6.04c Composite bodies: centre of mass

2 The rigid object shown in Fig. 2.1 is made of thin non-uniform rods. ABC is a straight line; \(\mathrm { BC } , \mathrm { BE }\) and ED form three sides of a rectangle. The centre of mass of the object is at G. The lengths are in centimetres. The weight of the object is 15 N . \begin{figure}[h] \end{figure} Initially, the object is suspended by light vertical strings attached to B and to C and hangs in equilibrium with AC horizontal.

1. Calculate the tensions in each of the strings. In a new situation the strings are removed. The object can rotate freely in a vertical plane about a fixed horizontal axis through A and perpendicular to ABCDE. The object is held in equilibrium with AC horizontal by a force of magnitude \(T \mathrm {~N}\) in the plane ABCDE acting at C at an angle of \(30 ^ { \circ }\) to CA . This situation is shown in Fig. 2.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ea3c0177-bf3b-4475-9ab1-ae628aeb0bf0-3_356_451_1292_808} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
2. Calculate \(T\). Calculate also the magnitude of the force exerted on the object by the axis at A . The object is now placed on a rough horizontal table and is in equilibrium with ABCDE in a vertical plane and DE in contact with the table. The coefficient of friction between the edge DE and the table is 0.65 . A force of slowly increasing magnitude (starting at 0 N ) is applied at A in the direction AB . Assume that the object remains in a vertical plane.
3. Determine whether the object slips before it tips.

3

1. You are given that the position of the centre of mass, G , of a right-angled triangle cut from thin uniform material in the position shown in Fig. 3.1 is at the point \(\left( \frac { 1 } { 3 } a , \frac { 1 } { 3 } b \right)\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ea3c0177-bf3b-4475-9ab1-ae628aeb0bf0-4_328_382_360_845} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
   \end{figure} A plane thin uniform sheet of metal is in the shape OABCDEFHIJO shown in Fig. 3.2. BDEA and CDIJ are rectangles and FEH is a right angle. The lengths of the sides are shown with each unit representing 1 cm . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ea3c0177-bf3b-4475-9ab1-ae628aeb0bf0-4_862_906_1032_584} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
   1. Calculate the coordinates of the centre of mass of the metal sheet, referred to the axes shown in Fig. 3.2. The metal sheet is freely suspended from corner B and hangs in equilibrium.
   2. Calculate the angle between BD and the vertical.
2. Part of a framework of light rigid rods freely pin-jointed at their ends is shown in Fig. 3.3. The framework is in equilibrium. All the rods meeting at the pin-joints at \(\mathrm { A } , \mathrm { B }\) and C are shown. The rods connected to \(\mathrm { A } , \mathrm { B }\) and C are connected to the rest of the framework at \(\mathrm { P } , \mathrm { Q } , \mathrm { R } , \mathrm { S }\) and T . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ea3c0177-bf3b-4475-9ab1-ae628aeb0bf0-5_499_734_493_662} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
   \end{figure} There is a tension of 18 N in rod AP and a thrust (compression) of 5 N in rod AQ.
   1. Show the forces internal to the rods acting on the pin-joints at \(\mathrm { A } , \mathrm { B }\) and C .
   2. Calculate the forces internal to the rods \(\mathrm { AB } , \mathrm { BC }\) and CA , stating whether each rod is in tension or compression. [You may leave your answers in surd form. Your working in this part should be consistent with your diagram in part (i).] \(4 P\) and \(Q\) are circular discs of mass 3 kg and 10 kg respectively which slide on a smooth horizontal surface. The discs have the same diameter and move in the line joining their centres with no resistive forces acting on them. The surface has vertical walls which are perpendicular to the line of centres of the discs. This information is shown in Fig. 4 together with the direction you should take as being positive. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{ea3c0177-bf3b-4475-9ab1-ae628aeb0bf0-6_430_1404_443_328} \captionsetup{labelformat=empty} \caption{Fig. 4}
      \end{figure}
      1. For what time must a force of 26 N act on P to accelerate it from rest to \(13 \mathrm {~ms} ^ { - 1 }\) ? P is travelling at \(13 \mathrm {~ms} ^ { - 1 }\) when it collides with Q , which is at rest. The coefficient of restitution in this collision is \(e\).
      2. Show that, after the collision, the velocity of P is \(( 3 - 10 e ) \mathrm { ms } ^ { - 1 }\) and find an expression in terms of \(e\) for the velocity of Q.
      3. For what set of values of \(e\) does the collision cause P to reverse its direction of motion?
      4. Determine the set of values of \(e\) for which P has a greater speed than Q immediately after the collision. You are now given that \(e = \frac { 1 } { 2 }\). After P and Q collide with one another, each has a perfectly elastic collision with a wall. P and Q then collide with one another again and in this second collision they stick together (coalesce).
      5. Determine the common velocity of P and Q .
      6. Determine the impulse of Q on P in this collision.

