6.04e Rigid body equilibrium: coplanar forces

7. \begin{figure}[h] \end{figure} Diagram not drawn to scale A uniform right circular solid cylinder has radius \(3 a\) and height \(2 a\). A right circular cone of height \(\frac { 3 a } { 2 }\) and base radius \(2 a\) is removed from the cylinder to form a solid \(S\), as shown in Figure 4. The plane face of the cone coincides with the upper plane face of the cylinder and the centre \(O\) of the plane face of the cone is also the centre of the upper plane face of the cylinder.

1. Show that the distance of the centre of mass of \(S\) from \(O\) is \(\frac { 69 a } { 64 }\). The point \(A\) is on the open face of \(S\) such that \(O A = 3 a\), as shown in Figure 4. The solid is now suspended from \(A\) and hangs freely in equilibrium.
2. Find the angle between \(O A\) and the horizontal.
   (3) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e5b08946-7311-4cf7-9c5f-5f309a1feed7-13_543_826_1653_557} \captionsetup{labelformat=empty} \caption{Figure 5}
   \end{figure} The solid is now placed on a rough inclined plane with the face through \(A\) in contact with the inclined plane, as shown in Figure 5. The solid rests in equilibrium on this plane. The coefficient of friction between the plane and \(S\) is 0.6 and the plane is inclined at an angle \(\phi ^ { \circ }\) to the horizontal. Given that \(S\) is on the point of sliding down the plane,
3. show that \(\phi = 31\) to 2 significant figures.

6. \begin{figure}[h] \end{figure} The shaded region \(R\) is bounded by part of the curve with equation \(y = x ^ { 2 } + 3\), the \(x\)-axis, the \(y\)-axis and the line with equation \(x = 2\), as shown in Figure 4. The unit of length on each axis is one centimetre. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid \(S\).
Using algebraic integration,

1. show that the volume of \(S\) is \(\frac { 202 } { 5 } \pi \mathrm {~cm} ^ { 3 }\),
2. show that, to 2 decimal places, the centre of mass of \(S\) is 1.30 cm from \(O\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b7cfcf0a-8f54-4350-8e07-a3b51d94d0f2-11_478_472_1407_762} \captionsetup{labelformat=empty} \caption{Figure 5}
   \end{figure} A uniform right circular solid cone, of base radius 7 cm and height 6 cm , is joined to \(S\) to form a solid \(T\). The base of the cone coincides with the larger plane face of \(S\), as shown in Figure 5. The vertex of the cone is \(V\).
   The mass per unit volume of \(S\) is twice the mass per unit volume of the cone.
3. Find the distance from \(V\) to the centre of mass of \(T\). The point \(A\) lies on the circumference of the base of the cone. The solid \(T\) is suspended from \(A\) and hangs freely in equilibrium.
4. Find the size of the angle between \(V A\) and the vertical.

2. \begin{figure}[h] \end{figure} A uniform solid right circular cone \(R\), with vertex \(V\), has base radius \(4 r\) and height \(4 h\). A right circular cone \(S\), also with vertex \(V\) and the same axis of symmetry as \(R\), has base radius \(3 r\) and height \(3 h\). The cone \(S\) is cut away from the cone \(R\) leaving a solid \(T\). The centre of the larger plane face of \(T\) is \(O\). Figure 1 shows the solid \(T\).

1. Find the distance from \(O\) to the centre of mass of \(T\). The point \(A\) lies on the circumference of the smaller plane face of \(T\). The solid is freely suspended from \(A\) and hangs in equilibrium. Given that \(h = r\)
2. find the size of the angle between \(O A\) and the downward vertical.

6. A uniform solid right circular cone has base radius \(r\) and height \(h\).

1. Use algebraic integration to show that the distance of the centre of mass of the cone from its vertex is \(\frac { 3 } { 4 } h\).
   [0pt] [You may assume that the volume of a cone of base radius \(r\) and height \(h\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) ] \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2273ca38-1e16-44ab-ae84-f4c576cbb8f9-20_394_716_632_621} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} A solid \(S\) is formed by joining a uniform right circular solid cone of mass \(5 m\) to a uniform solid hemisphere, of radius \(r\) and mass \(k m\) where \(k < 20\). The cone has base radius \(r\) and height \(6 r\). The plane face of the cone coincides with the plane face of the hemisphere. The centre of the plane face of the cone is \(O\) and the point \(A\) is on the circular edge of this plane face, as shown in Figure 3.
2. Find the distance from \(O\) to the centre of mass of \(S\). The solid is suspended from \(A\) and hangs freely in equilibrium. The angle between the axis of the cone and the horizontal is \(30 ^ { \circ }\).
3. Find, to the nearest whole number, the value of \(k\).

