6.04c Composite bodies: centre of mass

1. Three particles of masses \(3 m , 4 m\) and \(k m\) are positioned at the points with coordinates ( \(2 a , 3 a\) ), ( \(a , 5 a\) ) and ( \(2 \mu a , \mu a\) ) respectively, where \(k\) and \(\mu\) are constants.

The centre of mass of the three particles is at the point with coordinates \(( 2 a , 4 a )\).
Find (i) the value of \(k\) (ii) the value of \(\mu\)

1. \hspace{0pt} [In this question you may quote, without proof, the formula for the distance of the centre of mass of a uniform circular arc from its centre.]

\begin{figure}[h] \end{figure} Five pieces of a uniform wire are joined together to form the rigid framework \(O A B C O\) shown in Figure 1, where

• \(O A , O B\) and \(B C\) are straight, with \(O A = O B = B C = r\)
• arc \(A B\) is one quarter of a circle with centre \(O\) and radius \(r\)
• arc \(O C\) is one quarter of a circle of radius \(r\)
• all five pieces of wire lie in the same plane
  1. Show that the centre of mass of arc \(A B\) is a distance \(\frac { 2 r } { \pi }\) from \(O A\).

Given that the distance of the centre of mass of the framework from \(O A\) is \(d\),

show that \(\mathrm { d } = \frac { 7 r } { 2 ( 3 + ) }\) The framework is freely pivoted at \(A\).
The framework is held in equilibrium, with \(A O\) vertical, by a horizontal force of magnitude \(F\) which is applied to the framework at \(C\). Given that the weight of the framework is \(W\)

find \(F\) in terms of \(W\)

5. \begin{figure}[h] \end{figure} A uniform lamina \(O A B\) is modelled by the finite region bounded by the \(x\)-axis, the \(y\)-axis and the curve with equation \(y = 9 - x ^ { 2 }\), for \(x \geqslant 0\), as shown shaded in Figure 3. The unit of length on both axes is 1 m . The area of the lamina is \(18 \mathrm {~m} ^ { 2 }\)

1. Show that the centre of mass of the lamina is 3.6 m from \(\boldsymbol { O B }\).
   [0pt] [ Solutions relying on calculator technology are not acceptable.] A light string has one end attached to the lamina at \(O\) and the other end attached to the ceiling. A second light string has one end attached to the lamina at \(A\) and the other end attached to the ceiling.
   The lamina hangs in equilibrium with the strings vertical and \(O A\) horizontal.
   The weight of the lamina is \(W\) The tension in the string attached to the lamina at \(A\) is \(\lambda W\)
2. Find the value of \(\lambda\)

7. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} The shaded region shown in Figure 5 is bounded by the line with equation \(x = a\) and the curve with equation \(x ^ { 2 } + y ^ { 2 } = 4 a ^ { 2 }\) This shaded region is rotated through \(180 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution. This solid is used to model a dome with height \(a\) metres and base radius \(\sqrt { 3 } a\) metres.
The dome is modelled as being non-uniform with the mass per unit volume of the dome at the point \(( x , y , z )\) equal to \(\frac { \lambda } { x ^ { 2 } } \mathrm {~kg} \mathrm {~m} ^ { - 3 }\), where \(a \leqslant x \leqslant 2 a\) and \(\lambda\) is a constant.

1. Show that the distance of the centre of mass of the dome from the centre of its plane face is \(\left( 4 \ln 2 - \frac { 5 } { 2 } \right) a\) metres. A solid uniform right circular cone has base radius \(\sqrt { 3 } a\) metres and perpendicular height \(4 a\) metres. A toy is formed by attaching the plane surface of the dome to the plane surface of the cone, as shown in Figure 6. The weight of the cone is \(k W\) and the weight of the dome is \(2 W\) The centre of mass of the toy is a distance \(d\) metres from the plane face of the dome.
2. Show that \(d = \frac { | k + 5 - 8 \ln 2 | } { 2 + k } a\) The toy is suspended from a point on the circumference of the plane face of the dome and hangs freely in equilibrium with the plane face of the dome at an angle \(\alpha\) to the downward vertical.
   Given that \(\tan \alpha = \frac { 1 } { 2 \sqrt { 3 } }\)
3. find the exact value of \(k\).

2. \begin{figure}[h] \end{figure} A uniform rod of length \(28 a\) is cut into seven identical rods each of length \(4 a\). These rods are joined together to form the rigid framework \(A B C D E A\) shown in Figure 1. All seven rods lie in the same plane.
The distance of the centre of mass of the framework from \(E D\) is \(d\).

