6.04e Rigid body equilibrium: coplanar forces

\includegraphics{figure_2} A uniform rod \(AB\), of mass \(20\) kg and length \(4\) m, rests with one end \(A\) on rough horizontal ground. The rod is held in limiting equilibrium at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{3}{4}\), by a force acting at \(B\), as shown in Figure 2. The line of action of this force lies in the vertical plane which contains the rod. The coefficient of friction between the ground and the rod is \(0.5\). Find the magnitude of the normal reaction of the ground on the rod at \(A\). [7]

[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_1} The trapezium \(ABCD\) is a uniform lamina with \(AB = 4\) m and \(BC = CD = DA = 2\) m, as shown in Figure 1.

1. Show that the centre of mass of the lamina is \(\frac{4\sqrt{3}}{9}\) m from \(AB\). [5]

The lamina is freely suspended from \(D\) and hangs in equilibrium.

1. Find the angle between \(DC\) and the vertical through \(D\). [5]

\includegraphics{figure_2} A uniform rod \(AB\) has mass \(4\) kg and length \(1.4\) m. The end \(A\) is resting on rough horizontal ground. A light string \(BC\) has one end attached to \(B\) and the other end attached to a fixed point \(C\). The string is perpendicular to the rod and lies in the same vertical plane as the rod. The rod is in equilibrium, inclined at \(20°\) to the ground, as shown in Figure 2.

1. Find the tension in the string. [4]

Given that the rod is about to slip,

1. find the coefficient of friction between the rod and the ground. [7]

\includegraphics{figure_1} A uniform lamina \(L\) is formed by taking a uniform square sheet of material \(ABCD\), of side 10 cm, and removing the semi-circle with diameter \(AB\) from the square, as shown in Fig. 2.

1. Find, in cm to 2 decimal places, the distance of the centre of mass of the lamina \(L\) from the mid-point of \(AB\). [7]

[The centre of mass of a uniform semi-circular lamina, radius \(a\), is at a distance \(\frac{4a}{3\pi}\) from the centre of the bounding diameter.] The lamina is freely suspended from \(D\) and hangs at rest.

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

\includegraphics{figure_1} A uniform lamina \(ABCD\) is made by taking a uniform sheet of metal in the form of a rectangle \(ABED\), with \(AB = 3a\) and \(AD = 2a\), and removing the triangle \(BCE\), where \(C\) lies on \(DE\) and \(CE = a\), as shown in Fig. 1.

1. Find the distance of the centre of mass of the lamina from \(AD\). [5]

The lamina has mass \(M\). A particle of mass \(m\) is attached to the lamina at \(B\). When the loaded lamina is freely suspended from the mid-point of \(AB\), it hangs in equilibrium with \(AB\) horizontal.

1. Find \(m\) in terms of \(M\). [4]

Figure 1 \includegraphics{figure_1} Figure 1 shows four uniform rods joined to form a rigid rectangular framework \(ABCD\), where \(AB = CD = 2a\), and \(BC = AD = 3a\). Each rod has mass \(m\). Particles, of mass \(6m\) and \(2m\), are attached to the framework at points \(C\) and \(D\) respectively.

1. Find the distance of the centre of mass of the loaded framework from
   1. \(AB\),
   2. \(AD\).
   [7]

The loaded framework is freely suspended from \(B\) and hangs in equilibrium.

1. Find the angle which \(BC\) makes with the vertical. [3]

\includegraphics{figure_1} A triangular frame is formed by cutting a uniform rod into 3 pieces which are then joined to form a triangle \(ABC\), where \(AB = AC = 10\) cm and \(BC = 12\) cm, as shown in Figure 1.

1. Find the distance of the centre of mass of the frame from \(BC\). (5)

The frame has total mass \(M\). A particle of mass \(M\) is attached to the frame at the mid-point of \(BC\). The frame is then freely suspended from \(B\) and hangs in equilibrium.

1. Find the size of the angle between \(BC\) and the vertical. (4)

\includegraphics{figure_1} Figure 1 shows a uniform lamina \(ABCDE\) such that \(ABDE\) is a rectangle, \(BC = CD\), \(AB = 4a\) and \(AE = 2a\). The point \(F\) is the midpoint of \(BD\) and \(FC = a\).

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

The lamina is freely suspended from \(A\) and hangs in equilibrium.

1. Find the angle between \(AB\) and the downward vertical. [3]

\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]

\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]

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_1} A rod \(AB\), of mass \(2m\) and length \(2a\), is suspended from a fixed point \(C\) by two light strings \(AC\) and \(BC\). The rod rests horizontally in equilibrium with \(AC\) making an angle \(\alpha\) with the rod, where \(\tan \alpha = \frac{3}{4}\), and with \(AC\) perpendicular to \(BC\), as shown in Fig. 1.