4

1. Fig. 4.1 shows a framework constructed from 4 uniform heavy rigid rods \(\mathrm { OP } , \mathrm { OQ } , \mathrm { PR }\) and RS , rigidly joined at \(\mathrm { O } , \mathrm { P } , \mathrm { Q } , \mathrm { R }\) and S and with OQ perpendicular to PR . Fig. 4.1 also shows the dimensions of the rods and axes \(\mathrm { O } x\) and \(\mathrm { O } y\) : the units are metres. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-5_454_994_408_548} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure} Each rod has a mass of 0.8 kg per metre.
   1. Show that, referred to the axes in Fig. 4.1, the \(x\)-coordinate of the centre of mass of the framework is 1.5 and calculate the \(y\)-coordinate. The framework is freely suspended from S and a small object of mass \(m \mathrm {~kg}\) is attached to it at O . The framework is in equilibrium with OQ horizontal.
   2. Calculate \(m\).
2. Fig. 4.2 shows a framework in equilibrium in a vertical plane. The framework is made from 5 light, rigid rods \(\mathrm { OP } , \mathrm { OQ } , \mathrm { OR } , \mathrm { PQ }\) and QR . Its dimensions are indicated. PQ is horizontal and OR vertical. The rods are freely pin-jointed to each other at \(\mathrm { O } , \mathrm { P } , \mathrm { Q }\) and R . The pin-joint at O is fixed to a wall.
   Fig. 4.2 also shows the external forces acting on the framework: there are vertical loads of 120 N and 60 N at Q and P respectively; a horizontal string attached to Q has tension \(T \mathrm {~N}\); horizontal and vertical forces \(X \mathrm {~N}\) and \(Y \mathrm {~N}\) act on the framework from the pin-joint at O . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-6_566_453_625_788} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure}
   1. By considering only the pin-joint at R , explain why the rods OR and RQ must have zero internal force.
   2. Find the values of \(T , X\) and \(Y\).
   3. Using the diagram in your printed answer book, show all the forces acting on the pin-joints, including those internal to the rods.
      [0pt]
   4. Calculate the forces internal to the rods OP and PQ , stating whether each rod is in tension or compression (thrust). [You may leave answers in surd form. Your working in this part should correspond to your diagram in part (iii).]

2 Fig. 2.1 shows the positions of the points \(\mathrm { P } , \mathrm { Q } , \mathrm { R } , \mathrm { S } , \mathrm { T } , \mathrm { U } , \mathrm { V }\) and W which are at the vertices of a cube of side \(a\); Fig. 2.1 also shows coordinate axes, where O is the mid-point of PQ . \begin{figure}[h] \end{figure} An open box, A, is made from thin uniform material in the form of the faces of the cube with just the face TUVW missing.

