6.04c Composite bodies: centre of mass

\includegraphics{figure_2} A uniform object is made by attaching the base of a solid hemisphere to the base of a solid cone so that the object has an axis of symmetry. The base of the cone has radius \(0.3\text{ m}\), and the hemisphere has radius \(0.2\text{ m}\). The object is placed on a horizontal plane with a point \(A\) on the curved surface of the hemisphere and a point \(B\) on the circumference of the cone in contact with the plane (see diagram).

1. Given that the object is on the point of toppling about \(B\), find the distance of the centre of mass of the object from the base of the cone. [3]
2. Given instead that the object is on the point of toppling about \(A\), calculate the height of the cone. [3]

[The volume of a cone is \(\frac{1}{3}\pi r^2 h\). The volume of a hemisphere is \(\frac{2}{3}\pi r^3\).]

\includegraphics{figure_6} Fig. 1 shows the cross-section \(ABCDE\) through the centre of mass \(G\) of a uniform prism. The cross-section consists of a rectangle \(ABCF\) from which a triangle \(DEF\) has been removed; \(AB = 0.6\text{ m}\), \(BC = 0.7\text{ m}\) and \(DF = EF = 0.3\text{ m}\).

1. Show that the distance of \(G\) from \(BC\) is \(0.276\text{ m}\), and find the distance of \(G\) from \(AB\). [5]
2. The prism is placed with \(CD\) on a rough horizontal surface. A force of magnitude \(2\text{ N}\) acting in the plane of the cross-section is applied to the prism. The line of action of the force passes through \(G\) and is perpendicular to \(DE\) (see Fig. 2). The prism is on the point of toppling about the edge through \(D\). Calculate the weight of the prism. [3]

\includegraphics{figure_6} Fig. 1 shows the cross-section \(ABCDE\) through the centre of mass \(G\) of a uniform prism. The cross-section consists of a rectangle \(ABCF\) from which a triangle \(DEF\) has been removed; \(AB = 0.6\text{ m}\), \(BC = 0.7\text{ m}\) and \(DF = EF = 0.3\text{ m}\).

1. Show that the distance of \(G\) from \(BC\) is \(0.276\text{ m}\), and find the distance of \(G\) from \(AB\). [5] The prism is placed with \(CD\) on a rough horizontal surface. A force of magnitude \(2\text{ N}\) acting in the plane of the cross-section is applied to the prism. The line of action of the force passes through \(G\) and is perpendicular to \(DE\) (see Fig. 2). The prism is on the point of toppling about the edge through \(D\).
2. Calculate the weight of the prism. [3]

\includegraphics{figure_4} A uniform square lamina \(ABCD\) has sides of length \(10\text{cm}\). The point \(E\) is on \(BC\) with \(EC = 7.5\text{cm}\), and the point \(F\) is on \(DC\) with \(CF = x\text{cm}\). The triangle \(EFC\) is removed from \(ABCD\) (see diagram). The centre of mass of the resulting shape \(ABEFD\) is a distance \(\bar{x}\text{cm}\) from \(CB\) and a distance \(\bar{y}\text{cm}\) from \(CD\).

1. Show that \(\bar{x} = \frac{400 - x^2}{80 - 3x}\) and find a corresponding expression for \(\bar{y}\). [4]

The shape \(ABEFD\) is in equilibrium in a vertical plane with the edge \(DF\) resting on a smooth horizontal surface.

1. Find the greatest possible value of \(x\), giving your answer in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are constants to be determined. [3]

\includegraphics{figure_4} A uniform solid circular cone, of vertical height \(4r\) and radius \(2r\), is attached to a uniform solid cylinder, of height \(3r\) and radius \(kr\), where \(k\) is a constant less than 2. The base of the cone is joined to one of the circular faces of the cylinder so that the axes of symmetry of the two solids coincide (see diagram). The cone and the cylinder are made of the same material.

1. Show that the distance of the centre of mass of the combined solid from the vertex of the cone is $$\frac{(99k^2 + 96)r}{18k^2 + 32}.$$ [4]

\includegraphics{figure_4} A uniform solid circular cone has vertical height \(kh\) and radius \(r\). A uniform solid cylinder has height \(h\) and radius \(r\). The base of the cone is joined to one of the circular faces of the cylinder so that the axes of symmetry of the two solids coincide (see diagram, which shows a cross-section). The cone and the cylinder are made of the same material.