1. Show that \(d = \frac { 8 \sqrt { 3 } } { 7 } a\) The weight of the framework is \(W\).
   The framework is freely pivoted about a horizontal axis through \(C\).
   The framework is held in equilibrium in a vertical plane, with \(A C\) vertical and \(A\) below \(C\), by a horizontal force that is applied to the framework at \(A\). The force acts in the same vertical plane as the framework and has magnitude \(F\).
2. Find \(F\) in terms of \(W\).

4. \begin{figure}[h] \end{figure} A uniform lamina \(O A B\) is in the shape of the region \(R\).
Region \(R\) lies in the first quadrant and is bounded by the curve with equation \(\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 36 } = 1\), the \(x\)-axis, and the \(y\)-axis, as shown shaded in Figure 3. The point \(A\) is the point of intersection of the curve and the \(x\)-axis.
The point \(B\) is the point of intersection of the curve and the \(y\)-axis.
One unit on each axis represents 1 m .
The area of \(R\) is \(6 \pi\) The centre of mass of \(R\) lies at the point with coordinates \(( \bar { x } , \bar { y } )\)

1. Use algebraic integration to show that \(\bar { x } = \frac { 16 } { 3 \pi }\)
2. Use algebraic integration to find the exact value of \(\bar { y }\) The lamina is freely suspended from \(A\) and hangs in equilibrium with \(O A\) at angle \(\theta ^ { \circ }\) to the downward vertical.
3. Find the value of \(\theta\)

6. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} The shaded region, shown in Figure 4, is bounded by the \(x\)-axis, the line with equation \(x = 6\), the line with equation \(y = 2\) and the \(y\)-axis. This region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { x }\)-axis to form a solid of revolution. This solid is used to model a non-uniform cylinder of height 6 cm and radius 2 cm . The mass per unit volume of the cylinder at the point \(( x , y , z )\) is \(\lambda ( x + 2 ) \mathrm { kg } \mathrm { cm } ^ { - 3 }\), where \(0 \leqslant x \leqslant 6\) and \(\lambda\) is a constant.

1. Show that the mass of the cylinder is \(120 \lambda \pi \mathrm {~kg}\).
2. Show that the centre of mass of the cylinder is 3.6 cm from \(O\). The point \(O\) is the centre of one end of the cylinder. The point \(A\) is the centre of the other end of the cylinder. A uniform solid hemisphere of radius 3 cm has density \(\lambda \mathrm { kg } \mathrm { cm } ^ { - 3 }\). The hemisphere is attached to the cylinder with the centre of its circular face in contact with the point \(A\) on the cylinder to form the model shown in Figure 5. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c14975b7-6afa-44ce-beab-1cba2e82b249-20_309_673_1713_696} \captionsetup{labelformat=empty} \caption{Figure 5}
   \end{figure} The model is placed with the end containing \(O\) on a rough inclined plane which is inclined at angle \(\alpha ^ { \circ }\) to the horizontal. The plane is sufficiently rough to prevent the model from sliding. The model is on the point of toppling.
3. Find the value of \(\alpha\).

1. A flag pole is 15 m long.

The flag pole is non-uniform so that, at a distance \(x\) metres from its base, the mass per unit length of the flag pole, \(m \mathrm {~kg} \mathrm {~m} ^ { - 1 }\) is given by the formula \(m = 10 \left( 1 - \frac { x } { 25 } \right)\). The flag pole is modelled as a rod.

1. Show that the mass of the flag pole is 105 kg .
2. Find the distance of the centre of mass of the flag pole from its base.

3. \begin{figure}[h] \end{figure} A uniform solid cylinder has radius \(2 a\) and height \(h ( h > a )\).
A solid hemisphere of radius \(a\) is removed from the cylinder to form the vessel \(V\).
The plane face of the hemisphere coincides with the upper plane face of the cylinder.
The centre \(O\) of the hemisphere is also the centre of the upper plane face of the cylinder, as shown in Figure 2.

1. Show that the centre of mass of \(V\) is \(\frac { 3 \left( 8 h ^ { 2 } - a ^ { 2 } \right) } { 8 ( 6 h - a ) }\) from \(O\). The vessel \(V\) is placed on a rough plane which is inclined at an angle \(\phi\) to the horizontal. The lower plane circular face of \(V\) is in contact with the inclined plane. Given that \(h = 5 a\), the plane is sufficiently rough to prevent \(V\) from slipping and \(V\) is on the point of toppling,
2. find, to three significant figures, the size of the angle \(\phi\).