1. Find the \(z\)-coordinate of the centre of mass of A . Strips made of a thin heavy material are now fixed to the edges TW, WV and VU of box A, as shown in Fig. 2.2. Each of these three strips has the same mass as one face of the box. This new object is B. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{334b2170-3708-46db-bff7-bcad7d5fab00-3_488_476_1388_797} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
2. Find the \(x\)-and \(z\)-coordinates of the centre of mass of B and show that the \(y\)-coordinate is \(\frac { 9 a } { 16 }\). Object B is now placed on a plane which is inclined at \(\theta\) to the horizontal. B is positioned so that face PQRS is on the plane with SR at right angles to a line of greatest slope of the plane and with PQ higher than SR , as shown in Fig. 2.3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{334b2170-3708-46db-bff7-bcad7d5fab00-3_237_284_2087_1555} \captionsetup{labelformat=empty} \caption{Fig. 2.3}
   \end{figure}
3. Assuming that B does not slip, find \(\theta\) if B is on the point of tipping. B is now placed on a different plane which is inclined at \(30 ^ { \circ }\) to the horizontal. When B is released it accelerates down the plane at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
4. Calculate the coefficient of friction between B and the inclined plane.

3 A uniform heavy lamina occupies the region shaded in Fig. 3. This region is formed by removing a square of side 1 unit from a square of side \(a\) units (where \(a > 1\) ). \begin{figure}[h] \end{figure} Relative to the axes shown in Fig. 3, the centre of mass of the lamina is at \(( \bar { x } , \bar { y } )\).

1. Show that \(\bar { x } = \bar { y } = \frac { a ^ { 2 } + a + 1 } { 2 ( a + 1 ) }\).
   [0pt] [You may need to use the result \(\frac { a ^ { 3 } - 1 } { 2 \left( a ^ { 2 } - 1 \right) } = \frac { a ^ { 2 } + a + 1 } { 2 ( a + 1 ) }\).]
2. Show that the centre of mass of the lamina lies on its perimeter if \(a = \frac { 1 } { 2 } ( 1 + \sqrt { 5 } )\). In another situation, \(a = 4\).
   A particle of mass one third that of the lamina is attached to the lamina at vertex B ; the lamina with the particle is freely suspended from vertex A and hangs in equilibrium. The positions of A and B are shown in Fig. 3.
3. Calculate the angle that AB makes with the vertical.

4 Fig. 4.1 shows a hollow circular cylinder open at one end and closed at the other. The radius of the cylinder is 0.1 m and its height is \(h \mathrm {~m} . \mathrm { O }\) and C are points on the axis of symmetry at the centres of the open and closed ends, respectively. The thin material used for the closed end has four times the density of the thin material used for the curved surface. \begin{figure}[h] \end{figure} Cylinders of this type are made with different values of \(h\).

1. Show that the centres of mass of these cylinders are on the line OC at a distance \(\frac { 5 h ^ { 2 } + 2 h } { 2 + 10 h } \mathrm {~m}\) from O . Fig. 4.2 shows one of the cylinders placed with its open end on a slope inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 2 } { 3 }\). The cylinder does not slip but is on the point of tipping.
2. Show that \(50 h ^ { 2 } + 5 h - 3 = 0\) and hence that \(h = 0.2\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-5_383_497_1178_1402} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure} Fig. 4.3 shows another of the cylinders that has weight 42 N and \(h = 0.5\). This cylinder has its open end on a rough horizontal plane. A force of magnitude \(T \mathrm {~N}\) is applied to a point P on the circumference of the closed end. This force is at an angle \(\beta\) with the horizontal such that \(\tan \beta = \frac { 3 } { 4 }\) and the force is in the vertical plane containing \(\mathrm { O } , \mathrm { C }\) and P . The cylinder does not slip but is on the point of tipping. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-5_451_679_1955_685} \captionsetup{labelformat=empty} \caption{Fig. 4.3}
   \end{figure}
3. Calculate \(T\).