1. Show that the distance of the centre of mass of the combined solid from the base of the cylinder is \(\frac{h(k^2 + 4k + 6)}{4(3 + k)}\). [4]

The solid is placed on a plane that is inclined to the horizontal at an angle \(\theta\). The base of the cylinder is in contact with the plane. The plane is sufficiently rough to prevent sliding. It is given that \(3h = 2r\) and that the solid is on the point of toppling when \(\tan \theta = \frac{4}{3}\).

1. Find the value of \(k\). [3]

\includegraphics{figure_4} A uniform solid circular cone has vertical height \(kh\) and radius \(r\). A uniform solid cylinder has height \(h\) and radius \(r\). The base of the cone is joined to one of the circular faces of the cylinder so that the axes of symmetry of the two solids coincide (see diagram, which shows a cross-section). The cone and the cylinder are made of the same material.

1. Show that the distance of the centre of mass of the combined solid from the base of the cylinder is \(\frac{h(k^2 + 4k + 6)}{4(3 + k)}\). [4]

The solid is placed on a plane that is inclined to the horizontal at an angle \(\theta\). The base of the cylinder is in contact with the plane. The plane is sufficiently rough to prevent sliding. It is given that \(3h = 2r\) and that the solid is on the point of toppling when \(\tan \theta = \frac{1}{3}\).

1. Find the value of \(k\). [3]

\includegraphics{figure_1} A uniform lamina \(ABCD\) consists of two isosceles triangles \(ABD\) and \(BCD\). The diagonals of \(ABCD\) meet at the point \(O\). The length of \(AO\) is \(3a\), the length of \(OC\) is \(6a\) and the length of \(BD\) is \(16a\) (see diagram). Find the distance of the centre of mass of the lamina from \(DB\). [3]

\includegraphics{figure_4} An object is composed of a hemispherical shell of radius \(2a\) attached to a closed hollow circular cylinder of height \(h\) and base radius \(a\). The hemispherical shell and the hollow cylinder are made of the same uniform material. The axes of symmetry of the shell and the cylinder coincide. \(AB\) is a diameter of the lower end of the cylinder (see diagram).

1. Find, in terms of \(a\) and \(h\), an expression for the distance of the centre of mass of the object from \(AB\). [4]

The object is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac{2}{5}\). The object is in equilibrium with \(AB\) in contact with the plane and lying along a line of greatest slope of the plane.

1. Find the set of possible values of \(h\), in terms of \(a\). [4]

\includegraphics{figure_4} An object is formed from a solid hemisphere, of radius \(2a\), and a solid cylinder, of radius \(a\) and height \(d\). The hemisphere and the cylinder are made of the same material. The cylinder is attached to the plane face of the hemisphere. The line \(OC\) forms a diameter of the base of the cylinder, where \(C\) is the centre of the plane face of the hemisphere and \(O\) is common to both circumferences (see diagram). Relative to axes through \(O\), parallel and perpendicular to \(OC\) as shown, the centre of mass of the object is \((\bar{x}, \bar{y})\).

1. Show that \(\bar{x} = \frac{32a^2 + 3ad}{16a + 3d}\) and find an expression, in terms of \(a\) and \(d\), for \(\bar{y}\). [5]

The object is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\) where \(\sin\theta = \frac{1}{6}\). The object is in equilibrium with \(CO\) horizontal, where \(CO\) lies in a vertical plane through a line of greatest slope.

1. Find \(d\) in terms of \(a\). [3]

\includegraphics{figure_3} A uniform lamina is in the form of a triangle \(ABC\), with \(AC = 8a\), \(BC = 6a\) and angle \(ACB = 90°\). The point \(D\) on \(AC\) is such that \(AD = 3a\). The point \(E\) on \(CB\) is such that \(CE = x\) (see diagram). The triangle \(CDE\) is removed from the lamina.

1. Find, in terms of \(a\) and \(x\), the distance of the centre of mass of the remaining object \(ADEB\) from \(AC\). [4]

The object \(ADEB\) is on the point of toppling about the point \(E\) when the object is in the vertical plane with its edge \(EB\) on a smooth horizontal surface.

1. Find \(x\) in terms of \(a\). [3]

\includegraphics{figure_5} A uniform lamina is in the form of a triangle \(OBC\), with \(OC = 18a\), \(OB = 24a\) and angle \(COB = 90°\). The point \(A\) on \(OB\) is such that \(OA = x\) (see diagram). The triangle \(OAC\) is removed from the lamina.

