6.04e Rigid body equilibrium: coplanar forces

6.

1. A uniform lamina is in the shape of a quadrant of a circle of radius \(a\). Show, by integration, that the centre of mass of the lamina is at a distance of \(\frac { 4 a } { 3 \pi }\) from each of its straight edges. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{daa795f0-2c5e-4617-a295-fbe74c22be4a-10_809_802_484_571} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} A second uniform lamina \(A B C D E F A\) is shown shaded in Figure 3. The straight sides \(A C\) and \(A E\) are perpendicular and \(A C = A E = 2 a\). In the figure, the midpoint of \(A C\) is \(B\), the midpoint of \(A E\) is \(F\), and \(A B D F\) and \(D G E F\) are squares of side \(a\). \(B C D\) is a quadrant of a circle with centre \(B\). \(D G E\) is a quadrant of a circle with centre \(G\).
2. Find the distance of the centre of mass of the lamina from the side \(A E\). The lamina is smoothly hinged to a horizontal axis which passes through \(E\) and is perpendicular to the plane of the lamina. The lamina has weight \(W\) newtons. The lamina is held in equilibrium in a vertical plane, with \(A\) vertically above \(E\), by a horizontal force of magnitude \(X\) newtons applied at \(C\).
3. Find \(X\) in terms of \(W\).
   

3. \begin{figure}[h] \end{figure} Figure 2 shows a container in the shape of a uniform right circular conical shell of height 6r. The radius of the open circular face is \(r\). The container is suspended by two vertical strings attached to two points at opposite ends of a diameter of the open circular face. It hangs with the open circular face uppermost and axis vertical. Molten wax is poured into the container. The wax solidifies and adheres to the container, forming a uniform solid right circular cone. The depth of the wax in the container is \(2 r\). The container together with the wax forms a solid \(S\). The mass of the container when empty is \(m\) and the mass of the wax in the container is \(3 m\).

1. Find the distance of the centre of mass of the solid \(S\) from the vertex of the container. One of the strings is now removed and the solid \(S\) hangs freely in equilibrium suspended by the remaining vertical string.
2. Find the size of the angle between the axis of the container and the downward vertical.

5. \begin{figure}[h] \end{figure} Figure 2 shows a uniform solid spindle which is made by joining together the circular faces of two right circular cones. The common circular face has radius \(r\) and centre \(O\). The smaller cone has height \(h\) and the larger cone has height \(k h\). The point \(A\) lies on the circumference of the common circular face. The spindle is suspended from \(A\) and hangs freely in equilibrium with \(A O\) at an angle of \(30 ^ { \circ }\) to the vertical. Show that \(k = \frac { 4 r } { h \sqrt { 3 } } + 1\)

3. One end of a light elastic string, of natural length 1.5 m and modulus of elasticity 14.7 N ,
3. One end of a light elastic string, of natural length 1.5 m and modulus of elasticity 14.7 N , is attached to a fixed point \(O\) on a ceiling. A particle \(P\) of mass 0.6 kg is attached to the free end of the string. The particle is held at \(O\) and released from rest. The particle comes to instantaneous rest for the first time at the point \(A\). Find

1. the distance \(O A\),
2. the magnitude of the instantaneous acceleration of \(P\) at \(A\).
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4c1c51ff-6ae8-402d-b303-b656d26e4230-05_620_956_118_500} \captionsetup{labelformat=empty} \caption{igure 2}
   \end{figure} A uniform solid \(S\) consists of two right circular cones of base radius \(r\). The smaller cone has height \(2 h\) and the centre of the plane face of this cone is \(O\). The larger cone has height \(k h\) where \(k > 2\). The two cones are joined so that their plane faces coincide, as shown in Figure 2.
   1. Show that the distance of the centre of mass of \(S\) from \(O\) is $$\frac { h } { 4 } ( k - 2 )$$ The point \(A\) lies on the circumference of the base of one of the cones. The solid is suspended by a string attached at \(A\) and hangs freely in equilibrium. Given that \(r = 3 h\) and \(k = 6\)
   2. find the size of the angle between \(A O\) and the vertical.

4. \begin{figure}[h] \end{figure} A thin uniform right hollow cylinder, of radius \(a\) and height \(4 a\), has a base but no top. A thin uniform hemispherical shell, also of radius \(a\), is made of the same material as the cylinder. The hemispherical shell is attached to the open end of the cylinder forming a container \(C\). The open circular rim of the cylinder coincides with the rim of the hemispherical shell. The centre of the base of \(C\) is \(O\), as shown in Figure 3.