5. \begin{figure}[h] \end{figure} Figure 2 shows a uniform plane lamina \(A B C D E G\) in the shape of a letter ' \(L\) ' consisting of a rectangle \(A B F G\) joined to another rectangle \(C D E F\). The sides \(A B\) and \(D E\) are both 8 cm long and the sides \(E G\) and \(G A\) are of length 24 cm and 32 cm respectively.

1. Show that the centre of mass of the lamina lies on the line \(B F\).
2. Find the distance of the centre of mass from the line \(A B\). The uniform lamina in Figure 2 is a model of the letter ' \(L\) ' in a sign above a shop. The letter is normally suspended from a wall at \(A\) and \(B\) so that \(A B\) is horizontal but the fixing at \(B\) has broken and the letter hangs in equilibrium from the point \(A\).
3. Find, in degrees to one decimal place, the acute angle \(A G\) makes with the vertical.

7. \begin{figure}[h] \end{figure} Figure 2 shows a uniform lamina \(A B C D\) formed by removing an isosceles triangle \(B C D\) from an equilateral triangle \(A B D\) of side \(2 d\). The point \(C\) is the centroid of triangle \(A B D\).

1. Find the area of triangle \(B C D\) in terms of \(d\).
2. Show that the distance of the centre of mass of the lamina from \(B D\) is \(\frac { 4 } { 9 } \sqrt { 3 } d\).
   (8 marks)
   The lamina is freely suspended from the point \(B\) and hangs at rest.
3. Find in degrees, correct to 1 decimal place, the acute angle that the side \(A B\) makes with the vertical.

7. \begin{figure}[h] \end{figure} Figure 3 shows a hotel 'key' consisting of a rectangle \(O A B D\), where \(O A = 8 \mathrm {~cm}\) and \(O D = 4 \mathrm {~cm}\), joined to a semicircle whose diameter \(B C\) is 4 cm long. The thickness of the key is negligible and the same material is used throughout. The key is modelled as a uniform lamina.
Using this model,

1. find, correct to 3 significant figures, the distance of the centre of mass from
   1. OD ,
   2. \(O A\). A small circular hole of negligible diameter is made at the mid-point of \(B C\) so that the key can be hung on a smooth peg. When the key is freely suspended from the peg,
2. find, correct to 3 significant figures, the acute angle made by \(O A\) with the vertical.

4. \begin{figure}[h] \end{figure} Figure 2 shows an earring consisting of a uniform wire \(A B C D\) of length \(6 a\) bent to form right angles at \(B\) and \(C\) such that \(A B\) and \(C D\) are of length \(2 a\) and \(a\) respectively.

1. Find, in terms of \(a\), the distance of the centre of mass from
   1. \(\quad A B\),
   2. \(B C\). The earring is to be worn such that it hangs in equilibrium suspended from the point \(A\).
2. Find, to the nearest degree, the angle made by \(A B\) with the downward vertical.
   (4 marks)

6. \begin{figure}[h] \end{figure} Figure 3 shows a uniform rectangular lamina \(A B C D\) of mass \(8 m\) in which the sides \(A B\) and \(B C\) are of length \(a\) and \(2 a\) respectively. Particles of mass \(2 m , 6 m\) and \(4 m\) are fixed to the lamina at the points \(A , B\) and \(D\) respectively.

1. Write down the distance of the centre of mass from \(A D\).
2. Show that the distance of the centre of mass from \(A B\) is \(\frac { 4 } { 5 } a\). Another particle of mass \(k m\) is attached to the lamina at the point \(B\).
3. Show that the distance of the centre of mass from \(A D\) is now given by \(\frac { ( 10 + k ) a } { 20 + k }\).
   (4 marks)
   Given that when the lamina is suspended freely from the point \(A\) the side \(A B\) makes an angle of \(45 ^ { \circ }\) with the vertical,
4. find the value of \(k\).

