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

In this question you must show detailed reasoning. \includegraphics{figure_7} Fig. 7 shows the curve with equation \(y = \frac{2}{3}\ln x\). The region R, shown shaded in Fig. 7, is bounded by the curve and the lines \(x = 0\), \(y = 0\) and \(y = \ln 2\). A uniform solid of revolution is formed by rotating the region R completely about the \(y\)-axis. Find the exact \(x\)-coordinate of the centre of mass of the solid. [8]

[In this question you may use the facts that for a uniform solid right circular cone of height \(h\) and base radius \(r\) the volume is \(\frac{1}{3}\pi r^2 h\) and the centre of mass is \(\frac{1}{4}h\) above the base on the line from the centre of the base to the vertex.] \includegraphics{figure_9} The diagram shows the shaded region S bounded by the curve \(y = e^{\frac{1}{4}x}\) for \(0 \leq x \leq 2\), the \(x\)-axis, the \(y\)-axis, and the line \(y = \frac{1}{4}e(6-x)\). The line \(y = \frac{1}{4}e(6-x)\) meets the curve \(y = e^{\frac{1}{4}x}\) at the point A with coordinates \((2, e)\). The region S is rotated through \(2\pi\) radians about the \(x\)-axis to form a uniform solid of revolution T.

1. Show that the \(x\)-coordinate of the centre of mass of T is \(\frac{3(5e^2 + 1)}{7e^2 - 3}\). [8]

Solid T is freely suspended from A and hangs in equilibrium.

1. Determine the angle between AO, where O is the origin, and the vertical. [3]

\includegraphics{figure_8} The diagram shows the shaded region R bounded by the curve \(y = \sqrt{3x + 4}\), the \(x\)-axis, the \(y\)-axis, and the straight line that passes through the points \((k, 0)\) and \((4, 4)\), where \(0 < k < 4\). Region R is occupied by a uniform lamina.

1. Determine, in terms of \(k\), an expression for the \(y\)-coordinate of the centre of mass of the lamina. Give your answer in the form \(\frac{a + bk}{c + dk}\), where \(a\), \(b\), \(c\) and \(d\) are integers to be determined. [6]
2. Show that the \(y\)-coordinate of the centre of mass of the lamina cannot be \(\frac{3}{2}\). [2]

The region bounded by the curve \(y = x^3 - 3x^2 + 4\), the positive \(x\)-axis and the positive \(y\)-axis is occupied by a uniform lamina L. The vertices of L are O, A and B, where O is the origin, A is a point on the positive \(x\)-axis and B is a point on the positive \(y\)-axis (see diagram). \includegraphics{figure_7}

1. Determine the coordinates of the centre of mass of L. [5]

The lamina L is the cross-section through the centre of mass of a uniform solid prism M. The prism M is placed on an inclined plane, which makes an angle of \(30°\) with the horizontal, so that OA lies along a line of greatest slope of the plane with O lower down the plane than A. It is given that M does not slip on the plane.

1. Determine whether M will topple in this case. Give a reason to support your answer. [2]

The prism M is now placed on the same inclined plane so that OB lies along a line of greatest slope of the plane with O lower down the plane than B. It is given that M still does not slip on the plane.

1. Determine whether M will topple in this case. Give a reason to support your answer. [2]

[In this question, you may use the fact that the volume of a right circular cone of base radius \(r\) and height \(h\) is \(\frac{1}{3}\pi r^2 h\).]

1. By using integration, show that the centre of mass of a uniform solid right circular cone of height \(h\) and base radius \(r\) is at a distance \(\frac{3}{4}h\) from the vertex. [5]

\includegraphics{figure_8} Fig. 8 shows the side view of a toy formed by joining a uniform solid circular cylinder of radius \(r\) and height \(2r\) to a uniform solid right circular cone, made of the same material as the cylinder, of radius \(r\) and height \(r\). The toy is placed on a horizontal floor with the curved surface of the cone in contact with the floor.

1. Determine whether the toy will topple. [7]
2. Explain why it is not necessary to know whether the floor is rough or smooth in answering part (b). [1]

The region bounded by the \(x\)-axis and the curve \(y = \frac{1}{2}k(1-x^2)\) for \(-1 \leq x \leq 1\) is occupied by a uniform lamina, as shown in Fig. 11.1. \includegraphics{figure_11_1}

1. In this question you must show detailed reasoning. Show that the centre of mass of the lamina is at \(\left(0, \frac{1}{5}k\right)\). [7]

A shop sign is modelled as a uniform lamina in the form of the lamina in part (i) attached to a rectangle ABCD, where AB = 2 and BC = 1. The sign is suspended by two vertical wires attached at A and D, as shown in Fig. 11.2. \includegraphics{figure_11_2}

1. Show that the centre of mass of the sign is at a distance $$\frac{2k^2 + 10k + 15}{10k + 30}$$ from the midpoint of CD. [4]

The tension in the wire at A is twice the tension in the wire at D.

