6.04b Find centre of mass: using symmetry

3 A uniform rectangular lamina \(A B C D\), of mass 1.6 kg , has side \(A B\) of length 12 cm and side \(B C\) of length 8 cm . To create a logo, a uniform circular lamina, of mass 0.4 kg , is attached. The centre of the circular lamina is at the point \(C\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{088327c1-acd3-486d-b76f-1fe2560ffaff-3_520_780_593_630}

1. Find the distance of the centre of mass of the logo:
   1. from the line \(A B\);
   2. from the line \(A D\).
2. The logo is suspended in equilibrium, with \(A B\) horizontal, by two vertical strings. One string is attached at the point \(A\) and the other string is attached at the point \(B\). Find the tension in each of the two strings.

6. \includegraphics[max width=\textwidth, alt={}, center]{3c084e42-d304-4b77-afee-7e4bd801a03c-2_424_492_813_379} The diagram shows a uniform lamina \(A B C D E F\).

1. Calculate the distance of the centre of mass of the lamina from (i) \(A F\), (ii) \(A B\). The lamina is hung over a smooth peg at \(D\) and rests in equilibrium in a vertical plane.
2. Find the angle between \(C D\) and the vertical.

5. \includegraphics[max width=\textwidth, alt={}, center]{9e1d8a2f-0c35-4398-98ff-083ec76653ec-1_367_529_2122_383} A sign-board consists of a rectangular sheet of metal, of mass \(M\), which is 3 metres wide and 1 metre high, attached to two thin metal supports, each of mass \(m\) and length 2 metres. The board stands on horizontal ground.

1. Calculate the height above the ground of the centre of mass of the sign-board, in terms of \(M\) and \(m\). Given now that the centre of mass of the sign-board is \(2 \cdot 2\) metres above the ground, (b) find the ratio \(M : m\), in its simplest form. \section*{MECHANICS 2 (A) TEST PAPER 9 Page 2}

4 A golfer hits a ball from a point \(O\) on horizontal ground with a velocity of \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(\theta\) above the horizontal. The horizontal range of the ball is \(R\) metres and the time of flight is \(t\) seconds.

1. Express \(t\) in terms of \(\theta\), and hence show that \(R = 125 \sin 2 \theta\). The golfer hits the ball so that it lands 110 m from \(O\).
2. Calculate the two possible values of \(t\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6ae57fe9-3b6f-46c2-95b8-d48903ed796b-3_672_403_267_872} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} A toy is constructed by attaching a small ball of mass 0.01 kg to one end of a uniform rod of length 10 cm whose other end is attached to the centre of the plane face of a uniform solid hemisphere with radius 3 cm . The rod has mass 0.02 kg , the hemisphere has mass 0.5 kg and the rod is perpendicular to the plane face of the hemisphere (see Fig. 1).

3 Fig. 3.1 shows an object made up as follows. ABCD is a uniform lamina of mass \(16 \mathrm {~kg} . \mathrm { BE } , \mathrm { EF }\), FG, HI, IJ and JD are each uniform rods of mass 2 kg . ABCD, BEFG and HIJD are squares lying in the same plane. The dimensions in metres are shown in the figure. \begin{figure}[h] \end{figure}

1. Find the coordinates of the centre of mass of the object, referred to the axes shown in Fig.3.1. The rods are now re-positioned so that BEFG and HIJD are perpendicular to the lamina, as shown in Fig. 3.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5dd6ba0d-e516-4b9e-ba19-6e90520b171b-004_442_666_1510_722} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
2. Find the \(x\)-, \(y\)-and \(z\)-coordinates of the centre of mass of the object, referred to the axes shown in Fig. 3.2. Calculate the distance of the centre of mass from A . The object is now freely suspended from A and hangs in equilibrium with AC at \(\alpha ^ { \circ }\) to the vertical.
3. Calculate \(\alpha\).

