6.04e Rigid body equilibrium: coplanar forces

\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]

The point \(C\) is on the circumference of the base of the cone. When the combined solid is freely suspended from \(C\) and hanging in equilibrium, the diameter through \(C\) makes an angle \(\alpha\) with the downward vertical, where \(\tan \alpha = \frac{1}{5}\).

1. Given that the centre of mass of the combined solid is within the cylinder, find the value of \(k\). [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_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]

A ring of weight \(W\), with radius \(a\) and centre \(O\), is at rest on a rough surface that is inclined to the horizontal at an angle \(\alpha\) where \(\tan\alpha = \frac{1}{3}\). The plane of the ring is perpendicular to the inclined surface and parallel to a line of greatest slope of the surface. The point \(P\) on the circumference of the ring is such that \(OP\) is parallel to the surface. A light inextensible string is attached to \(P\) and to the point \(Q\), which is on the surface, such that \(PQ\) is horizontal (see diagram). The points \(O\), \(P\) and \(Q\) are in the same vertical plane. The system is in limiting equilibrium and the coefficient of friction between the ring and the surface is \(\mu\). \includegraphics{figure_4}

1. Find, in terms of \(W\), the tension in the string \(PQ\). [4]
2. Find the value of \(\mu\). [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} The diagram shows the cross-section \(ABCD\) of a uniform solid object which is formed by removing a cone with cross-section \(DCE\) from the top of a larger cone with cross-section \(ABE\). The perpendicular distance between \(AB\) and \(DC\) is \(h\), the diameter \(AB\) is \(6r\) and the diameter \(DC\) is \(2r\).

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

The object is freely suspended from the point \(B\) and hangs in equilibrium. The angle between \(AB\) and the downward vertical through \(B\) is \(\theta\).

1. Given that \(h = \frac{13}{4}r\), find the value of \(\tan\theta\). [2]

\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_4} An object is formed by removing a solid cylinder, of height \(ka\) and radius \(\frac{1}{2}a\), from a uniform solid hemisphere of radius \(a\). The axes of symmetry of the hemisphere and the cylinder coincide and one circular face of the cylinder coincides with the plane face of the hemisphere. \(AB\) is a diameter of the circular face of the hemisphere (see diagram).

1. Show that the distance of the centre of mass of the object from \(AB\) is \(\frac{3a(2-k^2)}{2(8-3k)}\). [4] When the object is freely suspended from the point \(A\), the line \(AB\) makes an angle \(\theta\) with the downward vertical, where \(\tan\theta = \frac{7}{18}\).
2. Find the possible values of \(k\). [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 square lamina of side \(2a\) and weight \(W\) is suspended from a light inextensible string attached to the midpoint \(E\) of the side \(AB\). The other end of the string is attached to a fixed point \(P\) on a rough vertical wall. The vertex \(B\) of the lamina is in contact with the wall. The string \(EP\) is perpendicular to the side \(AB\) and makes an angle \(\theta\) with the wall (see diagram). The string and the lamina are in a vertical plane perpendicular to the wall. The coefficient of friction between the wall and the lamina is \(\frac{1}{2}\). Given that the vertex \(B\) is about to slip up the wall, find the value of \(\tan\theta\). [8]

\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_7} A particle \(P\) of mass \(m\) is attached to one end of a light rod of length \(3a\). The other end of the rod is able to pivot smoothly about the fixed point \(A\). The particle is also attached to one end of a light spring of natural length \(a\) and modulus of elasticity \(kmg\). The other end of the spring is attached to a fixed point \(B\). The points \(A\) and \(B\) are in a horizontal line, a distance \(5a\) apart, and these two points and the rod are in a vertical plane. Initially, \(P\) is held in equilibrium by a vertical force \(F\) with the stretched length of the spring equal to \(4a\) (see diagram). The particle is released from rest in this position and has a speed of \(\frac{6}{5}\sqrt{2ag}\) when the rod becomes horizontal.

1. Find the value of \(k\). [5]
2. Find \(F\) in terms of \(m\) and \(g\). [2]
3. Find, in terms of \(m\) and \(g\), the tension in the rod immediately before it is released. [2]

When the object is suspended from \(A\), the angle between \(AB\) and the vertical is \(\theta\), where \(\tan\theta = \frac{1}{2}\).

1. Given that \(h = \frac{8}{3}a\), find the possible values of \(k\). [3]

\includegraphics{figure_4} The end \(A\) of a uniform rod \(AB\) of length \(6a\) and weight \(W\) is in contact with a rough vertical wall. One end of a light inextensible string of length \(3a\) is attached to the midpoint \(C\) of the rod. The other end of the string is attached to a point \(D\) on the wall, vertically above \(A\). The rod is in equilibrium when the angle between the rod and the wall is \(\theta\), where \(\tan \theta = \frac{3}{4}\). A particle of weight \(W\) is attached to the point \(E\) on the rod, where the distance \(AE\) is equal to \(ka\) (\(3 < k < 6\)) (see diagram). The rod and the string are in a vertical plane perpendicular to the wall. The coefficient of friction between the rod and the wall is \(\frac{1}{3}\). The rod is about to slip down the wall.

1. Find the value of \(k\). [5]
2. Find, in terms of \(W\), the magnitude of the frictional force between the rod and the wall. [2]

When the object is suspended from \(A\), the angle between \(AB\) and the vertical is \(\theta\), where \(\tan \theta = \frac{1}{2}\).

1. Given that \(h = \frac{8}{3}a\), find the possible values of \(k\). [3]

\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]

\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]

\includegraphics{figure_1} Figure 1 shows a template \(T\) made by removing a circular disc, of centre \(X\) and radius 8 cm, from a uniform circular lamina, of centre \(O\) and radius 24 cm. The point \(X\) lies on the diameter \(AOB\) of the lamina and \(AX = 16\) cm. The centre of mass of \(T\) is at the point \(G\).

1. Find \(AG\). [6]

The template \(T\) is free to rotate about a smooth fixed horizontal axis, perpendicular to the plane of \(T\), which passes through the mid-point of \(OB\). A small stud of mass \(\frac{1}{4}m\) is fixed at \(B\), and \(T\) and the stud are in equilibrium with \(AB\) horizontal. Modelling the stud as a particle,

1. find the mass of \(T\) in terms of \(m\). [4]

\includegraphics{figure_1} A set square \(S\) is made by removing a circle of centre \(O\) and radius 3 cm from a triangular piece of wood. The piece of wood is modelled as a uniform triangular lamina \(ABC\), with \(\angle ABC = 90°\), \(AB = 12\) cm and \(BC = 21\) cm. The point \(O\) is 5 cm from \(AB\) and 5 cm from \(BC\), as shown in Figure 1.

1. Find the distance of the centre of mass of \(S\) from
   1. \(AB\),
   2. \(BC\). [9]

The set square is freely suspended from \(C\) and hangs in equilibrium.

1. Find, to the nearest degree, the angle between \(CB\) and the vertical. [3]