Centre of mass of composite shapes

6 \includegraphics[max width=\textwidth, alt={}, center]{02da6b6a-6db1-4bc3-ad4e-537e4f61dcac-3_412_629_258_758} The diagram shows a metal plate made by fixing together two pieces, \(O A B C D\) (shaded) and \(O A E D\) (unshaded). The piece \(O A B C D\) is a minor sector of a circle with centre \(O\) and radius \(2 r\). The piece \(O A E D\) is a major sector of a circle with centre \(O\) and radius \(r\). Angle \(A O D\) is \(\alpha\) radians. Simplifying your answers where possible, find, in terms of \(\alpha , \pi\) and \(r\),

1. the perimeter of the metal plate,
2. the area of the metal plate. It is now given that the shaded and unshaded pieces are equal in area.
3. Find \(\alpha\) in terms of \(\pi\).

5 \includegraphics[max width=\textwidth, alt={}, center]{9cdb00a6-1e86-4185-bb73-ed3ecab981ba-3_560_506_258_822} The diagram shows a metal plate \(O A B C\), consisting of a right-angled triangle \(O A B\) and a sector \(O B C\) of a circle with centre \(O\). Angle \(A O B = 0.6\) radians, \(O A = 6 \mathrm {~cm}\) and \(O A\) is perpendicular to \(O C\).

1. Show that the length of \(O B\) is 7.270 cm , correct to 3 decimal places.
2. Find the perimeter of the metal plate.
3. Find the area of the metal plate.

5 A particle \(P\) of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 24 N . The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at the point 0.4 m vertically below \(O\).

1. Find the greatest speed of \(P\).
2. Calculate the greatest distance of \(P\) below \(O\).
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b8e52188-f9a6-46fc-90bf-97965c6dd324-10_608_611_258_767} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Fig. 1 shows the cross-section of a solid cylinder through which a cylindrical hole has been drilled to make a uniform prism. The radius of the cylinder is \(5 r\) and the radius of the hole is \(r\). The centre of the hole is a distance \(2 r\) from the centre of the cylinder.

6 \includegraphics[max width=\textwidth, alt={}, center]{fe5c198d-5d05-4241-98f5-894ba92f7afe-4_447_736_269_701} \(P\) is the vertex of a uniform solid cone of mass 5 kg , and \(O\) is the centre of its base. Strings are attached to the cone at \(P\) and at \(O\). The cone hangs in equilibrium with \(P O\) horizontal and the strings taut. The strings attached at \(P\) and \(O\) make angles of \(\theta ^ { \circ }\) and \(20 ^ { \circ }\), respectively, with the vertical (see diagram, which shows a cross-section).

1. By taking moments about \(P\) for the cone, find the tension in the string attached at \(O\).
2. Find the value of \(\theta\) and the tension in the string attached at \(P\).

7 \includegraphics[max width=\textwidth, alt={}, center]{81be887c-ab01-4327-a5df-f25c68a6fdb6-3_586_527_1030_810} A uniform lamina \(A B C\) is in the form of a major segment of a circle with centre \(O\) and radius 0.35 m . The straight edge of the lamina is \(A B\), and angle \(A O B = \frac { 2 } { 3 } \pi\) radians (see diagram).

1. Show that the centre of mass of the lamina is 0.0600 m from \(O\), correct to 3 significant figures. The weight of the lamina is 14 N . It is placed on a rough horizontal surface with \(A\) vertically above \(B\) and the lowest point of the arc \(B C\) in contact with the surface. The lamina is held in equilibrium in a vertical plane by a force of magnitude \(F \mathrm {~N}\) acting at \(A\).
2. Find \(F\) in each of the following cases:
   1. the force of magnitude \(F \mathrm {~N}\) acts along \(A B\);
   2. the force of magnitude \(F \mathrm {~N}\) acts along the tangent to the circular arc at \(A\).

4. [The centre of mass of a uniform semicircular lamina of radius \(r\) is \(\frac { 4 r } { 3 \pi }\) from the centre.] \begin{figure}[h] \end{figure} The uniform rectangular lamina \(A B C D E F\) has sides \(A C = F D = 6 a\) and \(A F = C D = 3 a\). The point \(B\) lies on \(A C\) with \(A B = 2 a\) and the point \(E\) lies on \(F D\) with \(F E = 2 a\). The template, \(T\), shown shaded in Figure 3, is formed by removing the semicircular lamina with diameter \(B C\) from the rectangular lamina and then fixing this semicircular lamina to the opposite side, \(F D\), of the rectangular lamina. The diameter of the semicircular lamina coincides with \(E D\) and the semicircular arc \(E D\) is outside the rectangle \(A B C D E F\). All points of \(T\) lie in the same plane.