3

1. A uniform lamina made from rectangular parts is shown in Fig. 3.1. All the dimensions are centimetres. All coordinates are referred to the axes shown in Fig. 3.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c1785fde-a6ce-4f8b-9948-4b4dd973ce84-4_691_529_427_762} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
   \end{figure}
   1. Show that the \(x\)-coordinate of the centre of mass of the lamina is 6.5 and find the \(y\)-coordinate. A square of side 2 cm is to be cut from the lamina. The sides of the square are to be parallel to the coordinate axes and the centre of the square is to be chosen so that the \(x\)-coordinate of the centre of mass of the new shape is 6.4
   2. Calculate the \(x\)-coordinate of the centre of the square to be removed. The \(y\)-coordinate of the centre of the square to be removed is now chosen so that the \(y\)-coordinate of the centre of mass of the final shape is as large as possible.
   3. Calculate the \(y\)-coordinate of the centre of mass of the lamina with the square removed, giving your answer correct to three significant figures.
2. Fig. 3.2 shows a framework made from light rods of length 2 m freely pin-jointed at \(\mathrm { A } , \mathrm { B } , \mathrm { C }\), D and E. The framework is in a vertical plane and is supported at A and C. There are loads of 120 N at B and at E . The force on the framework due to the support at A is \(R \mathrm {~N}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c1785fde-a6ce-4f8b-9948-4b4dd973ce84-5_448_741_459_662} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure} each rod is 2 m long
   1. Show that \(R = 150\).
   2. Draw a diagram showing all the forces acting at the points \(\mathrm { A } , \mathrm { B } , \mathrm { D }\) and E , including the forces internal to the rods. Calculate the internal forces in rods AE and EB , and determine whether each is a tension or a thrust. [You may leave your answers in surd form.]
   3. Without any further calculation of the forces in the rods, explain briefly how you can tell that rod ED is in thrust.

3 A fish slice consists of a blade and a handle as shown in Fig. 3.1. The rectangular blade ABCD is of mass 250 g and modelled as a lamina; this is 24 cm by 8 cm and is shown in the \(\mathrm { O } x y\) plane. The handle EF is of mass 125 g and is modelled as a thin rod; this is 30 cm long and E is attached to the mid-point of \(\mathrm { CD } . \mathrm { EF }\) is at right angles to CD and inclined at \(\alpha\) to the plane containing ABCD , where \(\sin \alpha = 0.6\) (and \(\cos \alpha = 0.8\) ). Coordinates refer to the axes shown in Fig. 3.1. Lengths are in centimetres. The \(y\) and \(z\)-coordinates of the centre of mass of the fish slice are \(\bar { y }\) and \(\bar { z }\) respectively. \begin{figure}[h] \end{figure}

1. Show that \(\bar { y } = 9 \frac { 1 } { 3 }\) and \(\bar { z } = 3\).
2. Suppose that the plane \(\mathrm { O } x y\) in Fig. 3.1 is horizontal and represents a table top and that the fish slice is placed on it as shown. Determine whether the fish slice topples. The 'superior' version of the fish slice has an extra mass of 125 g uniformly distributed over the existing handle for 10 cm from F towards E , as shown in Fig. 3.2. This section of the handle may still be modelled as a thin rod. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3865b4b3-97c7-412b-aabd-2705a954a847-4_513_1065_1683_539} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
3. In this new situation show that \(\bar { y } = 14\) and \(\bar { z } = 6\). A sales feature of the 'superior' version is the ability to suspend it using a very small hole in the blade. This situation is modelled as the fish slice hanging in equilibrium when suspended freely about an axis through O .
4. Indicate the position of the centre of mass on a diagram and calculate the angle of the line OE with the vertical.

4 In this question, coordinates refer to the axes shown in the figures and the units are centimetres.
Fig. 4.1 shows a lamina KLMNOP shaded. The lamina is made from uniform material and has the dimensions shown. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Show that the \(x\)-coordinate of the centre of mass of this lamina is 26 and calculate the \(y\)-coordinate. A uniform thin heavy wire KLMNOPQ is bent into the shape of part of the perimeter of the lamina KLMNOP with an extension of the side OP to Q, as shown in Fig. 4.2.
2. Show that the \(x\)-coordinate of the centre of mass of this wire is 23.2 and calculate the \(y\)-coordinate. The wire is freely suspended from Q and hangs in equilibrium.
3. Draw a diagram indicating the position of the centre of mass of the hanging wire and calculate the angle of QO with the vertical. A wall-mounted bin with an open top is shown in Fig. 4.3. The centre part has cross-section KLMNOPQ; the two ends are in the shape of the lamina KLMNOP. The ends are made from the same uniform, thin material and each has a mass of 1.5 kg . The centre part is made from different uniform, thin material and has a total mass of 7 kg . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2aaae62-a5f3-47da-afa5-1dd4b37ea2d6-5_499_540_2017_804} \captionsetup{labelformat=empty} \caption{Fig. 4.3}
   \end{figure}
4. Calculate the \(x\) - and \(y\)-coordinates of the centre of mass of the bin.