6 A uniform \(\operatorname { rod } A B\), of weight \(W\) and length \(2 l\) is in equilibrium at \(60 ^ { \circ }\) to the horizontal with \(A\) resting against a smooth vertical plane and \(B\) resting on a rough section of a horizontal plane. Another uniform rod \(C D\), of length \(\sqrt { 3 } l\) and weight \(W\), is freely jointed to the mid-point of \(A B\) at \(C\); its other end \(D\) rests on a smooth section of the horizontal plane. \(C D\) is inclined at \(30 ^ { \circ }\) to the horizontal (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{dfe477d4-eae6-40e1-b704-1a97485f4c7e-4_508_1075_438_495}

1. Show that the force exerted by the horizontal plane on \(C D\) is \(\frac { 1 } { 2 } W\). Find the normal component of the force exerted by the horizontal plane on \(A B\).
2. Find the magnitude and direction of the force exerted by \(C D\) on \(A B\).
3. Given that \(A B\) is in limiting equilibrium, find the coefficient of friction between \(A B\) and the horizontal plane.

3 \includegraphics[max width=\textwidth, alt={}, center]{09d3e8ca-0062-4f62-8453-7acaff591db5-2_661_711_918_717} Two uniform rods \(A B\) and \(A C\), of equal lengths, and of weights 200 N and 360 N respectively, are freely jointed at \(A\). The mid-points of the rods are joined by a taut light inextensible string. The rods are in equilibrium in a vertical plane with \(B\) and \(C\) in contact with a smooth horizontal surface. The point \(A\) is 2.1 m above the surface and \(B C = 1.4 \mathrm {~m}\) (see diagram).

1. Show that the force exerted on \(A B\) at \(B\) has magnitude 240 N and find the tension in the string.
2. Find the horizontal and vertical components of the force exerted on \(A B\) at \(A\) and state their directions.

5 \includegraphics[max width=\textwidth, alt={}, center]{a04e6d4e-2437-4761-87ee-43e6771fbbd9-3_549_447_253_849} Two uniform rods \(A B\) and \(B C\), each of length 1.4 m and weight 80 N , are freely jointed to each other at \(B\), and \(A B\) is freely jointed to a fixed point at \(A\). They are held in equilibrium with \(A B\) at an angle \(\alpha\) to the horizontal, and \(B C\) at an angle of \(60 ^ { \circ }\) to the horizontal, by a light string, perpendicular to \(B C\), attached to \(C\) (see diagram).

1. By taking moments about \(B\) for \(B C\), calculate the tension in the string. Hence find the horizontal and vertical components of the force acting on \(B C\) at \(B\).
2. Find \(\alpha\). \includegraphics[max width=\textwidth, alt={}, center]{a04e6d4e-2437-4761-87ee-43e6771fbbd9-3_691_665_1370_740} A circus performer \(P\) of mass 80 kg is suspended from a fixed point \(O\) by an elastic rope of natural length 5.25 m and modulus of elasticity \(2058 \mathrm {~N} . P\) is in equilibrium at a point 5 m above a safety net. A second performer \(Q\), also of mass 80 kg , falls freely under gravity from a point above \(P\). \(P\) catches \(Q\) and together they begin to descend vertically with initial speed \(3.5 \mathrm {~ms} ^ { - 1 }\) (see diagram). The performers are modelled as particles.

5 \includegraphics[max width=\textwidth, alt={}, center]{85402f4a-8d55-47d8-ba48-5b837609b0f4-3_581_903_267_621} Two uniform rods \(X A\) and \(X B\) are freely jointed at \(X\). The lengths of the rods are 1.5 m and 1.3 m respectively, and their weights are 150 N and 130 N respectively. The rods are in equilibrium in a vertical plane with \(A\) and \(B\) in contact with a rough horizontal surface. \(A\) and \(B\) are at distances horizontally from \(X\) of 0.9 m and 0.5 m respectively, and \(X\) is 1.2 m above the surface (see diagram).