1. Find, in terms of \(a\) and \(x\), the distance of the centre of mass of the remaining object \(ABC\) from \(OC\). [3]

The object \(ABC\) is suspended from \(C\). In its equilibrium position, the side \(AB\) makes an angle \(\theta\) with the vertical, where \(\tan\theta = \frac{8}{5}\).

1. Find \(x\) in terms of \(a\). [4]

\includegraphics{figure_4} A uniform lamina \(AECF\) is formed by removing two identical triangles \(BCE\) and \(CDF\) from a square lamina \(ABCD\). The square has side \(3a\) and \(EB = DF = h\) (see diagram).

1. Find the distance of the centre of mass of the lamina \(AECF\) from \(AD\) and from \(AB\), giving your answers in terms of \(a\) and \(h\). [5]

The lamina \(AECF\) is placed vertically on its edge \(AE\) on a horizontal plane.

1. Find, in terms of \(a\), the set of values of \(h\) for which the lamina remains in equilibrium. [3]

\includegraphics{figure_2} A uniform lamina is in the form of a triangle \(ABC\) in which angle \(B\) is a right angle, \(AB = 9a\) and \(BC = 6a\). The point \(D\) is on \(BC\) such that \(BD = x\) (see diagram). The region \(ABD\) is removed from the lamina. The resulting shape \(ADC\) is placed with the edge \(DC\) on a horizontal surface and the plane \(ADC\) is vertical. Find the set of values of \(x\), in terms of \(a\), for which the shape is in equilibrium. [6]

\includegraphics{figure_3} A uniform lamina is in the form of an isosceles triangle \(ABC\) in which \(AC = 2a\) and angle \(ABC = 90°\). The point \(D\) on \(AB\) is such that the ratio \(DB : AB = 1 : k\). The point \(E\) on \(CB\) is such that \(DE\) is parallel to \(AC\). The triangle \(DBE\) is removed from the lamina (see diagram).

1. Find, in terms of \(k\), the distance of the centre of mass of the remaining lamina \(ADEC\) from the midpoint of \(AC\). [4]

When the lamina \(ADEC\) is freely suspended from the vertex \(A\), the edge \(AC\) makes an angle \(\theta\) with the downward vertical, where \(\tan \theta = \frac{2}{15}\).

1. Find the value of \(k\). [3]

\includegraphics{figure_4} An object is formed by removing a cylinder of radius \(\frac{2}{3}a\) and height \(kh\) (\(k < 1\)) from a uniform solid cylinder of radius \(a\) and height \(h\). The vertical axes of symmetry of the two cylinders coincide. The upper faces of the two cylinders are in the same plane as each other. The points \(A\) and \(B\) are the opposite ends of a diameter of the upper face of the object (see diagram).

1. Find, in terms of \(h\) and \(k\), the distance of the centre of mass of the object from \(AB\). [4]

A child's toy consists of an iron disc of radius \(r\) and a vertical bead with \(3r\) at rail that is rigidly fixed to the disc so that the toy rocks as it rolls. The circumference of the disc is such that the disc and bead have the same material. Show that the centre of mass of the toy is at a distance \(\frac{27r}{10}\) from the centre of the disc. [4]

\includegraphics{figure_1} A seesaw in a playground consists of a beam \(AB\) of length \(4\) m which is supported by a smooth pivot at its centre \(C\). Jill has mass \(25\) kg and sits on the end \(A\). David has mass \(40\) kg and sits at a distance \(x\) metres from \(C\), as shown in Figure 1. The beam is initially modelled as a uniform rod. Using this model,

1. find the value of \(x\) for which the seesaw can rest in equilibrium in a horizontal position. [3]
2. State what is implied by the modelling assumption that the beam is uniform. [1]

David realises that the beam is not uniform as he finds that he must sit at a distance \(1.4\) m from \(C\) for the seesaw to rest horizontally in equilibrium. The beam is now modelled as a non-uniform rod of mass \(15\) kg. Using this model,

1. find the distance of the centre of mass of the beam from \(C\). [4]

\includegraphics{figure_1} A uniform lamina \(ABCD\) is formed by removing the isosceles triangle \(ADC\) of height \(h\) metres, where \(h < 2\sqrt{3}\), from a uniform lamina \(ABC\) in the shape of an equilateral triangle of side 4 m, as shown in Figure 1. The centre of mass of \(ABCD\) is at \(D\).