2 The shaded region shown in Fig. 2.1 is cut from a sheet of thin rigid uniform metal; LBCK and EFHI are rectangles; EF is perpendicular to CK . The dimensions shown in the figure are in centimetres. The Oy and Oz axes are also shown. \begin{figure}[h] \end{figure}

1. Calculate the coordinates of the centre of mass of the metal shape referred to the axes shown in Fig. 2.1. The metal shape is freely suspended from the point H and hangs in equilibrium.
2. Calculate the angle that HI makes with the vertical. The metal shape is now folded along OJ , AD and EI to give the object shown in Fig. 2.2; LOJK, ABCD and IEFH are all perpendicular to OADJ; LOJK and ABCD are on one side of OADJ and IEFH is on the other side of it. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a6297924-579e-4340-8fe6-2b43bd1a8698-3_542_929_1713_575} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
3. Referred to the axes shown in Fig. 2.2, show that the \(x\)-coordinate of the centre of mass of the object is - 0.1 and find the other two coordinates of the centre of mass. The object is placed on a rough inclined plane with LOAB in contact with the plane. OL is parallel to a line of greatest slope of the plane with L higher than O . The object does not slip but is on the point of tipping about the edge OA .
4. Calculate the angle of OL to the horizontal.

3 A thin rigid non-uniform beam AB of length 6 m has weight 800 N . Its centre of mass, G , is 2 m from B .
Initially the beam is horizontal and in equilibrium when supported by a small round peg at \(\mathrm { C } , 1 \mathrm {~m}\) from A , and a light vertical wire at B . This situation is shown in Fig. 3.1 where the lengths are in metres. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Calculate the tension in the wire and the normal reaction of the peg on the beam. The beam is now held horizontal and in equilibrium with the wire at \(70 ^ { \circ }\) to the horizontal, as shown in Fig. 3.2. The peg at C is rough and still supports the beam 1 m from A. The beam is on the point of slipping.
2. Calculate the new tension in the wire. Calculate also the coefficient of friction between the peg and the beam. The beam is now held in equilibrium at \(30 ^ { \circ }\) to the vertical with the wire at \(\theta ^ { \circ }\) to the beam, as shown in Fig. 3.3. A new small smooth peg now makes contact with the beam at C, still 1 m from A. The tension in the wire is now \(T \mathrm {~N}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a6297924-579e-4340-8fe6-2b43bd1a8698-4_456_353_1484_861} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
   \end{figure}
3. By taking moments about C , resolving in a suitable direction and obtaining two equations in terms of \(\theta\) and \(T\), or otherwise, calculate \(\theta\) and \(T\).

3 The object shown shaded in Fig. 3.1 is cut from a flat sheet of thin rigid uniform material; LMJK, OAIJ, AEFH and CDEB are rectangles. The grid-lines in Fig. 3.1 are 1 cm apart. \begin{figure}[h] \end{figure}

1. Calculate the coordinates of the centre of mass of the object referred to the axes shown in Fig. 3.1. [5] The object is freely suspended from the point K and hangs in equilibrium.
2. Calculate the angle that KI makes with the vertical. The mass of the object is 0.3 kg .
   A particle of mass \(m \mathrm {~kg}\) is attached to the object at a point on the line OJ so that the new centre of mass is at the centre of the square OAIJ.
3. Calculate the value of \(m\) and the position of the particle referred to the axes shown in Fig. 3.1. The extra particle is now removed and the object shown in Fig. 3.1 is folded: LMJK is folded along JM so that it is perpendicular to OAIJ; ABCDEFH is folded along AH so that it is perpendicular to OAIJ and on the same side of OAIJ as LMJK. The folded object is placed on a horizontal table with the edges KL and FED in contact with the table. A plan view and a 3D representation are shown in Fig. 3.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{42b6ee17-f0ae-4687-8392-281ba724a607-4_609_648_1836_246} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{42b6ee17-f0ae-4687-8392-281ba724a607-4_332_695_2001_1144} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
4. On the plan, indicate the region corresponding to positions of the centre of mass for which the folded object is stable. You are given that the \(x\)-coordinate of the centre of mass of the folded object is 1.7 . Determine whether the object is stable.

3 Fig. 3.1 shows an object made up as follows. ABCD is a uniform lamina of mass \(16 \mathrm {~kg} . \mathrm { BE } , \mathrm { EF }\), FG, HI, IJ and JD are each uniform rods of mass 2 kg . ABCD, BEFG and HIJD are squares lying in the same plane. The dimensions in metres are shown in the figure. \begin{figure}[h] \end{figure}

1. Find the coordinates of the centre of mass of the object, referred to the axes shown in Fig.3.1. The rods are now re-positioned so that BEFG and HIJD are perpendicular to the lamina, as shown in Fig. 3.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{43d5bbfb-8726-4bcd-a73d-01728d532e98-4_442_666_1510_722} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
2. Find the \(x\)-, \(y\)-and \(z\)-coordinates of the centre of mass of the object, referred to the axes shown in Fig. 3.2. Calculate the distance of the centre of mass from A . The object is now freely suspended from A and hangs in equilibrium with AC at \(\alpha ^ { \circ }\) to the vertical.
3. Calculate \(\alpha\).

