6.04e Rigid body equilibrium: coplanar forces

[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_12} The diagram shows a uniform square lamina ABCD, of weight \(W\) and side-length \(a\). The lamina is in equilibrium in a vertical plane that also contains the point O. The vertex A rests on a smooth plane inclined at an angle of 30° to the horizontal. The vertex B rests on a smooth plane inclined at an angle of 60° to the horizontal. OA is a line of greatest slope of the plane inclined at 30° to the horizontal and OB is a line of greatest slope of the plane inclined at 60° to the horizontal. The side AB is inclined at an angle \(\theta\) to the horizontal and the lamina is kept in equilibrium in this position by a clockwise couple of magnitude \(\frac{1}{8}aW\).

1. By resolving horizontally and vertically, determine, in terms of \(W\), the magnitude of the normal contact force between the plane and the lamina at B. [6]
2. By taking moments about A, show that \(\theta\) satisfies the equation $$2(\sqrt{3} + 2)\sin\theta - 2\cos\theta = 1.$$ [5]
3. Verify that \(\theta = 22.4°\), correct to 1 decimal place. [2]

\includegraphics{figure_11} The diagram shows the cross-section through the centre of mass of a uniform solid prism. The cross-section is a right-angled triangle ABC, with AB perpendicular to AC, which lies in a vertical plane. The length of AB is 3 cm, and the length of AC is 12 cm. The prism is resting in equilibrium on a horizontal surface and against a vertical wall. The side AC of the prism makes an angle \(\theta\) with the horizontal. A horizontal force of magnitude \(P\) N is now applied to the prism at B. This force acts towards the wall in the vertical plane which passes through the centre of mass G of the prism and is perpendicular to the wall. The weight of the prism is 15 N and the coefficients of friction between the prism and the surface, and between the prism and the wall, are each \(\frac{1}{2}\).

1. Show that the least value of \(P\) needed to move the prism is given by $$P = \frac{40 \cos \theta + 95 \sin \theta}{16 \sin \theta - 13 \cos \theta}.$$ [8]
2. Determine the range in which the value of \(\theta\) must lie. [4]

\includegraphics{figure_4} A uniform rod AB has mass 3 kg and length 4 m. The end A of the rod is in contact with rough horizontal ground. The rod rests in equilibrium on a smooth horizontal peg 1.5 m above the ground, such that the rod is inclined at an angle of \(25°\) to the ground (see diagram). The rod is in a vertical plane perpendicular to the peg.

1. Determine the magnitude of the normal contact force between the peg and the rod. [3]
2. Determine the range of possible values of the coefficient of friction between the rod and the ground. [5]

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]

\includegraphics{figure_8} As shown in the diagram, \(AB\) is a long thin rod which is fixed vertically with \(A\) above \(B\). One end of a light inextensible string of length 1 m is attached to \(A\) and the other end is attached to a particle \(P\) of mass \(m_1\) kg. One end of another light inextensible string of length 1 m is also attached to \(P\). Its other end is attached to a small smooth ring \(R\), of mass \(m_2\) kg, which is free to move on \(AB\). Initially, \(P\) moves in a horizontal circle of radius 0.6 m with constant angular velocity \(\omega \mathrm{rad s}^{-1}\). The magnitude of the tension in string \(AP\) is denoted by \(T_1\) N while that in string \(PR\) is denoted by \(T_2\) N.

1. By considering forces on \(R\), express \(T_2\) in terms of \(m_2\). [2]
2. Show that
   1. \(T_1 = \frac{49}{4}(m_1 + m_2)\), [2]
   2. \(\omega^2 = \frac{49(m_1 + 2m_2)}{4m_1}\). [3]
3. Deduce that, in the case where \(m_1\) is much bigger than \(m_2\), \(\omega \approx 3.5\). [2] In a different case, where \(m_1 = 2.5\) and \(m_2 = 2.8\), \(P\) slows down. Eventually the system comes to rest with \(P\) and \(R\) hanging in equilibrium.
4. Find the total energy lost by \(P\) and \(R\) as the angular velocity of \(P\) changes from the initial value of \(\omega \mathrm{rad s}^{-1}\) to zero. [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]

\includegraphics{figure_2} A uniform solid right circular cone has base radius \(a\) and semi-vertical angle \(\alpha\), where \(\tan \alpha = \frac{1}{3}\). The cone is freely suspended by a string attached at a point A on the rim of its base, and hangs in equilibrium with its axis of symmetry making an angle of \(\theta^0\) with the upward vertical, as shown in the diagram above. Find, to one decimal place, the value of \(\theta\). [5]

A uniform rod, PQ, of length \(2a\), rests with one end, P, on rough horizontal ground and a point T resting on a rough fixed prism of semi-circular cross-section of radius \(a\), as shown in the diagram. The rod is in a vertical plane which is parallel to the prism's cross-section. The coefficient of friction at both P and T is \(\mu\). \includegraphics{figure_6} The rod is on the point of slipping when it is inclined at an angle of 30\(^0\) to the horizontal. Find the value of \(\mu\). [8]

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]

\includegraphics{figure_1} A smooth solid hemisphere is fixed with its flat surface in contact with rough horizontal ground. The hemisphere has centre \(O\) and radius \(5a\). A uniform rod \(AB\), of length \(16a\) and weight \(W\), rests in equilibrium on the hemisphere with end \(A\) on the ground. The rod rests on the hemisphere at the point \(C\), where \(AC = 12a\) and angle \(CAO = \alpha\), as shown in Figure 1. Points \(A\), \(C\), \(B\) and \(O\) all lie in the same vertical plane.

1. Explain why \(AO = 13a\) [1]

The normal reaction on the rod at \(C\) has magnitude \(kW\)

1. Show that \(k = \frac{8}{13}\) [3]

The resultant force acting on the rod at \(A\) has magnitude \(R\) and acts upwards at \(\theta°\) to the horizontal.

1. Find
   1. an expression for \(R\) in terms of \(W\)
   2. the value of \(\theta\)
   [8]

\(A\) and \(B\) are two points a distance of 5 m apart on a horizontal ceiling. A particle \(P\) of mass \(m\) kg is attached to \(A\) and \(B\) by light elastic strings. The particle hangs in equilibrium at a distance of 4 m from \(A\) and 3 m from \(B\) so that angle \(APB = 90°\) (see diagram). \includegraphics{figure_4} The string joining \(P\) to \(A\) has natural length 2 m and modulus of elasticity \(\lambda_A\) N. The string joining \(P\) to \(B\) also has natural length 2 m but has modulus of elasticity \(\lambda_B\) N.

1. 1. Show that \(\lambda_B = \frac{3}{4}\lambda_A\). [4]
   2. Find an expression for \(\lambda_A\) in terms of \(m\) and \(g\). [3]
2. Find, in terms of \(m\) and \(g\), the total elastic potential energy stored in the strings. [2]

The string joining \(P\) to \(A\) is detached from \(A\) and a second particle, \(Q\), of mass \(0.3m\) kg is attached to the free end of the string. \(Q\) is then gently lowered into a position where the system hangs vertically in equilibrium.

1. Find the distance of \(Q\) below \(B\) in this equilibrium position. [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]