1. Show that the centre of mass of \(T\) is a distance \(\left( \frac { 9 + 2 \pi } { 6 } \right)\) a from \(A C\). The mass of \(T\) is \(M\). A particle of mass \(k M\) is attached to \(T\) at \(C\). The loaded template is freely suspended from \(A\) and hangs in equilibrium with \(A F\) at angle \(\phi\) to the downward vertical through \(A\). Given that \(\tan \phi = \frac { 3 } { 2 }\)
2. find the value of \(k\).
   \section*{\textbackslash section*\{Question 4 continued\}} \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-11_149_142_2604_1816}

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.

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 \(\theta\).

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 \(\theta\).

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 \includegraphics[max width=\textwidth, alt={}, center]{d6d87705-be4b-407d-b699-69fb441d88a7-2_710_572_721_788} A uniform solid hemisphere of weight 12 N and radius 6 cm is suspended by two vertical strings. One string is attached to the point \(O\), the centre of the plane face, and the other string is attached to the point \(A\) on the rim of the plane face. The hemisphere hangs in equilibrium and \(O A\) makes an angle of \(60 ^ { \circ }\) with the vertical (see diagram).

1. Find the horizontal distance from the centre of mass of the hemisphere to the vertical through \(O\).
2. Calculate the tensions in the strings.

8 A shape, \(S\), is formed by attaching a particle of mass \(2 m \mathrm {~kg}\) to the vertex of a uniform solid cone of mass \(8 m \mathrm {~kg}\). The height of the cone is \(h \mathrm {~m}\) and the radius of the base of the cone is 1.1 m .

1. Explain why the centre of mass of \(S\) must lie on the central axis of the cone. Two strings are attached to \(S\), one at the vertex of the cone and one at \(A\) which is a point on the edge of the base of \(S\). The other ends of the strings are attached to a horizontal ceiling in such a way that the strings are both vertical. The string attached to \(S\) at \(A\) is inextensible and has length 1.6 m . The string attached to \(S\) at the vertex is elastic with modulus of elasticity 8 mgN . Shape \(S\) is in equilibrium with its axis horizontal (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{05b479a4-4087-4332-924b-43b1aedbb4f2-6_654_1541_879_244}
2. Determine the natural length of the elastic string.

2 The position of the centre of mass, \(G\), of a uniform wire bent into the shape of an arc of a circle of radius \(r\) and centre C is shown in Fig. 2.1. \begin{figure}[h] \end{figure}

1. Use this information to show that the centre of mass, G , of the uniform wire bent into the shape of a semi-circular arc of radius 8 shown in Fig. 2.2 has coordinates \(\left( - \frac { 16 } { \pi } , 8 \right)\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-3_586_871_1016_806} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure} A walking-stick is modelled as a uniform rigid wire. The walking-stick and coordinate axes are shown in Fig. 2.3. The section from O to A is a semi-circular arc and the section OB lies along the \(x\)-axis. The lengths are in centimetres. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-3_394_958_1937_552} \captionsetup{labelformat=empty} \caption{Fig. 2.3}
   \end{figure}
2. Show that the coordinates of the centre of mass of the walking-stick are ( \(25.37,2.07\) ), correct to two decimal places. The walking-stick is now hung from a shelf as shown in Fig. 2.4. The only contact between the walking-stick and the shelf is at A . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-4_339_374_388_842} \captionsetup{labelformat=empty} \caption{Fig. 2.4}
   \end{figure}
3. When the walking-stick is in equilibrium, OB is at an angle \(\alpha\) to the vertical. Draw a diagram showing the position of the centre of mass of the walking-stick in relation to A .
   Calculate \(\alpha\).
4. The walking-stick is now held in equilibrium, with OB vertical and A still resting on the shelf, by means of a vertical force, \(F \mathrm {~N}\), at B . The weight of the walking-stick is 12 N . Calculate \(F\).