1. The normal components of the contact forces acting on the rods at \(A\) and \(B\) are \(R _ { A } \mathrm {~N}\) and \(R _ { B } \mathrm {~N}\) respectively. Show that \(R _ { A } = 125\) and find \(R _ { B }\).
2. Find the frictional components of the contact forces acting on the rods at \(A\) and \(B\).
3. Find the horizontal and vertical components of the force exerted on \(X A\) at \(X\), stating their directions.

4 The region between the curve \(y = 4 - x ^ { 2 }\) and the \(x\)-axis, from \(x = 0\) to \(x = 2\), is occupied by a uniform lamina. The units of the axes are metres.

1. Show that the coordinates of the centre of mass of this lamina are \(( 0.75,1.6 )\). This lamina and another exactly like it are attached to a uniform rod PQ , of mass 12 kg and length 8 m , to form a rigid body as shown in Fig. 4. Each lamina has mass 6.5 kg . The ends of the rod are at \(\mathrm { P } ( - 4,0 )\) and \(\mathrm { Q } ( 4,0 )\). The rigid body lies entirely in the \(( x , y )\) plane. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b7f8bdfd-33dc-4453-8f3a-ddd24be17372-4_511_956_1836_557} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
2. Find the coordinates of the centre of mass of the rigid body. The rigid body is freely suspended from the point \(\mathrm { A } ( 2,4 )\) and hangs in equilibrium.
3. Find the angle that PQ makes with the horizontal.

4 Fig. 4.1 shows the region \(R\) bounded by the curve \(y = x ^ { - \frac { 1 } { 3 } }\) for \(1 \leqslant x \leqslant 8\), the \(x\)-axis, and the lines \(x = 1\) and \(x = 8\). \begin{figure}[h] \end{figure}

1. Find the \(x\)-coordinate of the centre of mass of a uniform solid of revolution obtained by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis.
2. Find the coordinates of the centre of mass of a uniform lamina in the shape of the region \(R\).
3. Using your answer to part (ii), or otherwise, find the coordinates of the centre of mass of a uniform lamina in the shape of the region (shown shaded in Fig. 4.2) bounded by the curve \(y = x ^ { - \frac { 1 } { 3 } }\) for \(1 \leqslant x \leqslant 8\), the line \(y = \frac { 1 } { 2 }\) and the line \(x = 1\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c470e80e-b346-4335-9c08-beb5a46cc506-4_595_1015_1610_607} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure}

4

1. The region bounded by the \(x\)-axis and the semicircle \(y = \sqrt { a ^ { 2 } - x ^ { 2 } }\) for \(- a \leqslant x \leqslant a\) is occupied by a uniform lamina with area \(\frac { 1 } { 2 } \pi a ^ { 2 }\). Show by integration that the \(y\)-coordinate of the centre of mass of this lamina is \(\frac { 4 a } { 3 \pi }\).
2. A uniform solid cone is formed by rotating the region between the \(x\)-axis and the line \(y = m x\), for \(0 \leqslant x \leqslant h\), through \(2 \pi\) radians about the \(x\)-axis.
   1. Show that the \(x\)-coordinate of the centre of mass of this cone is \(\frac { 3 } { 4 } h\).
      [0pt] [You may use the formula \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) for the volume of a cone.]
      From such a uniform solid cone with radius 0.7 m and height 2.4 m , a cone of material is removed. The cone removed has radius 0.4 m and height 1.1 m ; the centre of its base coincides with the centre of the base of the original cone, and its axis of symmetry is also the axis of symmetry of the original cone. Fig. 4 shows the resulting object; the vertex of the original cone is V, and A is a point on the circumference of its base. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{b8573ee2-771c-4a93-88d9-346a9da94494-5_716_1228_1027_497} \captionsetup{labelformat=empty} \caption{Fig. 4}
      \end{figure}
   2. Find the distance of the centre of mass of this object from V . This object is suspended by a string attached to a point Q on the line VA, and hangs in equilibrium with VA horizontal.
   3. Find the distance VQ.