4.

1. Use algebraic integration to show that the centre of mass of a uniform solid hemisphere of radius \(a\) is a distance \(\frac { 3 } { 8 } a\) from the centre of its plane face.
   [0pt] [You may assume that the volume of a sphere of radius \(r\) is \(\frac { 4 } { 3 } \pi r ^ { 3 }\) ] \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ace84823-db30-463e-b24b-f0cd7df73746-08_444_764_539_591} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} A uniform solid hemisphere has mass \(m\) and radius \(a\). A particle of mass \(k m\) is attached to a point \(A\) on the circumference of the plane face of the hemisphere to form the loaded solid \(S\). The centre of the plane face of the hemisphere is the point \(O\), as shown in Figure 4. The loaded solid \(S\) is placed on a horizontal plane. The curved surface of \(S\) is in contact with the plane and \(S\) rests in equilibrium with \(O A\) making an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \sqrt { 3 }\)
2. Find the exact value of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{ace84823-db30-463e-b24b-f0cd7df73746-09_2255_50_314_34}
   

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1. A uniform solid right circular cone \(C\) has base radius \(r\), height \(H\) and vertex \(V\). A uniform solid \(S\), shown in Figure 3, is formed by removing from \(C\) a uniform solid right circular cone of height \(h ( h < H )\) that has the same base and axis of symmetry as \(C\).

\begin{figure}[h] \end{figure}

1. Show that the distance of the centre of mass of \(S\) from \(V\) is $$\frac { 1 } { 4 } ( 3 H - h )$$ The solid \(S\) is suspended by two vertical light strings. The first string is attached to \(S\) at \(V\) and the second string is attached to \(S\) at a point on the circumference of the circular base of \(S\).
   The solid \(S\) hangs freely in equilibrium with its axis of symmetry horizontal.
   The tension in the first string is \(T _ { 1 }\) and the tension in the second string is \(T _ { 2 }\)
2. Find \(\frac { T _ { 1 } } { T _ { 2 } }\), giving your answer in terms of \(H\) and \(h\), in its simplest form.

A uniform right solid circular cone \(C\) has radius \(r\) and height \(4 r\).

1. Show, using algebraic integration, that the distance of the centre of mass of \(C\) from its vertex is \(3 r\).
   [0pt] [You may assume that the volume of \(C\) is \(\frac { 4 } { 3 } \pi r ^ { 3 }\) ] A uniform solid \(S\), shown below in Figure 3, is formed by removing from \(C\) a uniform solid right circular cylinder of height \(r\) and radius \(\frac { 1 } { 2 } r\), where the centre of one end of the cylinder coincides with the centre of the plane face of \(C\) and the axis of the cylinder coincides with the axis of \(C\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{176ae8f8-7de9-4993-825a-6067614526ae-12_661_1194_861_440} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure}
2. Show that the distance of the centre of mass of \(S\) from the vertex of \(C\) is \(\frac { 75 } { 26 } r\) A rough plane is inclined at an angle \(\alpha\) to the horizontal.
   The solid \(S\) rests in equilibrium with its plane face in contact with the inclined plane.
   Given that \(S\) is on the point of toppling,
3. find the exact value of \(\tan \alpha\)

\hspace{0pt} [You may assume that the volume of a cone of height \(h\) and base radius \(r\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).] A uniform solid right circular cone \(C\), with vertex \(V\), has base radius \(r\) and height \(h\).

1. Show that the centre of mass of \(C\) is \(\frac { 3 } { 4 } h\) from \(V\) A solid \(F\), shown below in Figure 4, is formed by removing the solid right circular cone \(C ^ { \prime }\) from \(C\), where cone \(C ^ { \prime }\) has height \(\frac { 1 } { 3 } h\) and vertex \(V\) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{9777abb8-a564-40d5-8d96-d5649913737b-24_666_670_854_639} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure}
2. Show that the distance of the centre of mass of \(F\) from its larger plane face is \(\frac { 3 } { 13 } h\) The solid \(F\) rests in equilibrium with its curved surface in contact with a horizontal plane.
3. Show that \(13 r ^ { 2 } \leqslant 17 h ^ { 2 }\)
   \includegraphics[max width=\textwidth, alt={}]{9777abb8-a564-40d5-8d96-d5649913737b-28_2642_1844_116_114}

6. \begin{figure}[h] \end{figure} The shaded region \(R\) is bounded by part of the curve with equation \(y = x ^ { 2 } + 3\), the \(x\)-axis, the \(y\)-axis and the line with equation \(x = 2\), as shown in Figure 4. The unit of length on each axis is one centimetre. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid \(S\).
Using algebraic integration,

