6.04c Composite bodies: centre of mass

4. The diagram shows a uniform lamina consisting of a rectangular section \(G P Q E\) with a semi-circular section EFG of radius 4 cm . Quadrants \(A P B\) and \(C Q D\) each with radius 2 cm are removed. Dimensions in cm are as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{3efc4ef6-8a80-4267-8e95-733200e875c5-3_758_604_497_651}

1. Write down the distance of the centre of mass of the lamina \(A B C D E F G\) from \(A G\).
2. Determine the distance of the centre of mass of the lamina \(A B C D E F G\) from \(B C\).
3. The lamina \(A B C D E F G\) is suspended freely from the point \(E\) and hangs in equilibrium. Calculate the angle EG makes with the vertical.

3 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
A composite body consists of a uniform rod, \(A B\), and a particle.
The rod has length 4 metres and mass 22.5 kilograms.
The particle, \(P\), has mass 20 kilograms and is placed on the rod at a distance of 0.3 metres from \(B\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{4fdb2637-6368-422c-99da-85b80efe31c5-04_163_1323_767_402} 3

1. Find the distance of the centre of mass of the body from \(A\). 3
2. The body rests in equilibrium in a horizontal position on two supports, \(C\) and \(D\).
   The support \(C\) is 0.5 metres from \(A\) and the support \(D\) is 1 metre from \(B\). Find the magnitudes of the forces exerted on the body by the supports.
   [0pt] [4 marks]

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a solid doorstop made of wood. The doorstop is modelled as a tetrahedron. Relative to a fixed origin \(O\), the vertices of the tetrahedron are \(A ( 2,1,4 )\), \(B ( 6,1,2 ) , C ( 4,10,3 )\) and \(D ( 5,8 , d )\), where \(d\) is a positive constant and the units are in centimetres.

1. Find the area of the triangle \(A B C\). Given that the volume of the doorstop is \(21 \mathrm {~cm} ^ { 3 }\)
2. find the value of the constant \(d\).

1. Figure 1 A thin uniform rod, of total length \(30 a\) and mass \(M\), is bent to form a frame. The frame is in the shape of a triangle \(A B C\), where \(A B = 12 a , B C = 5 a\) and \(C A = 13 a\), as shown in Figure 1.

1. Show that the centre of mass of the frame is \(\frac { 3 } { 2 } a\) from \(A B\). The frame is freely suspended from \(A\). A horizontal force of magnitude \(k M g\), where \(k\) is a constant, is applied to the frame at \(B\). The line of action of the force lies in the vertical plane containing the frame. The frame hangs in equilibrium with \(A B\) vertical.
2. Find the value of \(k\).

3. \begin{figure}[h] \end{figure} The lamina \(L\), shown in Figure 2, consists of a uniform square lamina \(A B D F\) and two uniform triangular laminas \(B D C\) and \(F D E\). The square has sides of length \(2 a\). The two triangles are identical. The straight lines \(B D E\) and \(F D C\) are perpendicular with \(B D = D F = 2 a\) and \(D C = D E = a\).
The mass per unit of area of the square is \(M\).
The mass per unit area of each triangle is \(3 M\).
The centre of mass of \(L\) is at the point \(G\).

1. Without doing any calculations, explain why \(G\) lies on \(A D\).
2. Show that the distance of \(G\) from \(D\) is \(\frac { \sqrt { 2 } } { 2 } a\) The lamina \(L\) is freely suspended from \(B\) and hangs in equilibrium.
3. Find the size of the angle between \(B E\) and the downward vertical.

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1. \begin{figure}[h] \end{figure} Five identical uniform rods are joined together to form the rigid framework \(A B C D\) shown in Figure 1. Each rod has weight \(W\) and length 4a. The points \(A , B , C\) and \(D\) all lie in the same plane. The centre of mass of the framework is at the point \(G\).

1. Explain why \(G\) is the midpoint of \(A C\). The framework is suspended from the ceiling by two vertical light inextensible strings. One string is attached to the framework at \(A\) and the other string is attached to the framework at \(B\). The framework hangs freely in equilibrium with \(A B\) horizontal.
2. Find
   1. the tension in the string attached at \(A\),
   2. the tension in the string attached at \(B\). A particle of weight \(k W\) is now attached to the framework at \(D\) and a particle of weight \(2 k W\) is now attached to the framework at \(C\). The framework remains in equilibrium with \(A B\) horizontal and the strings vertical. Either string will break if the tension in it exceeds \(6 W\).
3. Find the greatest possible value of \(k\).