5. \begin{figure}[h] \end{figure} A shop sign is modelled as a uniform rectangular lamina \(A B C D\) with a semicircular lamina removed. The semicircle has radius \(a , B C = 4 a\) and \(C D = 2 a\).
The centre of the semicircle is at the point \(E\) on \(A D\) such that \(A E = d\), as shown in Figure 3.

1. Show that the centre of mass of the sign is \(\frac { 44 a } { 3 ( 16 - \pi ) }\) from \(A D\). The sign is suspended using vertical ropes attached to the sign at \(A\) and at \(B\) and hangs in equilibrium with \(A B\) horizontal. The weight of the sign is \(W\) and the ropes are modelled as light inextensible strings.
2. Find, in terms of \(W\) and \(\pi\), the tension in the rope attached at \(B\). The rope attached at \(B\) breaks and the sign hangs freely in equilibrium suspended from \(A\), with \(A D\) at an angle \(\alpha\) to the downward vertical. Given that \(\tan \alpha = \frac { 11 } { 18 }\)
3. find \(d\) in terms of \(a\) and \(\pi\).

1. Use an energy method to find the magnitude of the frictional force acting on the block. Calculate the coefficient of friction between the block and the plane.
2. Calculate the power of the tension in the string when the block has a speed of \(7 \mathrm {~ms} ^ { - 1 }\). Fig. 3.1 shows a thin planar uniform rigid rectangular sheet of metal, OPQR, of width 1.65 m and height 1.2 m . The mass of the sheet is \(M \mathrm {~kg}\). The sides OP and PQ have thin rigid uniform reinforcements attached with masses \(0.6 M \mathrm {~kg}\) and \(0.4 M \mathrm {~kg}\), respectively. Fig. 3.1 also shows coordinate axes with origin at O . The sheet with its reinforcements is to be used as an inn sign.

1. Calculate the coordinates of the centre of mass of the inn sign. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_421_492_210_1334} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
   \end{figure} The inn sign has a weight of 300 N . It hangs in equilibrium with QR horizontal when vertical forces \(Y _ { \mathrm { Q } } \mathrm { N }\) and \(Y _ { \mathrm { R } } \mathrm { N }\) act at Q and R respectively.
2. Calculate the value of \(Y _ { \mathrm { Q } }\) and show that \(Y _ { \mathrm { R } } = 120\). The inn sign is hung from a framework, ABCD , by means of two light vertical inextensible wires attached to the sign at Q and R and the framework at B and C , as shown in Fig. 3.2. QR and BC are horizontal. The framework is made from light rigid rods \(\mathrm { AB } , \mathrm { BC } , \mathrm { CA }\) and CD freely pin-jointed together at \(\mathrm { A } , \mathrm { B }\) and C and to a vertical wall at A and D . Fig. 3.3 shows the dimensions of the framework in metres as well as the external forces \(X _ { \mathrm { A } } \mathrm { N } , Y _ { \mathrm { A } } \mathrm { N }\) acting at A and \(X _ { \mathrm { D } } \mathrm { N } , Y _ { \mathrm { D } } \mathrm { N }\) acting at D . You are given that \(\sin \alpha = \frac { 5 } { 13 } , \cos \alpha = \frac { 12 } { 13 } , \sin \beta = \frac { 4 } { 5 }\) and \(\cos \beta = \frac { 3 } { 5 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_543_526_1420_253} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_629_793_1343_964} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
   \end{figure}
3. Mark on the diagram in your Printed Answer Book all the forces acting on the pin-joints at \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , including those internal to the rods, when the inn sign is hanging from the framework.
4. Show that \(X _ { \mathrm { D } } = 261\).
5. Calculate the forces internal to the rods \(\mathrm { AB } , \mathrm { BC }\) and CD , stating whether each rod is in tension or thrust (compression). Calculate also the values of \(Y _ { \mathrm { D } }\) and \(Y _ { \mathrm { A } }\). [Your working in this part should correspond to your diagram in part (iii).]

2 The region bounded by the \(x\)-axis and the curve \(y = a x ( 2 - x )\), where \(a\) is a constant, is occupied by a uniform lamina \(L _ { 1 }\) (see Fig. 1). Units on the axes are metres. \begin{figure}[h] \end{figure}

1. Write down the value of the \(x\)-coordinate of the centre of mass of \(L _ { 1 }\).
2. Show that the \(y\)-coordinate of the centre of mass of \(L _ { 1 }\) is \(\frac { 2 } { 5 } a\). The mass of \(L _ { 1 }\) is \(M \mathrm {~kg}\). A uniform rectangular lamina of width 2 m and height \(a \mathrm {~m}\) is made from a different material from that of \(L _ { 1 }\) and has a mass of \(2 M \mathrm {~kg}\). A new lamina, \(L _ { 2 }\), is formed by joining the straight edge of \(L _ { 1 }\) to an edge of the rectangular lamina of length 2 m (see Fig. 2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a8c9d007-e67f-4637-9e74-630ba9a91442-2_547_273_1772_890} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} \(L _ { 2 }\) is freely suspended from one of its right-angled corners and hangs in equilibrium with its edge of length 2 m making an angle of \(20 ^ { \circ }\) with the horizontal.
3. Find the value of \(a\), giving your answer correct to 3 significant figures.

