Lamina in equilibrium with applied force

2 A uniform semicircular lamina of radius 0.25 m has diameter \(A B\). It is freely suspended at \(A\) from a fixed point and hangs in equilibrium.

1. Find the distance of the centre of mass of the lamina from the diameter \(A B\).
2. Calculate the angle which the diameter \(A B\) makes with the vertical. The lamina is now held in equilibrium with the diameter \(A B\) vertical by means of a force applied at \(B\). This force has magnitude 6 N and acts at \(45 ^ { \circ }\) to the upward vertical in the plane of the lamina.
3. Calculate the weight of the lamina.

3 \includegraphics[max width=\textwidth, alt={}, center]{a20a6641-d771-4c89-b40f-168a0c61f99d-3_293_1045_267_550} A uniform lamina \(A B C D\) is in the form of a trapezium in which \(A B\) and \(D C\) are parallel and have lengths 2 m and 3 m respectively. \(B D\) is perpendicular to the parallel sides and has length 1 m (see diagram).

1. Find the distance of the centre of mass of the lamina from \(B D\). The lamina has weight \(W \mathrm {~N}\) and is in equilibrium, suspended by a vertical string attached to the lamina at \(B\). The lamina rests on a vertical support at \(C\). The lamina is in a vertical plane with \(A B\) and \(D C\) horizontal.
2. Find, in terms of \(W\), the tension in the string and the magnitude of the force exerted on the lamina at \(C\).

7 \includegraphics[max width=\textwidth, alt={}, center]{9daebcbe-826e-4eda-afa7-c935c6ea2bfc-10_451_574_258_781} \(A B C D\) is a uniform lamina in the shape of a trapezium which has centre of mass \(G\). The sides \(A D\) and \(B C\) are parallel and 1.8 m apart, with \(A D = 2.4 \mathrm {~m}\) and \(B C = 1.2 \mathrm {~m}\) (see diagram).

1. Show that the distance of \(G\) from \(A D\) is 0.8 m .
   The lamina is freely suspended at \(A\) and hangs in equilibrium with \(A D\) making an angle of \(30 ^ { \circ }\) with the vertical.
2. Calculate the distance \(A G\).
   With the lamina still freely suspended at \(A\) a horizontal force of magnitude 7 N acting in the plane of the lamina is applied at \(D\). The lamina is in equilibrium with \(A G\) making an angle of \(10 ^ { \circ }\) with the downward vertical.
3. Find the two possible values for the weight of the lamina.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6. (a) A uniform lamina is in the shape of a quadrant of a circle of radius \(a\). Show, by integration, that the centre of mass of the lamina is at a distance of \(\frac { 4 a } { 3 \pi }\) from each of its straight edges. \begin{figure}[h] \end{figure} A second uniform lamina \(A B C D E F A\) is shown shaded in Figure 3. The straight sides \(A C\) and \(A E\) are perpendicular and \(A C = A E = 2 a\). In the figure, the midpoint of \(A C\) is \(B\), the midpoint of \(A E\) is \(F\), and \(A B D F\) and \(D G E F\) are squares of side \(a\). \(B C D\) is a quadrant of a circle with centre \(B\). \(D G E\) is a quadrant of a circle with centre \(G\).
(b) Find the distance of the centre of mass of the lamina from the side \(A E\). The lamina is smoothly hinged to a horizontal axis which passes through \(E\) and is perpendicular to the plane of the lamina. The lamina has weight \(W\) newtons. The lamina is held in equilibrium in a vertical plane, with \(A\) vertically above \(E\), by a horizontal force of magnitude \(X\) newtons applied at \(C\).
(c) Find \(X\) in terms of \(W\).

5 The cover of a children's book is modelled as being a uniform lamina \(L . L\) occupies the region bounded by the \(x\)-axis, the curve \(\mathrm { y } = 6 + \sin \mathrm { x }\) and the lines \(x = 0\) and \(x = 5\) (see Fig. 5.1). The centre of mass of \(L\) is at the point ( \(\mathrm { x } , \mathrm { y }\) ). \begin{figure}[h] \end{figure}

1. Show that \(\bar { X } = 2.36\), correct to 3 significant figures.
2. Find \(\bar { y }\), giving your answer correct to 3 significant figures. The cover of the book weighs 6 N . \(A\) is the point on the cover with coordinates \(( 3 , \bar { y } )\) and \(B\) is the point on the cover with coordinates \(( 5 , \bar { y } )\). A small badge of weight 2 N is attached to the cover at \(A\). The side of \(L\) along the \(y\)-axis is attached to the rest of the book and the book is placed on a rough horizontal plane. The attachment of the cover to the book is modelled as a hinge. The cover is held in equilibrium at an angle of \(\frac { 1 } { 3 } \pi\) radians to the horizontal by a force of magnitude \(P N\) acting at \(B\) perpendicular to the cover (see Fig. 5.2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{831ba5da-df19-43bb-b163-02bbddb4e2b8-4_444_899_1889_246} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure}
3. State two additional modelling assumptions, one about the attachment of the cover and one about the badge, which are necessary to allow the value of \(P\) to be determined.
4. Using the modelling assumptions, determine the value of \(P\) giving your answer correct to 3 significant figures.

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.

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. A uniform solid of revolution \(S\) is formed by rotating the region enclosed between the \(x\)-axis and the curve \(y = x \sqrt { 4 - x }\) for \(0 \leqslant x \leqslant 4\) through \(2 \pi\) radians about the \(x\)-axis, as shown in Fig. 4.1. O is the origin and the end A of the solid is at the point \(( 4,0 )\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-5_520_625_408_703} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure}
   1. Find the \(x\)-coordinate of the centre of mass of the solid \(S\). The solid \(S\) has weight \(W\), and it is freely hinged to a fixed point at O . A horizontal force, of magnitude \(W\) acting in the vertical plane containing OA , is applied at the point A , as shown in Fig. 4.2. \(S\) is in equilibrium. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-5_346_512_1361_781} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
      \end{figure}
   2. Find the angle that OA makes with the vertical.
      [0pt] [Question 4(b) is printed overleaf]
2. Fig. 4.3 shows the region bounded by the \(x\)-axis, the \(y\)-axis, the line \(y = 8\) and the curve \(y = ( x - 2 ) ^ { 3 }\) for \(0 \leqslant y \leqslant 8\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-6_631_695_370_683} \captionsetup{labelformat=empty} \caption{Fig. 4.3}
   \end{figure} Find the coordinates of the centre of mass of a uniform lamina occupying this region.