2

1. A uniform solid hemisphere of volume \(\frac { 2 } { 3 } \pi a ^ { 3 }\) is formed by rotating the region bounded by the \(x\)-axis, the \(y\)-axis and the curve \(y = \sqrt { a ^ { 2 } - x ^ { 2 } }\) for \(0 \leqslant x \leqslant a\), through \(2 \pi\) radians about the \(x\)-axis. Show that the \(x\)-coordinate of the centre of mass of the hemisphere is \(\frac { 3 } { 8 } a\).
2. A uniform lamina is bounded by the \(x\)-axis, the line \(x = 1\), and the curve \(y = 2 - \sqrt { x }\) for \(1 \leqslant x \leqslant 4\). Its corners are \(\mathrm { A } ( 1,1 ) , \mathrm { B } ( 1,0 )\) and \(\mathrm { C } ( 4,0 )\).
   1. Find the coordinates of the centre of mass of the lamina. The lamina is suspended with AB vertical and BC horizontal by light vertical strings attached to A and C , as shown in Fig. 2. The weight of the lamina is \(W\). \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{023afdfb-21b6-40fe-9a09-e6769667ee7b-2_346_684_1672_772} \captionsetup{labelformat=empty} \caption{Fig. 2}
      \end{figure}
   2. Find the tensions in the two strings in terms of \(W\).

4

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2dd5719-bef3-45f2-afd2-c481e6a4b129-5_705_501_260_863} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure} The region \(R\), shown in Fig. 4.1, is bounded by the curve \(x ^ { 2 } - y ^ { 2 } = k ^ { 2 }\) for \(k \leqslant x \leqslant 4 k\) and the line \(x = 4 k\), where \(k\) is a positive constant. Find the \(x\)-coordinate of the centre of mass of the uniform solid of revolution formed when \(R\) is rotated about the \(x\)-axis.
2. A uniform lamina occupies the region bounded by the curve \(y = \frac { x ^ { 3 } } { a ^ { 2 } }\) for \(0 \leqslant x \leqslant 2 a\), the \(x\)-axis and the line \(x = 2 a\), where \(a\) is a positive constant. The vertices of the lamina are \(\mathrm { O } ( 0,0 ) , \mathrm { A } ( 2 a , 8 a )\) and \(\mathrm { B } ( 2 a , 0 )\), as shown in Fig. 4.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2dd5719-bef3-45f2-afd2-c481e6a4b129-5_714_509_1546_858} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure}
   1. Find the coordinates of the centre of mass of the lamina.
   2. The lamina is freely suspended from the point A and hangs in equilibrium. Find the angle that AB makes with the vertical.

4

1. The region \(T\) is bounded by the \(x\)-axis, the line \(y = k x\) for \(a \leqslant x \leqslant 3 a\), the line \(x = a\) and the line \(x = 3 a\), where \(k\) and \(a\) are positive constants. A uniform frustum of a cone is formed by rotating \(T\) about the \(x\)-axis. Find the \(x\)-coordinate of the centre of mass of this frustum.
2. A uniform lamina occupies the region (shown in Fig. 4) bounded by the \(x\)-axis, the curve \(y = 16 \left( 1 - x ^ { - \frac { 1 } { 3 } } \right)\) for \(1 \leqslant x \leqslant 8\) and the line \(x = 8\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{86d79489-aec1-4c94-bef6-45b007f818a0-4_368_519_1439_772} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
   1. Find the coordinates of the centre of mass of this lamina. A hole is made in the lamina by cutting out a circular disc of area 5 square units. This causes the centre of mass of the lamina to move to the point \(( 5,3 )\).
   2. Find the coordinates of the centre of the hole.