1. Find the distance from \(O\) to the centre of mass of \(C\). The container is placed with its circular base on a plane which is inclined at \(\theta ^ { \circ }\) to the horizontal. The plane is sufficiently rough to prevent \(C\) from sliding. The container is on the point of toppling.
2. Find the value of \(\theta\).

5. \includegraphics[max width=\textwidth, alt={}, center]{e256678d-89e8-48eb-aa8a-b8e027b62ef1-3_423_357_918_847} A uniform solid, \(S\), is placed with its plane face on horizontal ground. The solid consists of a right circular cylinder, of radius \(r\) and height \(r\), joined to a right circular cone, of radius \(r\) and height \(h\). The plane face of the cone coincides with one of the plane faces of the cylinder, as shown in Fig. 3.

1. Show that the distance of the centre of mass of \(S\) from the ground is $$\frac { 6 r ^ { 2 } + 4 r h + h ^ { 2 } } { 4 ( 3 r + h ) }$$ (8) The solid is now placed with its plane face on a rough plane which is inclined at an angle \(\alpha\) to the horizontal. The plane is rough enough to prevent \(S\) from sliding. Given that \(h = 2 r\), and that \(S\) is on the point of toppling,
2. find, to the nearest degree, the value of \(\alpha\).
   (5)

2. \begin{figure}[h] \end{figure} A thin uniform wire, of total length 20 cm , is bent to form a frame. The frame is in the shape of a trapezium \(A B C D\), where \(A B = A D = 4 \mathrm {~cm} , C D = 5 \mathrm {~cm}\), and \(A B\) is perpendicular to \(B C\) and \(A D\), as shown in Figure 1.

1. Find the distance of the centre of mass of the frame from \(A B\). The frame has mass \(M\). A particle of mass \(k M\) is attached to the frame at \(C\). When the frame is freely suspended from the mid-point of \(B C\), the frame hangs in equilibrium with \(B C\) horizontal.
2. Find the value of \(k\).

5. \begin{figure}[h] \end{figure} The uniform lamina \(A B C D E F\), shown in Figure 2, consists of two identical rectangles with sides of length \(a\) and \(3 a\). The mass of the lamina is \(M\). A particle of mass \(k M\) is attached to the lamina at \(E\). The lamina, with the attached particle, is freely suspended from \(A\) and hangs in equilibrium with \(A F\) at an angle \(\theta\) to the downward vertical. Given that \(\tan \theta = \frac { 4 } { 7 }\), find the value of \(k\).
(10)

6. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of mass \(3 m\) and length \(2 a\), is freely hinged at \(A\) to a fixed point on horizontal ground. A particle of mass \(m\) is attached to the rod at the end \(B\). The system is held in equilibrium by a force \(\mathbf { F }\) acting at the point \(C\), where \(A C = b\). The rod makes an acute angle \(\theta\) with the ground, as shown in Figure 3. The line of action of \(\mathbf { F }\) is perpendicular to the rod and in the same vertical plane as the rod.

1. Show that the magnitude of \(\mathbf { F }\) is \(\frac { 5 m g a } { b } \cos \theta\) The force exerted on the rod by the hinge at \(A\) is \(\mathbf { R }\), which acts upwards at an angle \(\phi\) above the horizontal, where \(\phi > \theta\).
2. Find
   1. the component of \(\mathbf { R }\) parallel to the rod, in terms of \(m , g\) and \(\theta\),
   2. the component of \(\mathbf { R }\) perpendicular to the rod, in terms of \(a , b , m , g\) and \(\theta\).
3. Hence, or otherwise, find the range of possible values of \(b\), giving your answer in terms of \(a\).

2. \begin{figure}[h] \end{figure} A thin uniform wire, of total length 20 cm , is bent to form a frame. The frame is in the shape of a trapezium \(A B C D\), where \(A B = A D = 4 \mathrm {~cm} , C D = 5 \mathrm {~cm}\), and \(A B\) is perpendicular to \(B C\) and \(A D\), as shown in Figure 1.

1. Find the distance of the centre of mass of the frame from \(A B\). The frame has mass \(M\). A particle of mass \(k M\) is attached to the frame at \(C\). When the frame is freely suspended from the mid-point of \(B C\), the frame hangs in equilibrium with \(B C\) horizontal.
2. Find the value of \(k\).

