder. The resulting solid \(S\) is shown in Figure 3.

1. Show that the centre of mass of \(S\) is at a distance \(\frac{85h}{168}\) from the plane face containing \(O\). [7]

The solid \(S\) is freely suspended from \(O\). In equilibrium the line \(OY\) is inclined at an angle arctan(17) to the horizontal.

1. Find \(r\) in terms of \(h\). [6]

\includegraphics{figure_4} A uniform square lamina \(ABCD\) of side 6 cm has a semicircular piece, with \(AB\) as diameter, removed (see diagram).

1. Find the distance of the centre of mass of the remaining shape from \(CD\). [6]

The remaining shape is suspended from a fixed point by a string attached at \(C\) and hangs in equilibrium.

1. Find the angle between \(CD\) and the vertical. [2]

\includegraphics{figure_1} A uniform lamina \(ABDC\) is bounded by two semicircular arcs \(AB\) and \(CD\), each with centre \(O\) and of radii \(3a\) and \(a\) respectively, and two straight edges, \(AC\) and \(DB\), which lie on the line \(AOB\) (see Fig. 1).

1. Show that the distance of the centre of mass of the lamina from \(O\) is \(\frac{13a}{3\pi}\). [5]

\includegraphics{figure_2} The lamina has mass 3 kg and is freely pivoted to a fixed point at \(A\). The lamina is held in equilibrium with \(AB\) vertical by means of a light string attached to \(B\). The string lies in the same plane as the lamina and is at an angle of \(40°\) below the horizontal (see Fig. 2).

1. Calculate the tension in the string. [3]
2. Find the direction of the force acting on the lamina at \(A\). [4]

A box is to be assembled in the shape of the cuboid shown in Fig. 3.1. The lengths are in centimetres. All the faces are made of the same uniform, rigid and thin material. All coordinates refer to the axes shown in this figure. \includegraphics{figure_3.1}

1. The four vertical faces OAED, ABFE, FGCB and CODG are assembled first to make an open box without a base or a top. Write down the coordinates of the centre of mass of this open box. [1]

The base OABC is added to the vertical faces.

1. Write down the \(x\)- and \(y\)-coordinates of the centre of mass of the box now. Show that the \(z\)-coordinate is now 1.875. [5]

The top face FGDE is now added. This is a lid hinged to the rest of the box along the line FG. The lid is open so that it hangs in a vertical plane touching the face FGCB.

1. Show that the coordinates of the centre of mass of the box in this situation are \((10, 2.4, 2.1)\). [6]

[This question is continued on the facing page.] The box, with the lid still touching face FGCB, is now put on a sloping plane with the edge OA horizontal and the base inclined at \(30°\) to the horizontal, as shown in Fig. 3.2. \includegraphics{figure_3.2} The weight of the box is 40 N. A force \(P\) N acts parallel to the plane and is applied to the mid-point of FG at \(90°\) to FG. This force tends to push the box down the plane. The box does not slip and is on the point of toppling about the edge AO.

1. Show that the clockwise moment of the weight of the box about the edge AO is about 0.411 Nm. [4]
2. Calculate the value of \(P\). [2]

A lamina is made from uniform material in the shape shown in Fig. 3.1. BCJA, DZOJ, ZEIO and FGHI are all rectangles. The lengths of the sides are shown in centimetres. \includegraphics{figure_3}

1. Find the coordinates of the centre of mass of the lamina, referred to the axes shown in Fig. 3.1. [5]

The rectangles BCJA and FGHI are folded through 90° about the lines CJ and FI respectively to give the fire-screen shown in Fig. 3.2.

1. Show that the coordinates of the centre of mass of the fire-screen, referred to the axes shown in Fig. 3.2, are (2.5, 0, 57.5). [4]

The \(x\)- and \(y\)-axes are in a horizontal floor. The fire-screen has a weight of 72 N. A horizontal force \(P\) N is applied to the fire-screen at the point Z. This force is perpendicular to the line DE in the positive \(x\) direction. The fire-screen is on the point of tipping about the line AH.

1. Calculate the value of \(P\). [5]

The coefficient of friction between the fire-screen and the floor is \(\mu\).