4

1. The region enclosed between the curve \(y = x ^ { 4 }\) and the line \(y = h\) (where \(h\) is positive) is rotated about the \(y\)-axis to form a uniform solid of revolution. Find the \(y\)-coordinate of the centre of mass of this solid.
2. The region \(A\) is bounded by the \(x\)-axis, the curve \(y = x + \sqrt { x }\) for \(0 \leqslant x \leqslant 4\), and the line \(x = 4\). The region \(B\) is bounded by the \(y\)-axis, the curve \(y = x + \sqrt { x }\) for \(0 \leqslant x \leqslant 4\), and the line \(y = 6\). These regions are shown in Fig. 4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3f674569-7e99-4ba8-84f1-a1eb438e30ed-3_572_513_1779_778} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
   1. A uniform lamina occupies the region \(A\). Show that the \(x\)-coordinate of the centre of mass of this lamina is 2.56 , and find the \(y\)-coordinate.
   2. Using your answer to part (i), or otherwise, find the coordinates of the centre of mass of a uniform lamina occupying the region \(B\).

4 The region bounded by the curve \(y = \sqrt { x }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution.

1. Find the \(x\)-coordinate of the centre of mass of this solid. From this solid, the cylinder with radius 1 and length 3 with its axis along the \(x\)-axis (from \(x = 1\) to \(x = 4\) ) is removed.
2. Show that the centre of mass of the remaining object, Q , has \(x\)-coordinate 3 . This object Q has weight 96 N and it is supported, with its axis of symmetry horizontal, by a string passing through the cylindrical hole and attached to fixed points A and B (see Fig. 4). AB is horizontal and the sections of the string attached to A and B are vertical. There is sufficient friction to prevent slipping. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5bb02383-91c0-4454-aaea-0bd6af6ba325-5_837_819_1034_628} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
3. Find the support forces, \(R\) and \(S\), acting on the string at A and B
   (A) when the string is light,
   (B) when the string is heavy and uniform with a total weight of 6 N .

4

1. The region bounded by the curve \(y = x ^ { 3 }\) for \(0 \leqslant x \leqslant 2\), the \(x\)-axis and the line \(x = 2\), is occupied by a uniform lamina. Find the coordinates of the centre of mass of this lamina. [8]
2. The region bounded by the circular arc \(y = \sqrt { 4 - x ^ { 2 } }\) for \(1 \leqslant x \leqslant 2\), the \(x\)-axis and the line \(x = 1\), is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution, as shown in Fig. 4.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{39e14918-5017-43c0-9b74-7c68717ad5f3-5_627_499_593_785} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure}
   1. Show that the \(x\)-coordinate of the centre of mass of this solid of revolution is 1.35 . This solid is placed on a rough horizontal surface, with its flat face in a vertical plane. It is held in equilibrium by a light horizontal string attached to its highest point and perpendicular to its flat face, as shown in Fig. 4.2. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{39e14918-5017-43c0-9b74-7c68717ad5f3-5_573_613_1662_728} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
      \end{figure}
   2. Find the least possible coefficient of friction between the solid and the horizontal surface.

4

1. A uniform lamina occupies the region bounded by the \(x\)-axis, the \(y\)-axis, the curve \(y = \mathrm { e } ^ { x }\) for \(0 \leqslant x \leqslant \ln 3\), and the line \(x = \ln 3\). Find, in an exact form, the coordinates of the centre of mass of this lamina.
2. A region is bounded by the \(x\)-axis, the curve \(y = \frac { 6 } { x ^ { 2 } }\) for \(2 \leqslant x \leqslant a\) (where \(a > 2\) ), the line \(x = 2\) and the line \(x = a\). This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution.
   1. Show that the \(x\)-coordinate of the centre of mass of this solid is \(\frac { 3 \left( a ^ { 3 } - 4 a \right) } { a ^ { 3 } - 8 }\).
   2. Show that, however large the value of \(a\), the centre of mass of this solid is less than 3 units from the origin.