4. \begin{figure}[h] \end{figure} The uniform triangular lamina \(A B C D E\) is such that angle \(C E A = 90 ^ { \circ } , C E = 9 a\) and \(E A = 6 a\). The point \(D\) lies on \(C E\), with \(D E = 3 a\). The point \(B\) on \(C A\) is such that \(D B\) is parallel to \(E A\) and \(D B = 4 a\). The triangular lamina is folded along the line \(D B\) to form the folded lamina \(A B D E C F\), as shown in Figure 2. The distance of the centre of mass of the triangular lamina from \(D C\) is \(d _ { 1 }\) The distance of the centre of mass of the folded lamina from \(D C\) is \(d _ { 2 }\)

1. Explain why \(d _ { 1 } = d _ { 2 }\) The folded lamina is freely suspended from \(B\) and hangs in equilibrium with \(B A\) inclined at an angle \(\alpha\) to the downward vertical through \(B\).
2. Find, to the nearest degree, the size of angle \(\alpha\).

1. \begin{figure}[h] \end{figure} Figure 1 shows a uniform rectangular lamina \(A B C D\) with \(A B = 2 a\) and \(A D = a\) The mass of the lamina is \(6 m\). A particle of mass \(2 m\) is attached to the lamina at \(A\), a particle of mass \(m\) is attached to the lamina at \(B\) and a particle of mass \(3 m\) is attached to the lamina at \(D\), to form a loaded lamina \(L\) of total mass \(12 m\).

1. Write down the distance of the centre of mass of \(L\) from \(A B\). You must give a reason for your answer.
2. Show that the distance of the centre of mass of \(L\) from \(A D\) is \(\frac { 2 a } { 3 }\) A particle of mass \(k m\) is now also attached to \(L\) at \(D\) to form a new loaded lamina \(N\).
3. Show that the distance of the centre of mass of \(N\) from \(A B\) is \(\frac { ( k + 6 ) a } { ( k + 12 ) }\) When \(N\) is freely suspended from \(A\) and is hanging in equilibrium, the side \(A B\) makes an angle \(\alpha\) with the vertical, where \(\tan \alpha = \frac { 3 } { 2 }\)
4. Find the value of \(k\).

1. \begin{figure}[h] \end{figure} A uniform rod of length \(72 a\) is cut into pieces. The pieces are used to make two rigid squares, \(A B C D\) and \(P Q R S\), with sides of length \(10 a\) and \(8 a\) respectively. The two squares are joined to form the rigid framework shown in Figure 1. The squares both lie in the same plane with the rod \(A B\) parallel to the rod \(P Q\).
Given that

• \(A D\) cuts \(P Q\) in the ratio \(3 : 5\)
• \(D C\) cuts \(Q R\) in the ratio 5:3
  1. explain why the centre of mass of square \(A B C D\) is at \(Q\).
  2. Find the distance of the centre of mass of the framework from \(B\).

3. \begin{figure}[h] \end{figure} The uniform lamina \(A B C D E F G H I J\) is shown in Figure 3.
The lamina has \(A J = 8 a , A B = 5 a\) and \(B C = D E = E F = F G = G H = H I = I J = 2 a\).
All the corners are right angles.

1. Show that the distance of the centre of mass of the lamina from \(A J\) is \(\frac { 49 } { 26 } a\) A light inextensible rope is attached to the lamina at \(A\) and another light inextensible rope is attached to the lamina at \(B\). The lamina hangs in equilibrium with both ropes vertical and \(A B\) horizontal. The weight of the lamina is \(W\).
2. Find, in terms of \(W\), the tension in the rope attached to the lamina at \(B\). The rope attached to \(B\) breaks and subsequently the lamina hangs freely in equilibrium, suspended from \(A\).
3. Find the size of the angle between \(A J\) and the downward vertical.

1. \begin{figure}[h] \end{figure} A uniform plane lamina is in the shape of an isosceles trapezium \(A B C D E F\), as shown shaded in Figure 1.

• \(B C E F\) is a square
• \(A B = C D = a\)
• \(B C = 3 a\)
  1. Show that the distance of the centre of mass of the lamina from \(A D\) is \(\frac { 11 a } { 8 }\)

The mass of the lamina is \(M\) The lamina is suspended by two light vertical strings, one attached to the lamina at \(A\) and the other attached to the lamina at \(F\) The lamina hangs freely in equilibrium, with \(B F\) horizontal.