6. \begin{figure}[h] \end{figure} A uniform pole \(A B\), of mass 30 kg and length 3 m , is smoothly hinged to a vertical wall at one end \(A\). The pole is held in equilibrium in a horizontal position by a light rod CD. One end \(C\) of the rod is fixed to the wall vertically below \(A\). The other end \(D\) is freely jointed to the pole so that \(\angle A C D = 30 ^ { \circ }\) and \(A D = 0.5 \mathrm {~m}\), as shown in Figure 2. Find

1. the thrust in the rod \(C D\),
2. the magnitude of the force exerted by the wall on the pole at \(A\). The rod \(C D\) is removed and replaced by a longer light rod \(C M\), where \(M\) is the mid-point of \(A B\). The rod is freely jointed to the pole at \(M\). The pole \(A B\) remains in equilibrium in a horizontal position.
3. Show that the force exerted by the wall on the pole at \(A\) now acts horizontally.

3 \includegraphics[max width=\textwidth, alt={}, center]{15ed1dfc-8188-4e20-9c0b-ce31af35f0b6-2_513_711_890_717} A uniform lamina of mass \(m\) is bounded by concentric circles with centre \(O\) and radii \(a\) and \(2 a\). The lamina is free to rotate about a fixed smooth horizontal axis \(T\) which is tangential to the outer rim (see diagram). Show that the moment of inertia of the lamina about \(T\) is \(\frac { 21 } { 4 } m a ^ { 2 }\). When hanging at rest, with \(O\) vertically below \(T\), the lamina is given an angular speed \(\omega\) about \(T\). The lamina comes to instantaneous rest in the subsequent motion. Neglecting air resistance, find the set of possible values of \(\omega\).

4 \includegraphics[max width=\textwidth, alt={}, center]{15ed1dfc-8188-4e20-9c0b-ce31af35f0b6-3_512_983_267_580} A uniform sphere rests on a horizontal plane. The sphere has centre \(O\), radius 0.6 m and weight 36 N . A uniform rod \(A B\), of weight 14 N and length 1 m , rests with \(A\) in contact with the plane and \(B\) in contact with the sphere at the end of a horizontal diameter. The point of contact of the sphere with the plane is \(C\), and \(A , B , C\) and \(O\) lie in the same vertical plane (see diagram). The contacts at \(A , B\) and \(C\) are rough and the system is in equilibrium. By taking moments about \(C\) for the system, show that the magnitude of the normal contact force at \(A\) is 10 N . Show that the magnitudes of the frictional forces at \(A , B\) and \(C\) are equal. The coefficients of friction at \(A , B\) and \(C\) are all equal to \(\mu\). Find the smallest possible value of \(\mu\).

Two uniform rods \(A B\) and \(A C\) have lengths \(2 a\) and \(4 a\) and weights \(W\) and \(2 W\) respectively. They are freely hinged together at \(A\) and rest in equilibrium in a vertical plane with \(B\) and \(C\) in contact with two rough parallel vertical walls. The plane containing the rods is perpendicular to the walls. The rods \(A B\) and \(A C\) each make an angle \(\beta\) with the vertical (see diagram). Show that the magnitude of the frictional force acting on \(A B\) at \(B\) is \(\frac { 5 } { 4 } W\). Given that the coefficient of friction at \(B\) and at \(C\) is \(\mu\), find the set of possible values of \(\mu\) in terms of \(\beta\).

3 \includegraphics[max width=\textwidth, alt={}, center]{e8a16ec8-b6b7-4b0c-b0c1-8f5f7a9e4fa6-2_355_695_1073_726} A uniform solid hemisphere, of radius \(a\) and mass \(M\), is placed with its curved surface in contact with a rough plane that is inclined at an angle \(\alpha\) to the horizontal. A particle \(P\) of mass \(m\) is attached to the rim of the hemisphere. The system rests in equilibrium with the rim of the hemisphere horizontal and \(P\) at the point on the rim that is closest to the inclined plane (see diagram). Given that the coefficient of friction between the plane and the hemisphere is \(\frac { 1 } { 2 }\), show that

1. \(\tan \alpha \leqslant \frac { 1 } { 2 }\),
2. \(m \leqslant \frac { M ( 1 + \sqrt { } 5 ) } { 4 }\).