1. show that the volume of \(S\) is \(\frac { 202 } { 5 } \pi \mathrm {~cm} ^ { 3 }\),
2. show that, to 2 decimal places, the centre of mass of \(S\) is 1.30 cm from \(O\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{bb73b211-7629-4ed7-9b71-91841c29bb85-20_483_469_1402_767} \captionsetup{labelformat=empty} \caption{Figure 5}
   \end{figure} A uniform right circular solid cone, of base radius 7 cm and height 6 cm , is joined to \(S\) to form a solid \(T\). The base of the cone coincides with the larger plane face of \(S\), as shown in Figure 5. The vertex of the cone is \(V\).
   The mass per unit volume of \(S\) is twice the mass per unit volume of the cone.
3. Find the distance from \(V\) to the centre of mass of \(T\). The point \(A\) lies on the circumference of the base of the cone. The solid \(T\) is suspended from \(A\) and hangs freely in equilibrium.
4. Find the size of the angle between \(V A\) and the vertical.

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   Q6

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2. \begin{figure}[h] \end{figure} 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 \(\theta ^ { \circ }\) with the upward vertical, as shown in Figure 1. Find, to one decimal place, the value of \(\theta\).

6. Figure 2 \includegraphics[max width=\textwidth, alt={}, center]{c4b453e7-8a32-458b-8041-58c9e4ef9533-5_691_1067_241_584} A uniform solid cylinder has radius \(2 a\) and height \(\frac { 3 } { 2 } a\). A hemisphere of radius \(a\) is removed from the cylinder. The plane face of the hemisphere coincides with the upper plane face of the cylinder, and the centre \(O\) of the hemisphere is also the centre of this plane face, as shown in Fig. 2. The remaining solid is \(S\).

1. Find the distance of the centre of mass of \(S\) from \(O\).
   (6) The lower plane face of \(S\) rests in equilibrium on a desk lid which is inclined at an angle \(\theta\) to the horizontal. Assuming that the lid is sufficiently rough to prevent \(S\) from slipping, and that \(S\) is on the point of toppling when \(\theta = \alpha\),
2. find the value of \(\alpha\).
   (3) Given instead that the coefficient of friction between \(S\) and the lid is 0.8 , and that \(S\) is on the point of sliding down the lid when \(\theta = \beta\),
3. find the value of \(\beta\).
   (3)

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 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 \(\theta ^ { \circ }\) with the upward vertical, as shown in Figure 1. Find, to one decimal place, the value of \(\theta\).

3. \begin{figure}[h] \end{figure} A uniform solid \(S\) is formed by taking a uniform solid right circular cone, of base radius \(2 r\) and height \(2 h\), and removing the cone, with base radius \(r\) and height \(h\), which has the same vertex as the original cone, as shown in Figure 1.

1. Show that the distance of the centre of mass of \(S\) from its larger plane face is \(\frac { 11 } { 28 } h\). The solid \(S\) lies with its larger plane face on a rough table which is inclined at an angle \(\theta ^ { \circ }\) to the horizontal. The table is sufficiently rough to prevent \(S\) from slipping. Given that \(h = 2 r\),
2. find the greatest value of \(\theta\) for which \(S\) does not topple.

6. \begin{figure}[h] \end{figure} The region \(R\) is bounded by part of the curve with equation \(y = 4 - x ^ { 2 }\), the positive \(x\)-axis and the positive \(y\)-axis, as shown in Figure 3. The unit of length on both axes is one metre. A uniform solid \(S\) is formed by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis.

1. Show that the centre of mass of \(S\) is \(\frac { 5 } { 8 } \mathrm {~m}\) from \(O\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8374fa0f-cb28-497f-8696-877d7d0762f1-09_702_584_1138_676} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} Figure 4 shows a cross section of a uniform solid \(P\) consisting of two components, a solid cylinder \(C\) and the solid \(S\). The cylinder \(C\) has radius 4 m and length \(l\) metres. One end of \(C\) coincides with the plane circular face of \(S\). The point \(A\) is on the circumference of the circular face common to \(C\) and \(S\). When the solid \(P\) is freely suspended from \(A\), the solid \(P\) hangs with its axis of symmetry horizontal.
2. Find the value of \(l\).

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

7. Diagram NOT accurately drawn \begin{figure}[h] \end{figure} The shaded region \(R\) is bounded by the curve with equation \(y = \frac { 1 } { 2 } x ( 6 - x )\), the \(x\)-axis and the line \(x = 2\), as shown in Figure 1. The unit of length on both axes is 1 cm . A uniform solid \(P\) is formed by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis.