Find, in terms of \(M\) and \(g\), the tension in the string attached at \(A\)

2. \begin{figure}[h] \end{figure} Uniform wire is used to form the framework shown in Figure 2.
In the framework

• \(A B C D\) is a rectangle with \(A D = 2 a\) and \(D C = a\)
• \(B E C\) is a semicircular arc of radius \(a\) and centre \(O\), where \(O\) lies on \(B C\)

The diameter of the semicircle is \(B C\) and the point \(E\) is such that \(O E\) is perpendicular to \(B C\). The points \(A , B , C , D\) and \(E\) all lie in the same plane.

1. Show that the distance of the centre of mass of the framework from \(B C\) is $$\frac { a } { 6 + \pi }$$ The framework is freely suspended from \(A\) and hangs in equilibrium with \(A E\) at an angle \(\theta ^ { \circ }\) to the downward vertical.
2. Find the value of \(\theta\). The mass of the framework is \(M\).
   A particle of mass \(k M\) is attached to the framework at \(B\).
   The centre of mass of the loaded framework lies on \(O A\).
3. Find the value of \(k\).

1. Three particles of masses \(4 m , 2 m\) and \(k m\) are placed at the points with coordinates \(( - 3 , - 1 ) , ( 6,1 )\) and \(( - 1,5 )\) respectively.

Given that the centre of mass of the three particles is at the point with coordinates \(( \bar { x } , \bar { y } )\)

1. show that \(\bar { x } = \frac { - k } { k + 6 }\)
2. find \(\bar { y }\) in terms of \(k\). Given that the centre of mass of the three particles lies on the line with equation \(y = 2 x + 3\)
3. find the value of \(k\). A fourth particle is placed at the point with coordinates \(( \lambda , 4 )\).
   Given that the centre of mass of the four particles also lies on the line with equation \(y = 2 x + 3\)
4. find the value of \(\lambda\).

4. \begin{figure}[h] \end{figure} A uniform triangular lamina \(A B C\) is isosceles, with \(A C = B C\). The midpoint of \(A B\) is \(M\). The length of \(A B\) is \(18 a\) and the length of \(C M\) is \(18 a\). The triangular lamina \(C D E\), with \(D E = 6 a\) and \(C D = 12 a\), has \(E D\) parallel to \(A B\) and \(M D C\) is a straight line. Triangle \(C D E\) is removed from triangle \(A B C\) to form the lamina \(L\), shown shaded in Figure 1. The distance of the centre of mass of \(L\) from \(M C\) is \(d\).

1. Show that \(d = \frac { 4 } { 7 } a\) The lamina \(L\) is suspended by two light inextensible strings. One string is attached to \(L\) at \(A\) and the other string is attached to \(L\) at \(B\).
   The lamina hangs in equilibrium in a vertical plane with the strings vertical and \(A B\) horizontal.
   The weight of \(L\) is \(W\)
2. Find, in terms of \(W\), the tension in the string attached to \(L\) at \(B\) The string attached to \(L\) at \(B\) breaks, so that \(L\) is now suspended from \(A\). When \(L\) is hanging in equilibrium in a vertical plane, the angle between \(A B\) and the downward vertical through \(A\) is \(\theta ^ { \circ }\)
3. Find the value of \(\theta\)

1. \begin{figure}[h] \end{figure} A uniform rod of length \(24 a\) is cut into seven pieces which are used to form the framework \(A B C D E F\) shown in Figure 1. It is given that

• \(A F = B E = C D = A B = F E = 4 a\)
• \(B C = E D = 2 a\)
• the rods \(A F , B E\) and \(C D\) are parallel
• the rods \(A B , B C , F E\) and \(E D\) are parallel
• \(A F\) is perpendicular to \(A B\)
• the rods all lie in the same plane

The distance of the centre of mass of the framework from \(A F\) is \(d\).

1. Show that \(d = \frac { 19 } { 6 } a\)
2. Find the distance of the centre of mass of the framework from \(A\).

4. \begin{figure}[h] \end{figure} The uniform triangular lamina \(A B C\) has \(A B\) perpendicular to \(A C\), \(A B = 9 a\) and \(A C = 6 a\). The point \(D\) on \(A B\) is such that \(A D = a\). The rectangle \(D E F G\), with \(D E = 2 a\) and \(E F = 3 a\), is removed from the lamina to form the template shown shaded in Figure 3. The distance of the centre of mass of the template from \(A C\) is \(d\).

