Composite solid with hemisphere and cylinder/cone

3 \begin{figure}[h] \end{figure} A uniform solid cylinder has mass 8 kg and height 16 cm . A uniform solid cone, whose base radius is the same as the radius of the cylinder, has mass 2 kg and height 12 cm . A composite solid is formed by joining the cylinder and cone so that the base of the cone coincides with one end of the cylinder (see Fig. 1).

1. Show that the centre of mass of the composite solid is 10.2 cm from its base. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{68acf474-5da2-4949-b3b2-fc42cd73bd4a-2_401_444_1877_849} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The composite solid is held with a point on the circumference of its base in contact with a horizontal table. The base makes an angle \(\theta ^ { \circ }\) with the table (see Fig. 2, which shows a cross-section). When the cone is released it moves towards the equilibrium position in which its base is in contact with the table.
2. Given that the radius of the base is 4 cm , find the greatest possible value of \(\theta\), correct to 1 decimal place.

3 An object consists of a uniform solid circular cone, of vertical height \(4 r\) and radius \(3 r\), and a uniform solid cylinder, of height \(4 r\) and radius \(3 r\). The circular base of the cone and one of the circular faces of the cylinder are joined together so that they coincide. The cone and the cylinder are made of the same material.

1. Find the distance of the centre of mass of the object from the end of the cylinder that is not attached to the cone.
2. Show that the object can rest in equilibrium with the curved surface of the cone in contact with a horizontal surface.

3 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held at the point \(A\), where \(O A\) makes an angle \(\theta\) with the downward vertical through \(O\), and with the string taut. The particle \(P\) is projected perpendicular to \(O A\) in an upwards direction with speed \(u\). It then starts to move along a circular path in a vertical plane. The string goes slack when \(P\) is at \(B\), where angle \(A O B\) is \(90 ^ { \circ }\) and the speed of \(P\) is \(\sqrt { \frac { 4 } { 5 } \mathrm { ag } }\).

1. Find the value of \(\sin \theta\).
2. Find, in terms of \(m\) and \(g\), the tension in the string when \(P\) is at \(A\). \includegraphics[max width=\textwidth, alt={}, center]{454be64a-204f-4fa4-a5fc-72fd88e1289f-06_846_767_258_689} An object is formed from a solid hemisphere, of radius \(2 a\), 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 \(O C\) 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 \(O C\) as shown, the centre of mass of the object is ( \(\mathrm { x } , \mathrm { y }\) ).

6 A solid object consists of a uniform hemisphere of radius 0.4 m attached to a uniform cylinder of radius 0.4 m so that the circumferences of their circular faces coincide. The hemisphere and cylinder each have weight 20 N . The centre of mass of the object lies at the centre \(O\) of their common circular face.

1. Show that the height of the cylinder is 0.3 m .
   A new object is made by cutting the cylinder in half and removing the half not attached to the hemisphere. The cut is perpendicular to the axis of symmetry, so the new object consists of a hemisphere and a cylinder half the height of the original cylinder.
2. Find the distance of the centre of mass of the new object from \(O\).
   The new object is placed with its hemispherical part on a rough horizontal surface. The new object is held in equilibrium by a force of magnitude \(P \mathrm {~N}\) acting along its axis of symmetry, which is inclined at \(30 ^ { \circ }\) to the horizontal.
3. Find \(P\).

2 A uniform solid cone has height 0.6 m and base radius 0.2 m . A uniform hollow cylinder, open at both ends, has the same dimensions. An object is made by putting the cone inside the cylinder so that the base of the cone coincides with one end of the cylinder (see diagram, which shows a cross-section). The total weight of the object is 60 N and its centre of mass is 0.25 m from the base of the cone. Calculate the weight of the cone.

2 A uniform solid object is made by attaching a cone to a cylinder so that the circumferences of the base of the cone and a plane face of the cylinder coincide. The cone and the cylinder each have radius 0.3 m and height 0.4 m .

1. Calculate the distance of the centre of mass of the object from the vertex of the cone.
   [0pt] [The volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
   The object has weight \(W \mathrm {~N}\) and is placed with its plane circular face on a rough horizontal surface. A force of magnitude \(k W \mathrm {~N}\) acting at \(30 ^ { \circ }\) to the upward vertical is applied to the vertex of the cone. The object does not slip.
2. Find the greatest possible value of \(k\) for which the object does not topple.

5. A solid \(S\) consists of a uniform solid hemisphere of radius \(r\) and a uniform solid circular cylinder of radius \(r\) and height \(3 r\). The circular face of the hemisphere is joined to one of the circular faces of the cylinder, so that the centres of the two faces coincide. The other circular face of the cylinder has centre \(O\). The mass per unit volume of the hemisphere is \(3 k\) and the mass per unit volume of the cylinder is \(k\).