A rigid body is made from uniform wire of negligible thickness and is in the form of a square \(A B C D\) of mass \(M\) enclosed within a circular ring of radius \(a\) and mass \(2 M\). The centres of the square and the circle coincide at \(O\) and the corners of the square are joined to the circle (see diagram). Show that the moment of inertia of the body about an axis through \(O\), perpendicular to the plane of the body, is \(\frac { 8 } { 3 } M a ^ { 2 }\). Hence find the moment of inertia of the body about an axis \(l\), through \(A\), in the plane of the body and tangential to the circle. A particle \(P\) of mass \(M\) is now attached to the body at \(C\). The system is able to rotate freely about the fixed axis \(l\), which is horizontal. The system is released from rest with \(A C\) making an angle of \(60 ^ { \circ }\) with the upward vertical. Find, in terms of \(a\) and \(g\), the greatest speed of \(P\) in the subsequent motion.

3 \includegraphics[max width=\textwidth, alt={}, center]{020ebd88-b920-40ce-84cf-5c26d45e2935-2_355_695_1073_726} A uniform solid hemisphere, of radius \(a\) and mass \(M\), is placed with its curved surface in contact with a rough plane that is inclined at an angle \(\alpha\) to the horizontal. A particle \(P\) of mass \(m\) is attached to the rim of the hemisphere. The system rests in equilibrium with the rim of the hemisphere horizontal and \(P\) at the point on the rim that is closest to the inclined plane (see diagram). Given that the coefficient of friction between the plane and the hemisphere is \(\frac { 1 } { 2 }\), show that

1. \(\tan \alpha \leqslant \frac { 1 } { 2 }\),
2. \(m \leqslant \frac { M ( 1 + \sqrt { } 5 ) } { 4 }\).

A rigid body is made from uniform wire of negligible thickness and is in the form of a square \(A B C D\) of mass \(M\) enclosed within a circular ring of radius \(a\) and mass \(2 M\). The centres of the square and the circle coincide at \(O\) and the corners of the square are joined to the circle (see diagram). Show that the moment of inertia of the body about an axis through \(O\), perpendicular to the plane of the body, is \(\frac { 8 } { 3 } M a ^ { 2 }\). Hence find the moment of inertia of the body about an axis \(l\), through \(A\), in the plane of the body and tangential to the circle. A particle \(P\) of mass \(M\) is now attached to the body at \(C\). The system is able to rotate freely about the fixed axis \(l\), which is horizontal. The system is released from rest with \(A C\) making an angle of \(60 ^ { \circ }\) with the upward vertical. Find, in terms of \(a\) and \(g\), the greatest speed of \(P\) in the subsequent motion.

3 \includegraphics[max width=\textwidth, alt={}, center]{3daca234-9b7f-41d4-bbaa-d35615a120fc-2_419_1102_1859_520} The diagram shows two uniform rods \(B A\) and \(A C\), smoothly hinged at \(A\). The rod \(B A\) has length \(8 a\) and weight \(W\); the rod \(A C\) has length \(6 a\) and weight \(2 W\). The rods are in equilibrium in a vertical plane with \(B\) and \(C\) resting on a rough horizontal floor and angle \(C A B\) equal to \(90 ^ { \circ }\). Show that the normal contact force at \(B\) is \(\frac { 26 } { 25 } W\). The coefficient of friction between each rod and the floor is \(\mu\). Find the least possible value of \(\mu\).

The diagram shows a uniform rod \(A B\), of length \(4 a\) and weight \(W\), resting in equilibrium with its end \(A\) on rough horizontal ground. The rod rests at \(C\) on the surface of a smooth cylinder whose axis is horizontal. The cylinder rests on the ground and is fixed to it. The rod is in a vertical plane perpendicular to the axis of the cylinder and is inclined at an angle \(\theta\) to the horizontal, where \(\cos \theta = \frac { 3 } { 5 }\). A particle of weight \(k W\) is attached to the rod at \(B\). Given that \(A C = 3 a\), show that the least possible value of the coefficient of friction \(\mu\) between the rod and the ground is \(\frac { 8 ( 2 k + 1 ) } { 13 k + 19 }\). Given that \(\mu = \frac { 9 } { 10 }\), find the set of values of \(k\) for which equilibrium is possible.

1 \includegraphics[max width=\textwidth, alt={}, center]{137d2806-f45c-4121-8ee9-bf89580e1cca-2_684_714_246_717} A uniform \(\operatorname { rod } A B\), of mass \(m\) and length \(4 a\), rests with the end \(A\) on rough horizontal ground. The point \(C\) on \(A B\) is such that \(A C = 3 a\). A light inextensible string has one end attached to the point \(P\) which is at a distance \(5 a\) vertically above \(A\), and the other end attached to \(C\). The rod and the string are in the same vertical plane and the system is in equilibrium with angle \(A C P\) equal to \(90 ^ { \circ }\) (see diagram). The coefficient of friction between the rod and the ground is \(\mu\). Show that the least possible value of \(\mu\) is \(\frac { 24 } { 43 }\).