1. Show that \(d = \frac { 23 } { 7 } a\) The template is freely suspended from \(A\) and hangs in equilibrium with \(A B\) at an angle \(\theta ^ { \circ }\) to the downward vertical through \(A\).
2. Find the value of \(\theta\) A new piece, of exactly the same size and shape as the template, is cut from a lamina of a different uniform material. The template and the new piece are joined together to form the model shown in Figure 4. Both parts of the model lie in the same plane. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fd8bc7b5-adee-4d67-b15d-571255b00b83-12_369_1185_1667_440} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} The weight of \(C P Q R S T A\) is \(W\) The weight of \(A D G F E B C\) is \(4 W\) The model is freely suspended from \(A\).
   A horizontal force of magnitude \(X\), acting in the same vertical plane as the model, is now applied to the model at \(T\) so that \(A C\) is vertical, as shown in Figure 4.
3. Find \(X\) in terms of \(W\).

3. \begin{figure}[h] \end{figure} Figure 1 shows the shape and dimensions of a template \(O P Q R S T U V\) made from thin uniform metal. \(O P = 5 \mathrm {~m} , P Q = 2 \mathrm {~m} , Q R = 1 \mathrm {~m} , R S = 1 \mathrm {~m} , T U = 2 \mathrm {~m} , U V = 1 \mathrm {~m} , V O = 3 \mathrm {~m}\).
Figure 1 also shows a coordinate system with \(O\) as origin and the \(x\)-axis and \(y\)-axis along \(O P\) and \(O V\) respectively. The unit of length on both axes is the metre. The centre of mass of the template has coordinates \(( \bar { x } , \bar { y } )\).

1. 1. Show that \(\bar { y } = 1\)
   2. Find the value of \(\bar { x }\). A new design requires the template to have its centre of mass at the point (2.5,1). In order to achieve this, two circular discs, each of radius \(r\) metres, are removed from the template which is shown in Figure 1, to form a new template \(L\). The centre of the first disc is ( \(0.5,0.5\) ) and the centre of the second disc is ( \(0.5 , a\) ) where \(a\) is a constant.
2. Find the value of \(r\).
3. 1. Explain how symmetry can be used to find the value of \(a\).
   2. Find the value of \(a\). The template \(L\) is now freely suspended from the point \(U\) and hangs in equilibrium.
4. Find the size of the angle between the line \(T U\) and the horizontal.

A flagpole, \(A B\), is 4 m long. The flagpole is modelled as a non-uniform rod so that, at a distance \(x\) metres from \(A\), the mass per unit length of the flagpole, \(m \mathrm {~kg} \mathrm {~m} ^ { - 1 }\), is given by \(m = 18 - 3 x\).

1. Show that the mass of the flagpole is 48 kg . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{9b995178-a4be-4d5a-95f8-6c2978ff01b3-12_515_439_502_806} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} The end \(A\) of the flagpole is fixed to a point on a vertical wall. A cable has one end attached to the midpoint of the flagpole and the other end attached to a point on the wall that is vertically above \(A\). The cable is perpendicular to the flagpole. The flagpole and the cable lie in the same vertical plane that is perpendicular to the wall. A small ball of mass 4 kg is attached to the flagpole at \(B\). The cable holds the flagpole and ball in equilibrium, with the flagpole at \(45 ^ { \circ }\) to the wall, as shown in Figure 3. The tension in the cable is \(T\) newtons.
   The cable is modelled as a light inextensible string and the ball is modelled as a particle.
2. Using the model, find the value of \(T\).
3. Give a reason why the answer to part (b) is not likely to be the true value of \(T\).

4. \begin{figure}[h] \end{figure} A uniform solid cylinder of base radius \(r\) and height \(\frac { 4 } { 3 } r\) has the same density as a uniform solid hemisphere of radius \(r\). The plane face of the hemisphere is joined to a plane face of the cylinder to form the composite solid \(S\) shown in Figure 3. The point \(O\) is the centre of the plane face of \(S\).

1. Show that the distance from \(O\) to the centre of mass of \(S\) is \(\frac { 73 } { 72 } r\) The solid \(S\) is placed with its plane face on a rough horizontal plane. The coefficient of friction between \(S\) and the plane is \(\mu\). A horizontal force \(P\) is applied to the highest point of \(S\). The magnitude of \(P\) is gradually increased.
2. Find the range of values of \(\mu\) for which \(S\) will slide before it starts to tilt.

