Hemisphere or sphere resting on plane or wall

2 \includegraphics[max width=\textwidth, alt={}, center]{fb79f949-567c-4dbb-8533-7b7278cad21c-2_839_330_539_906} \(A B\) is a diameter of a uniform solid hemisphere with centre \(O\), radius 10 cm and weight 12 N . One end of a light inextensible string is attached to the hemisphere at \(B\) and the other end is attached to a fixed point \(C\) of a vertical wall. The hemisphere is in equilibrium with \(A\) in contact with the wall at a point vertically below \(C\). The centre of mass \(G\) of the hemisphere is at the same horizontal level as \(A\), and angle \(A B C\) is a right angle (see diagram). Calculate the tension in the string.

3 \includegraphics[max width=\textwidth, alt={}, center]{9d377c95-09b8-4893-b29f-8517a5016e8b-2_786_1249_1455_447} A smooth hemispherical shell, with centre \(O\), weight 12 N and radius 0.4 m , rests on a horizontal plane. A particle of weight \(W \mathrm {~N}\) lies at rest on the inner surface of the hemisphere vertically below \(O\). A force of magnitude \(F \mathrm {~N}\) acting vertically upwards is applied to the highest point of the hemisphere, which is in equilibrium with its axis of symmetry inclined at \(20 ^ { \circ }\) to the horizontal (see diagram).

1. Show, by taking moments about \(O\), that \(F = 16.48\) correct to 4 significant figures.
2. Find the normal contact force exerted by the plane on the hemisphere in terms of \(W\). Hence find the least possible value of \(W\).

2 A uniform hemispherical shell of weight 8 N and a uniform solid hemisphere of weight 12 N are joined along their circumferences to form a non-uniform sphere of radius 0.2 m .

1. Show that the distance between the centre of mass of the sphere and the centre of the sphere is 0.005 m . This sphere is placed on a horizontal surface with its axis of symmetry horizontal. The equilibrium of the sphere is maintained by a force of magnitude \(F \mathrm {~N}\) acting parallel to the axis of symmetry applied to the highest point of the sphere.
2. Calculate \(F\).

2 \includegraphics[max width=\textwidth, alt={}, center]{334b4bdf-6d9c-4208-9032-572eb7c5f9ee-2_295_805_484_671} A uniform solid hemisphere of weight 60 N and radius 0.8 m rests in limiting equilibrium with its curved surface on a rough horizontal plane. The axis of symmetry of the hemisphere is inclined at an angle of \(\theta\) to the horizontal, where \(\cos \theta = 0.28\). Equilibrium is maintained by a horizontal force of magnitude \(P\) N applied to the lowest point of the circular rim of the hemisphere (see diagram).

1. Show that \(P = 8.75\).
2. Find the coefficient of friction between the hemisphere and the plane.

3. \begin{figure}[h] \end{figure} A bowl \(B\) consists of a uniform solid hemisphere, of radius \(r\) and centre \(O\), from which is removed a solid hemisphere, of radius \(\frac { 2 } { 3 } r\) and centre \(O\), as shown in Figure 1.

1. Show that the distance of the centre of mass of \(B\) from \(O\) is \(\frac { 65 } { 152 } r\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d831556d-fdf3-4639-9a89-6d3b372d3446-05_526_1014_1292_478} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} The bowl \(B\) has mass \(M\). A particle of mass \(k M\) is attached to a point \(P\) on the outer rim of \(B\). The system is placed with a point \(C\) on its outer curved surface in contact with a horizontal plane. The system is in equilibrium with \(P , O\) and \(C\) in the same vertical plane. The line \(O P\) makes an angle \(\theta\) with the horizontal as shown in Figure 2. Given that \(\tan \theta = \frac { 4 } { 5 }\),
2. find the exact value of \(k\). January 2010

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. \(m \leqslant \frac { M ( 1 + \sqrt { } 5 ) } { 4 }\).

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. \(m \leqslant \frac { M ( 1 + \sqrt { } 5 ) } { 4 }\).

