Folded lamina

4. \begin{figure}[h] \end{figure} The uniform square lamina \(A B C D\) shown in Figure 2 has sides of length 4a. The points \(E\) and \(F\), on \(D A\) and \(D C\) respectively, are both at a distance \(3 a\) from \(D\). The portion \(D E F\) of the lamina is folded through \(180 ^ { \circ }\) about \(E F\) to form the folded lamina \(A B C F E\) shown in Figure 3 below. \begin{figure}[h] \end{figure}

1. Show that the distance from \(A B\) of the centre of mass of the folded lamina is \(\frac { 55 } { 32 } a\).
   (6) The folded lamina is freely suspended from \(E\) and hangs in equilibrium.
2. Find the size of the angle between \(E D\) and the downward vertical.

3. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} The uniform triangular lamina \(A B C\), shown in Figure 1, has height \(9 y\), base \(B C = 6 x\), and \(A B = A C\) The points \(P\) and \(Q\) are such that \(A P : P C = A Q : Q B = 2 : 1\) The lamina is folded along \(P Q\) to form the folded lamina \(F\) The distance of the centre of mass of \(F\) from \(P Q\) is \(d\)

1. Show that \(d = \frac { 16 } { 9 } y\) The folded lamina is suspended from \(P\) and hangs freely in equilibrium with \(P Q\) at an angle \(\alpha\) to the downward vertical.
   Given that \(\tan \alpha = \frac { 64 } { 81 }\)
2. find \(x\) in terms of \(y\)

5. \begin{figure}[h] \end{figure} Figure 3 shows a uniform rectangular lamina \(A B C D\) with sides of length \(3 a\) and \(k a\), where \(k > 3\). The point \(E\) on side \(A D\) is such that \(D E = 3 a\). Rectangle \(A B C D\) is folded along the line \(C E\) to produce the folded lamina \(L\) shown in Figure 4. \begin{figure}[h] \end{figure} Find, in terms of \(a\) and \(k\),

1. the distance of the centre of mass of \(L\) from \(A B\),
2. the distance of the centre of mass of \(L\) from \(A E\). The folded lamina \(L\) is freely suspended from \(A\) and hangs in equilibrium with \(A B\) at \(45 ^ { \circ }\) to the downward vertical.
3. Find, to 3 significant figures, the value of \(k\).

3. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} The uniform rectangular lamina \(A B D E\), shown in Figure 1, has side \(A B\) of length \(2 a\) and side \(B D\) of length \(6 a\). The point \(C\) divides \(B D\) in the ratio 1:2 and the point \(F\) divides \(E A\) in the ratio \(1 : 2\). The rectangular lamina is folded along \(F C\) to produce the folded lamina \(L\), shown in Figure 2.

1. Show that the centre of mass of \(L\) is \(\frac { 16 } { 9 } a\) from \(E F\). The folded lamina, \(L\), is freely suspended from \(C\) and hangs in equilibrium.
2. Find the size of the angle between \(C F\) and the downward vertical.

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 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.

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.

7 The vertices of a uniform triangular lamina, which is in the \(x - y\) plane, are at the origin and the points \(( 20,60 )\) and \(( 100,0 )\).

1. Determine the coordinates of the lamina's centre of mass. Fig. 7.1 shows a uniform lamina consisting of a triangular section and two identical rectangular sections. The coordinates of some of the vertices of the lamina are given in Fig. 7.1. The rectangular sections are then folded at right-angles to the triangular section, to give the rigid three-dimensional object illustrated in Fig. 7.2. Two of the edges, \(E _ { 1 }\) and \(E _ { 2 }\), are marked on both figures. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-7_933_739_799_164} \captionsetup{labelformat=empty} \caption{Fig. 7.1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-7_924_725_808_1133} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
   \end{figure}
2. Show that the \(x\)-coordinate of the centre of mass of the folded object is 43.6, and determine the \(y\) - and \(z\)-coordinates.
3. Determine whether it is possible for the folded object to rest in equilibrium with edges \(E _ { 1 }\) and \(E _ { 2 }\) in contact with a horizontal surface.

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 \(\alpha\).

\includegraphics{figure_2} A uniform triangular lamina \(ABC\) of mass \(M\) is such that \(AB = AC\), \(BC = 2a\) and the distance of \(A\) from \(BC\) is \(h\). A line, parallel to \(BC\) and at a distance \(\frac{2h}{3}\) from \(A\), cuts \(AB\) at \(D\) and cuts \(AC\) at \(E\), as shown in Figure 2. It is given that the mass of the trapezium \(BCED\) is \(\frac{5M}{9}\).

1. Show that the centre of mass of the trapezium \(BCED\) is \(\frac{7h}{45}\) from \(BC\). [5]

\includegraphics{figure_3} The portion \(ADE\) of the lamina is folded through 180° about \(DE\) to form the folded lamina shown in Figure 3.

1. Find the distance of the centre of mass of the folded lamina from \(BC\). [4]

The folded lamina is freely suspended from \(D\) and hangs in equilibrium. The angle between \(DE\) and the downward vertical is \(\alpha\).

1. Find tan \(\alpha\) in terms of \(a\) and \(h\). [4]

A lamina is made from uniform material in the shape shown in Fig. 3.1. BCJA, DZOJ, ZEIO and FGHI are all rectangles. The lengths of the sides are shown in centimetres. \includegraphics{figure_3}

1. Find the coordinates of the centre of mass of the lamina, referred to the axes shown in Fig. 3.1. [5]

The rectangles BCJA and FGHI are folded through 90° about the lines CJ and FI respectively to give the fire-screen shown in Fig. 3.2.

1. Show that the coordinates of the centre of mass of the fire-screen, referred to the axes shown in Fig. 3.2, are (2.5, 0, 57.5). [4]

The \(x\)- and \(y\)-axes are in a horizontal floor. The fire-screen has a weight of 72 N. A horizontal force \(P\) N is applied to the fire-screen at the point Z. This force is perpendicular to the line DE in the positive \(x\) direction. The fire-screen is on the point of tipping about the line AH.

1. Calculate the value of \(P\). [5]

The coefficient of friction between the fire-screen and the floor is \(\mu\).

1. For what values of \(\mu\) does the fire-screen slide before it tips? [4]