1. Show that the distance of the centre of mass of \(S\) from \(O\) is \(\frac { 9 r } { 4 }\) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2c0bb9ea-31a6-42f1-9e2e-d792eee8fd10-07_501_1082_653_422} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} The solid \(S\) is held in equilibrium by a horizontal force of magnitude \(P\). The circular face of \(S\) has one point in contact with a fixed rough horizontal plane and is inclined at an angle \(\alpha\) to the horizontal. The force acts through the highest point of the circular face of \(S\) and in the vertical plane through the axis of the cylinder, as shown in Figure 2. The coefficient of friction between \(S\) and the plane is \(\mu\). Given that \(S\) is on the point of slipping along the plane in the same direction as \(P\),
2. show that \(\mu = \frac { 1 } { 8 } ( 9 - 4 \cot \alpha )\).

4. \begin{figure}[h] \end{figure} A body consists of a uniform solid circular cylinder \(C\), together with a uniform solid hemisphere \(H\) which is attached to \(C\). The plane face of \(H\) coincides with the upper plane face of \(C\), as shown in Figure 2. The cylinder \(C\) has base radius \(r\), height \(h\) and mass 3M. The mass of \(H\) is \(2 M\). The point \(O\) is the centre of the base of \(C\).

1. Show that the distance of the centre of mass of the body from \(O\) is $$\frac { 14 h + 3 r } { 20 } .$$ The body is placed with its plane face on a rough plane which is inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 4 } { 3 }\). The plane is sufficiently rough to prevent slipping. Given that the body is on the point of toppling,
2. find \(h\) in terms of \(r\).

2. \begin{figure}[h] \end{figure} A uniform solid consists of a right circular cone of radius \(r\) and height \(k r\), where \(k > \sqrt { } 3\), 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 1.

1. Show that the distance of the centre of mass of the solid from \(O\) is $$\frac { \left( k ^ { 2 } - 3 \right) r } { 4 ( k + 2 ) }$$ 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. The angle between \(A O\) and the vertical is \(\theta\), where \(\tan \theta = \frac { 11 } { 14 }\)
2. Find the value of \(k\).

3. \begin{figure}[h] \end{figure} A solid consists of a uniform solid right cylinder of height \(5 l\) and radius \(3 l\) joined to a uniform solid hemisphere of radius \(3 l\). The plane face of the hemisphere coincides with a circular end of the cylinder and has centre \(O\), as shown in Figure 2. The density of the hemisphere is twice the density of the cylinder.

1. Find the distance of the centre of mass of the solid from \(O\).
   (5) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{826ad8ff-6e5c-4224-88ba-e78b79d1bc21-04_618_807_1327_571} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} The solid is now placed with its circular face on a plane inclined at an angle \(\theta ^ { \circ }\) to the horizontal, as shown in Figure 3. The plane is sufficiently rough to prevent the solid slipping. The solid is on the point of toppling.
2. Find the value of \(\theta\).

5 A uniform solid is made of a hemisphere with centre \(O\) and radius 0.6 m , and a cylinder of radius 0.6 m and height 0.6 m . The plane face of the hemisphere and a plane face of the cylinder coincide. (The formula for the volume of a sphere is \(\frac { 4 } { 3 } \pi r ^ { 3 }\).)

1. Show that the distance of the centre of mass of the solid from \(O\) is 0.09 m .
2. \includegraphics[max width=\textwidth, alt={}, center]{941c0c81-a74f-49c0-acb7-1c23266fc2c8-03_636_1036_982_593} The solid is placed with the curved surface of the hemisphere on a rough horizontal surface and the axis inclined at \(45 ^ { \circ }\) to the horizontal. The equilibrium of the solid is maintained by a horizontal force of 2 N applied to the highest point on the circumference of its plane face (see diagram). Calculate
   1. the mass of the solid,
   2. the set of possible values of the coefficient of friction between the surface and the solid.

2 \begin{figure}[h] \end{figure} A child's toy is a uniform solid consisting of a hemisphere of radius \(r \mathrm {~cm}\) joined to a cone of base radius \(r \mathrm {~cm}\). The curved surface of the cone makes an angle \(\alpha\) with its base. The two shapes are joined at the plane faces with their circumferences coinciding (see Fig. 1). The distance of the centre of mass of the toy above the common circular plane face is \(x \mathrm {~cm}\).
[0pt] [The volume of a sphere is \(\frac { 4 } { 3 } \pi r ^ { 3 }\) and the volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]

1. Show that \(x = \frac { r \left( \tan ^ { 2 } \alpha - 3 \right) } { 8 + 4 \tan \alpha }\). The toy is placed on a horizontal surface with the hemisphere in contact with the surface. The toy is released from rest from the position in which the common plane circular face is vertical (see Fig. 2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5addd79d-d502-455c-936f-27005483164e-2_193_670_1827_699} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
2. Find the set of values of \(\alpha\) such that the toy moves to the upright position.