1. \begin{figure}[h] \end{figure} A letter P from a shop sign is modelled as a uniform plane lamina which consists of a rectangular lamina, \(O A B D E\), joined to a semicircular lamina, \(B C D\), along its diameter \(B D\). $$O A = E D = a , A B = 2 a , O E = 4 a \text {, and the diameter } B D = 2 a \text {, as shown in Figure } 1 .$$ Using the model,

1. find, in terms of \(\pi\) and \(a\), the distance of the centre of mass of the letter P ,
   from (i) \(O E\) (ii) \(O A\) The letter P is freely suspended from \(O\) and hangs in equilibrium. The angle between \(O E\) and the downward vertical is \(\alpha\). Using the model,
2. find the exact value of \(\tan \alpha\)

3. \begin{figure}[h] \end{figure} A uniform solid hemisphere \(H\) has radius \(2 a\). A solid hemisphere of radius \(a\) is removed from the hemisphere \(H\) to form a bowl. The plane faces of the hemispheres coincide and the centres of the two hemispheres coincide at the point \(O\), as shown in Figure 2. The centre of mass of the bowl is at the point \(G\).

1. Show that \(O G = \frac { 45 a } { 56 }\) Figure 3 below shows a cross-section of the bowl which is resting in equilibrium with a point \(P\) on its curved surface in contact with a rough plane. The plane is inclined to the horizontal at an angle \(\alpha\) and is sufficiently rough to prevent the bowl from slipping. The line \(O G\) is horizontal and the points \(O , G\) and \(P\) lie in a vertical plane which passes through a line of greatest slope of the inclined plane. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d4fc2ea6-3ffc-42f2-b462-9694adfe2ec1-10_812_1086_1667_493} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure}
2. Find the size of \(\alpha\), giving your answer in degrees to 3 significant figures.

1. Three particles of masses \(2 m , 3 m\) and \(k m\) are placed at the points with coordinates (3a, 2a), (a, -4a) and (-3a, 4a) respectively.

The centre of mass of the three particles lies at the point with coordinates \(( \bar { x } , \bar { y } )\).

1. 1. Find \(\bar { x }\) in terms of \(a\) and \(k\)
   2. Find \(\bar { y }\) in terms of \(a\) and \(k\) Given that the distance of the centre of mass of the three particles from the point ( 0,0 ) is \(\frac { 1 } { 3 } a\)
2. find the possible values of \(k\)

3. \begin{figure}[h] \end{figure} Nine uniform rods are joined together to form the rigid framework \(A B C D E F A\), with \(A B = B C = D F = 3 a , B F = C D = D E = 4 a\) and \(A F = F E = C F = 5 a\), as shown in Figure 1. All nine rods lie in the same plane. The mass per unit length of each of the rods \(B F , C F\) and \(D F\) is twice the mass per unit length of each of the other six rods.

1. Find the distance of the centre of mass of the framework from \(A C\) The mass of the framework is \(M\). A particle of mass \(k M\) is attached to the framework at \(E\) to form a loaded framework. When the loaded framework is freely suspended from \(F\), it hangs in equilibrium with \(C E\) horizontal.
2. Find the exact value of \(k\)

5. \begin{figure}[h] \end{figure} The uniform plane lamina shown in Figure 3 is formed from two squares, \(A B C O\) and \(O D E F\), and a sector \(O D C\) of a circle with centre \(O\). Both squares have sides of length \(3 a\) and \(A O\) is perpendicular to \(O F\). The radius of the sector is \(3 a\) [0pt] [In part (a) you may use, without proof, any of the centre of mass formulae given in the formulae booklet.]

1. Show that the distance of the centre of mass of the sector \(O D C\) from \(O C\) is \(\frac { 4 a } { \pi }\)
2. Find the distance of the centre of mass of the lamina from \(F C\) The lamina is freely suspended from \(F\) and hangs in equilibrium with \(F C\) at an angle \(\theta ^ { \circ }\) to the downward vertical.
3. Find the value of \(\theta\)

6. \begin{figure}[h] \end{figure} The shaded region shown in Figure 4 is bounded by the \(x\)-axis, the line with equation \(x = 9\) and the line with equation \(y = \frac { 1 } { 3 } x\). This shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution. This solid of revolution is used to model a solid right circular cone of height 9 cm and base radius 3 cm . The cone is non-uniform and the mass per unit volume of the cone at the point ( \(x , y , z\) ) is \(\lambda x \mathrm {~kg} \mathrm {~cm} ^ { - 3 }\), where \(0 \leqslant x \leqslant 9\) and \(\lambda\) is constant.

1. Find the distance of the centre of mass of the cone from its vertex. A toy is made by joining the circular plane face of the cone to the circular plane face of a uniform solid hemisphere of radius 3 cm , so that the centres of the two plane surfaces coincide. The weight of the cone is \(W\) newtons and the weight of the hemisphere is \(k W\) newtons.
   When the toy is placed on a smooth horizontal plane with any point of the curved surface of the hemisphere in contact with the plane, the toy will remain at rest.
2. Find the value of \(k\)