1 \includegraphics[max width=\textwidth, alt={}, center]{a473cbb8-877f-48df-8751-c76d96396734-2_684_714_246_717} A uniform \(\operatorname { rod } A B\), of mass \(m\) and length \(4 a\), rests with the end \(A\) on rough horizontal ground. The point \(C\) on \(A B\) is such that \(A C = 3 a\). A light inextensible string has one end attached to the point \(P\) which is at a distance \(5 a\) vertically above \(A\), and the other end attached to \(C\). The rod and the string are in the same vertical plane and the system is in equilibrium with angle \(A C P\) equal to \(90 ^ { \circ }\) (see diagram). The coefficient of friction between the rod and the ground is \(\mu\). Show that the least possible value of \(\mu\) is \(\frac { 24 } { 43 }\).

A uniform rod \(A B\) rests in limiting equilibrium in a vertical plane with the end \(A\) on rough horizontal ground and the end \(B\) against a rough vertical wall that is perpendicular to the plane of the rod. The angle between the rod and the ground is \(\theta\). The coefficient of friction between the rod and the wall is \(\mu\), and the coefficient of friction between the rod and the ground is \(2 \mu\). Show that \(\tan \theta = \frac { 1 - 2 \mu ^ { 2 } } { 4 \mu }\). Given that \(\theta \leqslant 45 ^ { \circ }\), find the set of possible values of \(\mu\).

4 A uniform \(\operatorname { rod } A B\) has length \(2 a\) and weight \(W\). The end \(A\) rests on rough horizontal ground and the end \(B\) rests against a smooth vertical wall. The rod is in a vertical plane that is perpendicular to the wall. The angle between the rod and the horizontal is \(\theta\). A particle of weight \(5 W\) hangs from the rod at the point \(C\), with \(A C = x a\), where \(0 < x < 1\).

1. By taking moments about \(A\), show that the magnitude of the normal reaction at \(B\) is \(\frac { W ( 5 x + 1 ) } { 2 \tan \theta }\).
   [0pt] [3]
   The particle of weight \(5 W\) is now moved a distance \(a\) up the rod, so that \(A C = ( x + 1 ) a\). This results in the magnitude of the normal reaction at \(B\) being double its previous value. The system remains in equilibrium with the rod at angle \(\theta\) with the horizontal.
2. Show that \(x = \frac { 4 } { 5 }\).
   The coefficient of friction between the rod and the ground is \(\frac { 2 } { 3 }\).
3. Given that the rod is about to slip when the particle of weight \(5 W\) is in its second position, find the value of \(\tan \theta\).
   \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Axis \(l\)} \includegraphics[alt={},max width=\textwidth]{0eb3892f-628f-449a-b022-b38170754d89-08_462_693_301_731}
   \end{figure} Three thin uniform rings \(A , B\) and \(C\) are joined together, so that each ring is in contact with each of the other two rings. Ring \(A\) has radius \(2 a\) and mass \(3 M\); rings \(B\) and \(C\) each have radius \(3 a\) and mass \(2 M\). The rings lie in the same plane and the centres of the rings are at the vertices of an isosceles triangle. The object consisting of the three rings is free to rotate about the horizontal axis \(l\) which is tangential to ring \(A\), in the plane of the object and perpendicular to the line of symmetry of the object (see diagram).

An object is formed from a square frame \(A B C D\) with a square lamina attached in one corner of the frame. The frame consists of four identical thin rods, each of mass \(M\) and length \(2 a\). The lamina has mass \(k M\) and edges of length \(a\). It has one vertex at \(C\) and adjacent sides in contact with \(C B\) and \(C D\) (see diagram).

1. Show that the moment of inertia of the object about an axis \(l\) through \(A\) perpendicular to the plane of the object is \(\frac { 2 } { 3 } M a ^ { 2 } ( 7 k + 20 )\).
   The object is released from rest with the edge \(A B\) horizontal and \(D\) vertically above \(A\). The object rotates freely about the fixed axis \(l\). The angular speed of the object is \(\frac { 1 } { 2 } \sqrt { } \left( \frac { 5 g } { a } \right)\) when \(D\) is first vertically below \(A\).
2. Find the value of \(k\).