8 \begin{figure}[h] \end{figure} An object consists of a uniform solid hemisphere of weight 40 N and a uniform solid cylinder of weight 5 N . The cylinder has height \(h \mathrm {~m}\). The solids have the same base radius 0.8 m and are joined so that the hemisphere's plane face coincides with one of the cylinder's faces. The centre of the common face is the point \(O\) (see Fig. 1). The centre of mass of the object lies inside the hemisphere and is at a distance of 0.2 m from \(O\).

1. Show that \(h = 1.2\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{9951c978-37e6-4d89-9fe3-c1e5e28b221e-4_620_1065_1297_541} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} One end of a light inextensible string is attached to a point on the circumference of the upper face of the cylinder. The string is horizontal and its other end is tied to a fixed point on a rough plane. The object rests in equilibrium on the plane with its axis of symmetry vertical. The plane makes an angle of \(30 ^ { \circ }\) with the horizontal (see Fig. 2). The tension in the string is \(T \mathrm {~N}\) and the frictional force acting on the object is \(F \mathrm {~N}\).
2. By taking moments about \(O\), express \(F\) in terms of \(T\).
3. Find another equation connecting \(T\) and \(F\). Hence calculate the tension and the frictional force.

3 Fig. 3.1 shows the curve with equation \(y = x ^ { 2 } + 3\). The region \(R\), shown shaded, is bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 2\). A uniform solid of revolution S is formed by rotating the region R through \(2 \pi\) about the \(x\)-axis. The volume of \(S\) is \(\frac { 202 } { 5 } \pi\). \begin{figure}[h] \end{figure} \section*{(a) In this question you must show detailed reasoning.} Show that the \(x\)-coordinate of the centre of mass of S is \(\frac { 395 } { 303 }\). S is fixed to a cylinder of base radius 3 units and height 2 units to form the uniform solid D . The smaller circular face of S is joined to the top circular face of the cylinder, as shown in Fig. 3.2. \begin{figure}[h] \end{figure} (b) Find the distance of the centre of mass of D from its smaller circular face. D is placed with its smaller circular face in contact with a rough plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. It is given that D does not slip.
(c) Determine whether D topples.

3 In this question you must show detailed reasoning. [In this question you may use the formula: Volume of cone \(= \frac { 1 } { 3 } \times\) base area × height.]
The region between the line \(\mathrm { y } = - 3 \mathrm { x } + 3 \mathrm { a }\), where \(a > 0\), the \(x\)-axis and the \(y\)-axis is rotated about the \(y\)-axis to form a uniform right circular cone C with base radius \(a\).

1. Show that the centre of mass of C is \(\frac { 3 } { 4 } a\) from its base. The cone C is fixed on top of a uniform cube, B , of edge length \(2 a\), so that there is no overlap. Fig. 3.1 shows a side view of C and B fixed together; Fig. 3.2 shows a view of C and B from above. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{37798594-8cb0-48aa-8401-090f09e25dff-3_570_323_785_246} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{37798594-8cb0-48aa-8401-090f09e25dff-3_309_319_982_753} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure} The centre of mass of the combined shape lies on the boundary of C and B .
   The density of \(B\) is not equal to the density of \(C\).
2. Determine the exact value of \(\frac { \text { density of } \mathrm { C } } { \text { density of } \mathrm { B } }\).
   [0pt] [3]

\includegraphics{figure_7}

\includegraphics{figure_4} The diagram shows the cross-section of a uniform solid consisting of a cylinder of radius \(0.4\) m and height \(1.5\) m with a hemisphere of radius \(0.4\) m on top.

1. Find the distance of the centre of mass above the base of the cylinder. [5]
2. The solid can just rest in equilibrium on a plane inclined at angle \(\alpha\) to the horizontal. Find \(\alpha\). [3]

\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_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]

1. Prove that the centre of mass of a uniform solid hemisphere of radius \(r\) is at a distance \(\frac{3r}{8}\) from its plane face. [7 marks]

A solid cylinder of radius \(\frac{3r}{4}\) and height \(kr\), where \(k < 1\), is welded to a uniform hemisphere of radius \(r\) made of the same material, so that their axes of symmetry coincide. The figure shows the cross section of the resulting solid. If the centre of mass of this solid is at \(O\), the centre of the plane face of the hemisphere, \includegraphics{figure_5}

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