1. For what values of \(\mu\) does the fire-screen slide before it tips? [4]

You are given that the centre of mass, G, of a uniform lamina in the shape of an isosceles triangle lies on its axis of symmetry in the position shown in Fig. 4.1. \includegraphics{figure_4_1} Fig. 4.2 shows the cross-section OABCD of a prism made from uniform material. OAB is an isosceles triangle, where OA = AB, and OBCD is a rectangle. The distance OD is \(h\) cm, where \(h\) can take various positive values. All coordinates refer to the axes Ox and Oy shown. The units of the axes are centimetres. \includegraphics{figure_4_2}

1. Write down the coordinates of the centre of mass of the triangle OAB. [1]
2. Show that the centre of mass of the region OABCD is \(\left(\frac{12-h^2}{2(h+3)}, 2.5\right)\). [6]

The \(x\)-axis is horizontal. The prism is placed on a horizontal plane in the position shown in Fig. 4.2.

1. Find the values of \(h\) for which the prism would topple. [3]

The following questions refer to the case where \(h = 3\) with the prism held in the position shown in Fig. 4.2. The cross-section OABCD contains the centre of mass of the prism. The weight of the prism is 15 N. You should assume that the prism does not slide.

1. Suppose that the prism is held in this position by a vertical force applied at A. Given that the prism is on the point of tipping clockwise, calculate the magnitude of this force. [3]
2. Suppose instead that the prism is held in this position by a force in the plane of the cross-section OABCD, applied at 30° below the horizontal at C, as shown in Fig. 4.3. Given that the prism is on the point of tipping anti-clockwise, calculate the magnitude of this force. [4]

\includegraphics{figure_4_3}

A uniform solid right circular cone has height \(h\) and base radius \(r\). The top part of the cone is removed by cutting through the cone parallel to the base at a height \(\frac{h}{2}\). \includegraphics{figure_5}

1. Show that the centre of mass of the remaining solid is at a height \(\frac{11h}{56}\) above the base, along its axis of symmetry. [7 marks]

The remaining part of the solid is suspended from the point \(D\) on the circumference of its smaller circular face, and the axis of symmetry then makes an angle \(\alpha\) with the vertical, where \(\tan \alpha = \frac{1}{2}\).

1. Find the value of the ratio \(h : r\). [5 marks]

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

The figure shows the vertical cross-section of a rough solid hemisphere at rest on a rough inclined plane inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{10}\). \includegraphics{figure_7} \begin{enumerate}[label=(\alph*)] \setcounter{enumi}{1} \item Indicate on a copy of the figure the three forces acting on the hemisphere, clearly stating what they are, and paying special attention to their lines of action. [3 marks] \item Given that the plane face containing the diameter \(AB\) makes an angle \(\alpha\) with the vertical, show that \(\cos \alpha = \frac{4}{5}\). [6 marks] \end{enumerate]

A uniform solid sphere, of radius \(a\), is divided into two sections by a plane at a distance \(\frac{a}{2}\) from the centre and parallel to a diameter.

1. Show that the centre of gravity of the smaller cap from its plane face is \(\frac{7a}{40}\). [9 marks]

This smaller cap is now placed on an inclined plane whose angle of inclination to the horizontal is \(\theta\). The plane is rough enough to prevent slipping and the cap rests with its curved surface in contact with the plane.

1. If the maximum value of \(\theta\) for which this is possible without the cap turning over is 30°, find the corresponding maximum inclination of the axis of symmetry of the cap to the vertical. [6 marks]

A container consists of two sections made from the same material: a hollow portion formed by removing a cone (shaded in the figure) from a solid cylinder of radius \(r\) and height \(h\), and a solid hemisphere of radius \(r\). The vertex of the removed cone coincides with the centre \(O\) of the horizontal plane face of the hemisphere. \(CD\) is a diameter of this plane face. \includegraphics{figure_7}

1. Show that the distance of the centre of mass of the container from the plane face of the hemisphere is \(\left|\frac{3}{8}(h-r)\right|\). Explain why the modulus sign is necessary. [9 marks]
2. Find the ratio \(h : r\) in each of the following cases:
   1. When the container is suspended from the point \(C\), the angle made by \(CD\) with the vertical is equal to the angle which \(CD\) would make with the vertical if the hemisphere alone were suspended from \(C\). [4 marks]
   2. The container is able to stand without toppling in any position when it is placed with the surface of the hemispherical part in contact with a smooth horizontal table. [3 marks]