1. Show that \(h = \sqrt{3}\) [7]

The weight of the lamina \(ABCD\) is \(W\) newtons. The lamina is freely suspended from \(A\). A horizontal force of magnitude \(F\) newtons is applied at \(B\) so that the lamina is in equilibrium with \(AB\) vertical. The horizontal force acts in the vertical plane containing the lamina.

1. Find \(F\) in terms of \(W\). [4]

A thin uniform wire of mass \(12m\) is bent to form a right-angled triangle \(ABC\). The lengths of the sides \(AB\), \(BC\) and \(AC\) are \(3a\), \(4a\) and \(5a\) respectively. A particle of mass \(2m\) is attached to the triangle at \(B\) and a particle of mass \(3m\) is attached to the triangle at \(C\). The bent wire and the two particles form the system \(S\). The system \(S\) is freely suspended from \(A\) and hangs in equilibrium. Find the size of the angle between \(AB\) and the downward vertical. [10]

\includegraphics{figure_1} Figure 1 shows a decoration which is made by cutting 2 circular discs from a sheet of uniform card. The discs are joined so that they touch at a point \(D\) on the circumference of both discs. The discs are coplanar and have centres \(A\) and \(B\) with radii 10 cm and 20 cm respectively.

1. Find the distance of the centre of mass of the decoration from \(B\). [5]

The point \(C\) lies on the circumference of the smaller disc and \(\angle CAB\) is a right angle. The decoration is freely suspended from \(C\) and hangs in equilibrium.

1. Find, in degrees to one decimal place, the angle between \(AB\) and the vertical. [4]

\includegraphics{figure_1} Figure 1 shows a template made by removing a square \(WXYZ\) from a uniform triangular lamina \(ABC\). The lamina is isosceles with \(CA = CB\) and \(AB = 12a\). The midpoint of \(AB\) is \(N\) and \(NC = 8a\). The centre \(O\) of the square lies on \(NC\) and \(ON = 2a\). The sides \(WX\) and \(ZY\) are parallel to \(AB\) and \(WZ = 2a\). The centre of mass of the template is at \(G\).

1. Show that \(NG = \frac{8}{7}a\). [7]

The template has mass \(M\). A small metal stud of mass \(kM\) is attached to the template at \(C\). The centre of mass of the combined template and stud lies on \(YZ\). By modelling the stud as a particle,

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

\includegraphics{figure_1} Figure 1 shows a template made by removing a square \(WXYZ\) from a uniform triangular lamina \(ABC\). The lamina is isosceles with \(CA = CB\) and \(AB = 12a\). The mid-point of \(AB\) is \(N\) and \(NC = 8a\). The centre \(O\) of the square lies on \(NC\) and \(ON = 2a\). The sides \(WX\) and \(ZY\) are parallel to \(AB\) and \(WZ = 2a\). The centre of mass of the template is at \(G\).

1. Show that \(NG = \frac{30}{11}a\). [7]

The template has mass \(M\). A small metal stud of mass \(kM\) is attached to the template at \(C\). The centre of mass of the combined template and stud lies on \(YZ\). By modelling the stud as a particle,

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

\includegraphics{figure_2} Figure 2 shows a uniform lamina \(ABCDE\) such that \(ABDE\) is a rectangle, \(BC = CD\), \(AB = 8a\) and \(AE = 6a\). The point \(X\) is the mid-point of \(BD\) and \(XC = 4a\). The centre of mass of the lamina is at \(G\).

1. Show that \(GX = \frac{14}{15}a\). [6]

The mass of the lamina is \(M\). A particle of mass \(\lambda M\) is attached to the lamina at \(C\). The lamina is suspended from \(B\) and hangs freely under gravity with \(AB\) horizontal.

1. Find the value of \(\lambda\). [3]

\includegraphics{figure_1} Figure 1 shows a triangular lamina \(ABC\). The coordinates of \(A\), \(B\) and \(C\) are \((0, 4)\), \((9, 0)\) and \((0, -4)\) respectively. Particles of mass \(4m\), \(6m\) and \(2m\) are attached at \(A\), \(B\) and \(C\) respectively.

1. Calculate the coordinates of the centre of mass of the three particles, without the lamina. [4]

The lamina \(ABC\) is uniform and of mass \(km\). The centre of mass of the combined system consisting of the three particles and the lamina has coordinates \((4, \lambda)\).

1. Show that \(k = 6\). [3]
2. Calculate the value of \(\lambda\). [2]

The combined system is freely suspended from \(O\) and hangs at rest.

1. Calculate, in degrees to one decimal place, the angle between \(AC\) and the vertical. [3]