Two identical uniform rough spheres \(A\) and \(B\), each of weight \(W\) and radius \(a\), are at rest on a rough horizontal plane, and are not in contact with each other. A third identical sphere \(C\) rests on \(A\) and \(B\) with its centre in the same vertical plane as the centres of \(A\) and \(B\). The line joining the centres of \(A\) and \(C\) and the line joining the centres of \(B\) and \(C\) are each inclined at an angle \(\theta\) to the vertical (see diagram). The coefficient of friction between each sphere and the plane is \(\mu\). The coefficient of friction between \(C\) and \(A\), and between \(C\) and \(B\), is \(\mu ^ { \prime }\). The system remains in equilibrium. Show that $$\mu \geqslant \frac { \sin \theta } { 3 ( 1 + \cos \theta ) } \quad \text { and } \quad \mu ^ { \prime } \geqslant \frac { \sin \theta } { 1 + \cos \theta } .$$

Two identical uniform rough spheres \(A\) and \(B\), each of weight \(W\) and radius \(a\), are at rest on a rough horizontal plane, and are not in contact with each other. A third identical sphere \(C\) rests on \(A\) and \(B\) with its centre in the same vertical plane as the centres of \(A\) and \(B\). The line joining the centres of \(A\) and \(C\) and the line joining the centres of \(B\) and \(C\) are each inclined at an angle \(\theta\) to the vertical (see diagram). The coefficient of friction between each sphere and the plane is \(\mu\). The coefficient of friction between \(C\) and \(A\), and between \(C\) and \(B\), is \(\mu ^ { \prime }\). The system remains in equilibrium. Show that $$\mu \geqslant \frac { \sin \theta } { 3 ( 1 + \cos \theta ) } \quad \text { and } \quad \mu ^ { \prime } \geqslant \frac { \sin \theta } { 1 + \cos \theta } .$$

4 \includegraphics[max width=\textwidth, alt={}, center]{b486decd-75b8-44bd-889f-2472f1163871-3_567_575_258_785} A uniform circular disc, with centre \(O\) and weight \(W\), rests in equilibrium on a horizontal floor and against a vertical wall. The plane of the disc is vertical and perpendicular to the wall. The disc is in contact with the floor at \(A\) and with the wall at \(B\). A force of magnitude \(P\) acts tangentially on the disc at the point \(C\) on the edge of the disc, where the radius \(O C\) makes an angle \(\theta\) with the upward vertical, and \(\tan \theta = \frac { 4 } { 3 }\) (see diagram). The coefficient of friction between the disc and the floor and between the disc and the wall is \(\frac { 1 } { 2 }\). Show that the sum of the magnitudes of the frictional forces at \(A\) and \(B\) is equal to \(P\). Given that the equilibrium is limiting at both \(A\) and \(B\),

1. show that \(P = \frac { 15 } { 34 } \mathrm {~W}\),
2. find the ratio of the magnitude of the normal reaction at \(A\) to the magnitude of the normal reaction at \(B\).

8 A capsule consists of a uniform hollow right circular cylinder of radius \(r\) and length \(2 h\) attached to two uniform hollow hemispheres of radius \(r\).
The centres of the plane faces of the hemispheres coincide with the centres, A and B , of the ends of the cylinder. \begin{figure}[h] \end{figure} Fig. 8 represents a vertical cross-section through a plane of symmetry of the capsule as it rests in limiting equilibrium with a point C of one hemisphere on a rough horizontal floor and a point D of the other hemisphere against a rough vertical wall. The total weight of the capsule is \(W\) and acts at a point midway between A and B . The plane containing \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D is vertical, with AB making an acute angle \(\theta\) with the downward vertical.

1. Complete the copy of Fig. 8 in the Printed Answer Booklet to show all the remaining forces acting on the capsule. The coefficient of friction at each point of contact is \(\frac { 1 } { 3 }\).
2. By resolving vertically and horizontally, determine the magnitude of the normal contact force between the floor and the capsule in terms of \(W\).
3. By determining an expression for \(r\) in terms of \(h\) and \(\theta\), show that \(\tan \theta > \frac { 3 } { 4 }\).