1. Show that the centre of mass of \(P\) is, to 3 significant figures, 1.42 cm from its plane face. The uniform solid \(P\) is placed with its plane face on an inclined plane which makes an angle \(\theta\) with the horizontal. Given that the plane is sufficiently rough to prevent \(P\) from sliding and that \(P\) is on the point of toppling when \(\theta = \alpha\),
2. find the angle \(\alpha\). Given instead that \(P\) is on the point of sliding down the plane when \(\theta = \beta\) and that the coefficient of friction between \(P\) and the plane is 0.3 ,
3. find the angle \(\beta\).

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 \(k\).

7. A particle \(P\) of mass 1.5 kg is attached to the mid-point of a light elastic string of natural length 0.30 m and modulus of elasticity \(\lambda\) newtons. The ends of the string are attached to two fixed points \(A\) and \(B\), where \(A B\) is horizontal and \(A B = 0.48 \mathrm {~m}\). Initially \(P\) is held at rest at the mid-point, \(M\), of the line \(A B\) and the tension in the string is 240 N .

1. Show that \(\lambda = 400\) The particle is now held at rest at the point \(C\), where \(C\) is 0.07 m vertically below \(M\). The particle is released from rest at \(C\).
2. Find the magnitude of the initial acceleration of \(P\).
3. Find the speed of \(P\) as it passes through \(M\).

5. \begin{figure}[h] \end{figure} A toy is formed by joining a uniform solid right circular cone, of base radius \(3 r\) and height \(4 r\), to a uniform solid cylinder, also of radius \(3 r\) and height \(4 r\). The cone and the cylinder are made from the same material, and the plane face of the cone coincides with a plane face of the cylinder, as shown in Fig. 2. The centre of this plane face is \(O\).

1. Find the distance of the centre of mass of the toy from \(O\). The point \(A\) lies on the edge of the plane face of the cylinder which forms the base of the toy. The toy is suspended from \(A\) and hangs in equilibrium.
2. Find, in degrees to one decimal place, the angle between the axis of symmetry of the toy and the vertical. The toy is placed with the curved surface of the cone on horizontal ground.
3. Determine whether the toy will topple.
   (4)

2. A closed container \(C\) consists of a thin uniform hollow hemispherical bowl of radius \(a\), together with a lid. The lid is a thin uniform circular disc, also of radius \(a\). The centre \(O\) of the disc coincides with the centre of the hemispherical bowl. The bowl and its lid are made of the same material.

1. Show that the centre of mass of \(C\) is at a distance \(\frac { 1 } { 3 } a\) from \(O\). The container \(C\) has mass \(M\). A particle of mass \(\frac { 1 } { 2 } M\) is attached to the container at a point \(P\) on the circumference of the lid. The container is then placed with a point of its curved surface in contact with a horizontal plane. The container rests in equilibrium with \(P , O\) and the point of contact in the same vertical plane.
2. Find, to the nearest degree, the angle made by the line \(P O\) with the horizontal.

4. \begin{figure}[h] \end{figure} A uniform solid hemisphere, of radius \(6 a\) and centre \(O\), has a solid hemisphere of radius \(2 a\), and centre \(O\), removed to form a bowl \(B\) as shown in Figure 3.

1. Show that the centre of mass of \(B\) is \(\frac { 30 } { 13 } a\) from \(O\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-07_735_614_1126_662} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} The bowl \(B\) is fixed to a plane face of a uniform solid cylinder made from the same material as \(B\). The cylinder has radius \(2 a\) and height \(6 a\) and the combined solid \(S\) has an axis of symmetry which passes through \(O\), as shown in Figure 4.
2. Show that the centre of mass of \(S\) is \(\frac { 201 } { 61 } a\) from \(O\). The plane surface of the cylindrical base of \(S\) is placed on a rough plane inclined at \(12 ^ { \circ }\) to the horizontal. The plane is sufficiently rough to prevent slipping.
3. Determine whether or not \(S\) will topple. \section*{
   \includegraphics[max width=\textwidth, alt={}]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-08_56_366_251_178}
   }

4. \begin{figure}[h] \end{figure} A container is formed by removing a right circular solid cone of height \(4 l\) from a uniform solid right circular cylinder of height \(6 l\). 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 \(2 l\) and the base of the cone has radius \(l\).

1. Find the distance of the centre of mass of the container from \(O\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{049ea68c-d15f-41f8-860e-0816d36a2748-07_460_588_1254_676} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} The container is placed on a plane which is inclined at an angle \(\theta ^ { \circ }\) 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.
2. Find the value of \(\theta\).

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 \(\theta\).