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

The triangular region shown below is rotated through \(360°\) around the \(x\)-axis, to form a solid cone. \includegraphics{figure_1} The coordinates of the vertices of the triangle are \((0, 0)\), \((8, 0)\) and \((0, 4)\). All units are in centimetres.

1. State an assumption that you should make about the cone in order to find the position of its centre of mass. [1 mark]
2. Using integration, prove that the centre of mass of the cone is \(2\)cm from its plane face. [5 marks]
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.
   1. Find the angle between the board and the horizontal when the cone topples, giving your answer to the nearest degree. [2 marks]
   2. Find the range of possible values for the coefficient of friction between the cone and the board. [3 marks]

Numerical (calculator) integration is not acceptable in this question. \includegraphics{figure_4} The shaded region \(OAB\) in Figure 2 is bounded by the \(x\)-axis, the line with equation \(x = 4\) and the curve with equation \(y = \frac{1}{4}(x-2)^3 + 2\). The point \(A\) has coordinates \((4, 4)\) and the point \(B\) has coordinates \((4, 0)\). A uniform lamina \(L\) has the shape of \(OAB\). The unit of length on both axes is one centimetre. The centre of mass of \(L\) is at the point with coordinates \((\bar{x}, \bar{y})\). Given that the area of \(L\) is \(8\)cm²,

1. show that \(\bar{y} = \frac{8}{7}\). [4]
2. The lamina is freely suspended from \(A\) and hangs in equilibrium with \(AB\) at an angle \(\theta°\) to the downward vertical. Find the value of \(\theta\). [7]

The diagram shows the cross-section through the centre of mass of a uniform solid prism. The cross-section is a trapezium ABCD with AB and CD perpendicular to AD. The lengths of AB and AD are each 5 cm and the length of CD is \((a + 5)\) cm. \includegraphics{figure_7}

1. Show the distance of the centre of mass of the prism from AD is $$\frac{a^2 + 15a + 75}{3(a + 10)} \text{ cm.}$$ [5]

The prism is placed with the face containing AB in contact with a horizontal surface.

1. Find the greatest value of \(a\) for which the prism does not topple. [3]

The prism is now placed on an inclined plane which makes an angle \(\theta^o\) with the horizontal. AB lies along a line of greatest slope with B higher than A.

1. Using the value for \(a\) found in part (ii), and assuming the prism does not slip down the plane, find the great value of \(\theta\) for which the prism does not topple. [6]

A uniform solid hemisphere has radius 0.4 m. A uniform solid cone, made of the same material, has base radius 0.4 m and height 1.2 m. A solid, \(S\), is formed by joining the hemisphere and the cone so that their circular faces coincide. \(O\) is the centre of the joint circular face and \(V\) is the vertex of the cone. \(G\) is the centre of mass of \(S\).

1. Explain briefly why \(G\) lies on the line through \(O\) and \(V\). [1]
2. Show that the distance of \(G\) from \(O\) is 0.12 m. (The volumes of a hemisphere and cone are \(\frac{2}{3}\pi r^3\) and \(\frac{1}{3}\pi r^2 h\) respectively.) [5]

\includegraphics{figure_7} \(S\) is suspended from two light vertical strings, one attached to \(V\) and the other attached to a point on the circumference of the joint circular face, and hangs in equilibrium with \(OV\) horizontal (see diagram).

1. The weight of \(S\) is \(W\). Find the magnitudes of the tensions in the strings in terms of \(W\). [3]

1 \includegraphics[max width=\textwidth, alt={}, center]{c1aae41e-530c-4db4-8959-8afe223c4dbc-2_547_423_260_861} Three identical uniform rods, \(A B , B C\) and \(C D\), each of mass \(M\) and length \(2 a\), are rigidly joined to form three sides of a square. A uniform circular disc, of mass \(\frac { 2 } { 3 } M\) and radius \(a\), has the opposite ends of one of its diameters attached to \(A\) and \(D\) respectively. The disc and the rods all lie in the same plane (see diagram). Find the moment of inertia of the system about the axis \(A D\).