1. Give a reason why the rod cannot be uniform. [1]
2. Show that the tension in \(BC\) is \(\frac{4}{5}mg\) and find the tension in \(AC\). [5]

The string \(BC\) is elastic, with natural length \(a\) and modulus of elasticity \(kmg\), where \(k\) is constant.

1. Find the value of \(k\). [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_1} A child's toy consists of a uniform solid hemisphere, of mass \(M\) and base radius \(r\), joined to a uniform solid right circular cone of mass \(m\), where \(2m < M\). The cone has vertex \(O\), base radius \(r\) and height \(3r\). Its plane face, with diameter \(AB\), coincides with the plane face of the hemisphere, as shown in Figure 1.

1. Show that the distance of the centre of mass of the toy from \(AB\) is $$\frac{3(M - 2m)}{8(M + m)}r.$$ [5]

The toy is placed with \(OA\) on a horizontal surface. The toy is released from rest and does not remain in equilibrium.

1. Show that \(M > 26m\). [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_1} A toy is formed by joining a uniform solid hemisphere, of radius \(r\) and mass \(4m\), to a uniform right circular solid cone of mass \(km\). The cone has vertex \(A\), base radius \(r\) and height \(2r\). The plane face of the cone coincides with the plane face of the hemisphere. The centre of the plane face of the hemisphere is \(O\) and \(OB\) is a radius of its plane face as shown in Figure 1. The centre of mass of the toy is at \(O\).

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

A metal stud of mass \(2m\) is attached to the toy at \(A\). The toy is now suspended by a light string attached to \(B\) and hangs freely at rest. The angle between \(OB\) and the vertical is \(30°\).

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

\includegraphics{figure_3} An ornament \(S\) is formed by removing a solid right circular cone, of radius \(r\) and height \(\frac{1}{2}h\), from a solid uniform cylinder, of radius \(r\) and height \(h\), as shown in Fig. 3.

1. Show that the distance of the centre of mass \(S\) from its plane face is \(\frac{19}{30}h\). [7]

The ornament is suspended from a point on the circular rim of its open end. It hangs in equilibrium with its axis of symmetry inclined at an angle \(\alpha\) to the horizontal. Given that \(h = 4r\),

1. find, in degrees to one decimal place, the value of \(\alpha\). [4]

\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 plank \(XY\) has length 7 m and mass 2 kg. It is placed with the portion \(ZY\) in contact with a horizontal surface, where \(ZY = 2.8\) m. To prevent the plank from toppling, a stone is placed on the plank at \(Y\). \includegraphics{figure_2}

1. Find the smallest possible mass of the stone. [4 marks]
2. State, with a reason, whether your answer to part (a) would be greater or smaller if a shorter portion of the plank were in contact with the surface. [2 marks]

A uniform yoke \(AB\), of mass 4 kg and length 4\(a\) m, rests on the shoulders \(S\) and \(T\) of two oxen. \(AS = TB = a\) m. A bucket of mass \(x\) kg is suspended from \(A\). \includegraphics{figure_4}

1. Show that the vertical force on the yoke at \(T\) has magnitude \((2 - \frac{1}{4}x)g\) N and find, in terms of \(x\) and \(g\), the vertical force on the yoke at \(S\). [7 marks]
2. If the ratio of these vertical forces is \(5 : 1\), find the value of \(x\). [3 marks]
3. Find the maximum value of \(x\) for which the yoke will remain horizontal. [2 marks]

The diagram shows a uniform lamina \(ABCDEFGHIJKL\). \includegraphics{figure_3}

1. Explain why the centre of mass of the lamina is \(35\) cm from \(AL\). [1 mark]
2. Find the distance of the centre of mass from \(AF\). [4 marks]
3. The lamina is freely suspended from \(A\). Find the angle between \(AB\) and the vertical when the lamina is in equilibrium. [3 marks]
4. Explain, briefly, how you have used the fact that the lamina is uniform. [1 mark]

A uniform wire \(ABCD\) is bent into the shape shown, where the sections \(AB\), \(BC\) and \(CD\) are straight and of length \(3a\), \(10a\) and \(5a\) respectively and \(AD\) is parallel to \(BC\). \includegraphics{figure_6}

1. Show that the cosine of angle \(BCD\) is \(\frac{3}{5}\). [2 marks]
2. Find the distances of the centre of mass of the bent wire from (i) \(AB\), (ii) \(BC\). [6 marks]

The wire is hung over a smooth peg at \(B\) and rests in equilibrium.

1. Find, to the nearest 0.1°, the angle between \(BC\) and the vertical in this position. [4 marks]