6.04c Composite bodies: centre of mass

\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}

The figure show a wine glass consisting of a hemispherical cup of radius \(r\), a cylindrical solid stem of height \(r\) and a circular base of radius \(r\). The cup has mass \(M\) and the stem has mass \(m\). Modelling the cup as a thin, uniform hemispherical shell, the base as a uniform lamina made of the same thin material as the cup, and the stem as a uniform solid cylinder,

1. show that the mass of the circular base is \(\frac{1}{2}M\). [1 mark]

Given that the centre of mass of the glass is at a distance \(\frac{13r}{14}\) from the base along the vertical axis of symmetry,

1. express \(M\) in terms of \(m\). [6 marks]

If the cup is now filled with liquid whose mass is \(2M\),

1. show that the position of the centre of mass rises through a distance \(\frac{13r}{35}\). [6 marks]
2. State an assumption that you have made about the liquid. [1 mark]

\includegraphics{figure_6}

\(ABCD\) is a uniform lamina in the shape of a kite with \(BA = BC = 0.37\) m, \(DA = DC = 0.91\) m and \(AC = 0.7\) m (see diagram). The centre of mass of \(ABCD\) is \(G\). \includegraphics{figure_4}

1. Explain why \(G\) lies on \(BD\). [1]
2. Show that the distance of \(G\) from \(B\) is \(0.36\) m. [4]

The lamina \(ABCD\) is freely suspended from the point \(A\).

1. Determine the acute angle that \(CD\) makes with the horizontal, stating which of \(C\) or \(D\) is higher. [4]

\includegraphics{figure_4} Fig. 4 shows a uniform lamina ABCDE such that ABDE is a rectangle and BCD is an isosceles triangle. AB = 5a, AE = 4a and BC = CD. The point F is the midpoint of BD and FC = a.

1. Find, in terms of \(a\), the exact distance of the centre of mass of the lamina from AE. [4]

The lamina is freely suspended from B and hangs in equilibrium.

1. Find the angle between AB and the downward vertical. [2]

\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]

\includegraphics{figure_3} A circular hole with centre C and radius \(r\) m, where \(r < 0.5\), is cut in a uniform circular disc with centre O and radius 0.5 m. The hole touches the rim of the disc at A (see diagram). The centre of mass, G, of the remainder of the disc is on the rim of the hole. Determine the value of \(r\). [5]

[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 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]

\includegraphics{figure_2} The uniform L-shaped lamina \(OABCDE\), shown in Figure 2, is made from two identical rectangles. Each rectangle is 4 metres long and \(a\) metres wide. Giving each answer in terms of \(a\), find the distance of the centre of mass of the lamina from

1. \(OE\). [4]
2. \(OA\). [4]

The lamina is freely suspended from \(O\) and hangs in equilibrium with \(OE\) at an angle \(\theta\) to the downward vertical through \(O\), where \(\tan \theta = \frac{4}{3}\).

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

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]