9 Two blocks have square cross sections. One block has mass 9 kg and its cross section has sides of length 20 cm
The other block has mass 1 kg and its cross section has sides of length 4 cm
The blocks are fixed together to form the composite body shown in Figure 1. \begin{figure}[h] \end{figure} 9

1. Find the distance of the centre of mass of the composite body from \(A F\) [0pt] [2 marks]
   Question 9 continues on the next page 9
2. A uniform rod has mass 12 kg and length 1 metre. One end of the rod rests against a smooth vertical wall.
   The other end of the rod rests on the composite body at point \(B\) The composite body is on a horizontal surface.
   The coefficient of friction between the composite body and the horizontal surface is 0.3 The angle between the rod and \(A B\) is \(60 ^ { \circ }\) A particle of mass \(m \mathrm {~kg}\) is fixed to the rod at a distance of 75 cm from \(B\) The rod, particle and composite body are shown in Figure 2. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{0afe3ff2-0af5-4aeb-98c5-1346fa803388-14_939_1020_1133_511}
   \end{figure} 9 (b) (i) Write down the magnitude of the vertical reaction force acting on the rod at \(B\) in terms of \(m\) and \(g\) [0pt] [1 mark] 9 (b) (ii) Show that the magnitude of the horizontal reaction force acting on the rod at \(B\) is $$\frac { g ( 6 + 0.75 m ) } { \sqrt { 3 } }$$ 9 (b) (iii) Find the maximum value of \(m\) for which the composite body does not slide or topple. Fully justify your answer.

4. \begin{figure}[h] \end{figure} The uniform lamina \(A B C D\) shown in Figure 2 is in the shape of an isosceles trapezium.

• \(B C\) is parallel to \(A D\) and angle \(B A D\) is equal to angle \(A D C\)
• \(B C = 5 a\) and \(A D = 7 a\)
• the perpendicular distance between \(B C\) and \(A D\) is \(3 a\)
• the distance of the centre of mass of \(A B C D\) from \(A D\) is \(d\)
  1. Show that \(d = \frac { 17 } { 12 } a\)

The uniform lamina \(P Q R S\) is a rectangle with \(P Q = 5 a\) and \(Q R = 9 a\).
The lamina \(A B C D\) in Figure 2 is used to cut a hole in \(P Q R S\) to form the template shown shaded in Figure 3. \begin{figure}[h] \end{figure}

The template is freely suspended from \(P\) and hangs in equilibrium with \(P S\) at an angle of \(\theta ^ { \circ }\) to the downward vertical.

Find the value of \(\theta\)

\includegraphics{figure_6} Fig. 1 shows the cross-section \(ABCDE\) through the centre of mass \(G\) of a uniform prism. The cross-section consists of a rectangle \(ABCF\) from which a triangle \(DEF\) has been removed; \(AB = 0.6\text{ m}\), \(BC = 0.7\text{ m}\) and \(DF = EF = 0.3\text{ m}\).

1. Show that the distance of \(G\) from \(BC\) is \(0.276\text{ m}\), and find the distance of \(G\) from \(AB\). [5] The prism is placed with \(CD\) on a rough horizontal surface. A force of magnitude \(2\text{ N}\) acting in the plane of the cross-section is applied to the prism. The line of action of the force passes through \(G\) and is perpendicular to \(DE\) (see Fig. 2). The prism is on the point of toppling about the edge through \(D\).
2. Calculate the weight of the prism. [3]

\includegraphics{figure_4} A uniform square lamina \(ABCD\) has sides of length \(10\text{cm}\). The point \(E\) is on \(BC\) with \(EC = 7.5\text{cm}\), and the point \(F\) is on \(DC\) with \(CF = x\text{cm}\). The triangle \(EFC\) is removed from \(ABCD\) (see diagram). The centre of mass of the resulting shape \(ABEFD\) is a distance \(\bar{x}\text{cm}\) from \(CB\) and a distance \(\bar{y}\text{cm}\) from \(CD\).

1. Show that \(\bar{x} = \frac{400 - x^2}{80 - 3x}\) and find a corresponding expression for \(\bar{y}\). [4]

The shape \(ABEFD\) is in equilibrium in a vertical plane with the edge \(DF\) resting on a smooth horizontal surface.

1. Find the greatest possible value of \(x\), giving your answer in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are constants to be determined. [3]

The point \(C\) is on the circumference of the base of the cone. When the combined solid is freely suspended from \(C\) and hanging in equilibrium, the diameter through \(C\) makes an angle \(\alpha\) with the downward vertical, where \(\tan \alpha = \frac{1}{5}\).

1. Given that the centre of mass of the combined solid is within the cylinder, find the value of \(k\). [4]

\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]