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 \(F\).

4 In this question you may use the following facts: as illustrated in Fig. 4.1, the centre of mass, G, of a uniform thin open hemispherical shell is at the mid-point of OA on its axis of symmetry; the surface area of this shell is \(2 \pi r ^ { 2 }\), where \(r\) is the distance OA. \begin{figure}[h] \end{figure} A perspective view and a cross-section of a dog bowl are shown in Fig. 4.2. The bowl is made throughout from thin uniform material. An open hemispherical shell of radius 8 cm is fitted inside an open circular cylinder of radius 8 cm so that they have a common axis of symmetry and the rim of the hemisphere is at one end of the cylinder. The height of the cylinder is \(k \mathrm {~cm}\). The point O is on the axis of symmetry and at the end of the cylinder. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Show that the centre of mass of the bowl is a distance \(\frac { 64 + k ^ { 2 } } { 16 + 2 k } \mathrm {~cm}\) from O . A version of the bowl for the 'senior dog' has \(k = 12\) and an end to the cylinder, as shown in Fig. 4.3. The end is made from the same material as the original bowl.
2. Show that the centre of mass of this bowl is a distance \(6 \frac { 1 } { 3 } \mathrm {~cm}\) from O . This bowl is placed on a rough slope inclined at \(\theta\) to the horizontal.
3. Assume that the bowl is prevented from sliding and is on the point of toppling. Draw a diagram indicating the position of the centre of mass of the bowl with relevant lengths marked. Calculate the value of \(\theta\).
4. If the bowl is not prevented from sliding, determine whether it will slide when placed on the slope when there is a coefficient of friction between the bowl and the slope of 1.5.

3 A bracket is being made from a sheet of uniform thin metal. Firstly, a plate is cut from a square of the sheet metal in the shape OABCDEFHJK, shown shaded in Fig. 3.1. The dimensions shown in the figure are in centimetres; axes \(\mathrm { O } x\) and \(\mathrm { O } y\) are also shown. \begin{figure}[h] \end{figure}

1. Show that, referred to the axes given in Fig. 3.1, the centre of mass of the plate OABCDEFHJK has coordinates (0.8, 2.5). The plate is hung using light vertical strings attached to \(\mathbf { J }\) and \(\mathbf { H }\). The edge \(\mathbf { J H }\) is horizontal and the plate is in equilibrium. The weight of the plate is 3.2 N .
2. Calculate the tensions in each of the strings. The plate is now bent to form the bracket. This is shown in Fig. 3.2: the rectangle IJKO is folded along the line IA so that it is perpendicular to the plane ABCGHI ; the rectangle DEFG is folded along the line DG so it is also perpendicular to the plane ABCGHI but on the other side of it. Fig. 3.2 also shows the axes \(\mathrm { O } x , \mathrm { O } y\) and \(\mathrm { O } z\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-4_611_782_1713_678} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
3. Show that, referred to the axes given in Fig. 3.2, the centre of mass of the bracket has coordinates ( \(1,2.7,0\) ). The bracket is now hung freely in equilibrium from a string attached to O .
4. Calculate the angle between the edge OI and the vertical.

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 \(T\).

3. \begin{figure}[h] \end{figure} Figure 1 shows a uniform triangular lamina \(A B C\) placed with edge \(B C\) along the line of greatest slope of a plane inclined at an angle \(\theta\) to the horizontal. The lengths \(A C\) and \(B C\) are 15 cm and 9 cm respectively and \(\angle A B C\) is a right angle.

1. Find the distance of the centre of mass of the lamina from
   1. \(\quad A B\),
   2. \(B C\). Assuming that the plane is rough enough to prevent the lamina from slipping,
2. find in degrees, correct to 1 decimal place, the maximum value of \(\theta\) for which the lamina remains in equilibrium.
   (4 marks)

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)

2 A straight \(\operatorname { rod } A B\) has length \(a\). The rod has variable density, and at a distance \(x\) from \(A\) its mass per unit length is given by \(k \left( 4 - \sqrt { \frac { x } { a } } \right)\), where \(k\) is a constant. Find the distance from \(A\) of the centre of mass of the rod.

3 The region bounded by the \(y\)-axis and the curves \(y = \sin 2 x\) and \(y = \sqrt { 2 } \cos x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\) is occupied by a uniform lamina. Find the exact value of the \(x\)-coordinate of the centre of mass of the lamina.