7 Particles \(A , B\) and \(C\) of masses \(2 \mathrm {~kg} , 3 \mathrm {~kg}\) and 5 kg respectively are joined by light rigid rods to form a triangular frame. The frame is placed at rest on a horizontal plane with \(A\) at the point ( 0,0 ), \(B\) at the point ( \(0.6,0\) ) and \(C\) at the point ( \(0.4,0.2\) ), where distances in the coordinate system are measured in metres (see Fig. 1). \begin{figure}[h] \end{figure} \(G\), which is the centre of mass of the frame, is at the point \(( \bar { x } , \bar { y } )\).

1. - Show that \(\bar { x } = 0.38\).
   A rough plane, \(\Pi\), is inclined at an angle \(\theta\) to the horizontal where \(\sin \theta = \frac { 3 } { 5 }\). The frame is placed on \(\Pi\) with \(A B\) vertical and \(B\) in contact with \(\Pi\). \(C\) is in the same vertical plane as \(A B\) and a line of greatest slope of \(\Pi . C\) is on the down-slope side of \(A B\). The frame is kept in equilibrium by a horizontal light elastic string whose natural length is \(l \mathrm {~m}\) and whose modulus of elasticity is \(g \mathrm {~N}\). The string is attached to \(A\) at one end and to a fixed point on \(\Pi\) at the other end (see Fig. 2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{493f11f4-e25c-4eeb-a0ab-20ec6d7a7a7d-5_611_842_1649_248} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The coefficient of friction between \(B\) and \(\Pi\) is \(\mu\).
2. Show that \(l = 0.3\).
3. Show that \(\mu \geqslant \frac { 14 } { 27 }\). \section*{OCR} Oxford Cambridge and RSA

4 A uniform T-shaped lamina is formed by rigidly joining two rectangles \(A B C H\) and \(D E F G\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-4_748_652_456_644}

1. Show that the centre of mass of the lamina is 26 cm from the edge \(A B\).
2. Explain why the centre of mass of the lamina is 5 cm from the edge \(G F\).
3. The point \(X\) is on the edge \(A B\) and is 7 cm from \(A\), as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-4_697_534_1576_753} The lamina is freely suspended from \(X\) and hangs in equilibrium.
   Find the angle between the edge \(A B\) and the vertical, giving your answer to the nearest degree.
   (4 marks)

4 A uniform rectangular lamina \(A B C D\) has a mass of 8 kg . The side \(A B\) has length 20 cm , the side \(B C\) has length 10 cm , and \(P\) is the mid-point of \(A B\). A uniform circular lamina, of mass 2 kg and radius 5 cm , is fixed to the rectangular lamina to form a sign. The centre of the circular lamina is 5 cm from each of \(A B\) and \(B C\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{851cb2a3-5bc8-4af9-b1fc-a143d37beebe-3_661_1200_589_406}

1. Find the distance of the centre of mass of the sign from \(A D\).
2. Write down the distance of the centre of mass of the sign from \(A B\).
3. The sign is freely suspended from \(P\). Find the angle between \(A D\) and the vertical when the sign is in equilibrium.
4. Explain how you have used the fact that each lamina is uniform in your solution to this question.

1 A rigid rod, \(A B\), has mass 2 kg and length 4 metres.
Two particles of masses 5 kg and 3 kg are fixed to \(A\) and \(B\) respectively to create a composite body, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b0d0c552-71cb-4e5a-b545-de8a9052def0-02_120_730_769_653} Find the distance of the centre of mass of the composite body from \(B\). Circle your answer.
1.5 metres
1.6 metres
2.4 metres
2.5 metres

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.

3 A uniform disc has mass 6 kg and diameter 8 cm A uniform rectangular lamina, \(A B C D\), has mass 4 kg , width 8 cm and length 20 cm
The disc is fixed to the lamina to form a composite body as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-03_448_881_587_577} The sides \(A B , A D\) and \(C D\) are tangents to the disc.
Calculate the distance of the centre of mass of the composite body from \(A D\) Circle your answer.
4 cm
5.6 cm
6.4 cm
8.8 cm