L-shaped or composite rectangular lamina

3. \begin{figure}[h] \end{figure} The uniform lamina \(O A B C D\) is shown in Figure 1, with \(O A = 6 a , A B = 3 a , C D = 2 a\) and \(D O = 6 a\) and with right angles at \(O , A\) and \(D\).

1. Find the distance of the centre of mass of the lamina
   1. from \(O D\),
   2. from \(O A\). The lamina is suspended from \(C\) and hangs freely in equilibrium with \(C B\) inclined at an angle \(\alpha\) to the vertical.
2. Find, to the nearest degree, the size of the angle \(\alpha\).

2. \begin{figure}[h] \end{figure} Figure 1 shows a template where

• PQUY is a uniform square lamina with sides of length \(4 a\)
• RSTU is a uniform square lamina with sides of length \(2 a\)
• VWXY is a uniform square lamina with sides of length \(2 a\)
• the three squares all lie in the same plane
• the mass per unit area of \(V W X Y\) is double the mass per unit area of \(P Q U Y\)
• the mass per unit area of \(R S T U\) is double the mass per unit area of \(P Q U Y\)
• the distance of the centre of mass of the template from \(P X\) is \(d\)
  1. Show that \(d = \frac { 5 } { 2 } a\)

The template is freely pivoted about \(Q\) and hangs in equilibrium with \(P Q\) at an angle of \(\theta\) to the downward vertical.

Find the value of \(\tan \theta\) The mass of the template is \(M\) The template is still freely pivoted about \(Q\), but it is now held in equilibrium, with \(P Q\) vertical, by a horizontal force of magnitude \(F\) which acts on the template at \(X\). The line of action of the force lies in the same plane as the template.

Find \(F\) in terms of \(M\) and \(g\)

5. \begin{figure}[h] \end{figure} The uniform L-shaped lamina \(A B C D E F\), shown in Figure 2, has sides \(A B\) and \(F E\) parallel, and sides \(B C\) and \(E D\) parallel. The pairs of parallel sides are 9 cm apart. The points \(A , F\), \(D\) and \(C\) lie on a straight line. \(A B = B C = 36 \mathrm {~cm} , F E = E D = 18 \mathrm {~cm} . \angle A B C = \angle F E D = 90 ^ { \circ }\), and \(\angle B C D = \angle E D F = \angle E F D = \angle B A C = 45 ^ { \circ }\).

1. Find the distance of the centre of mass of the lamina from
   1. side \(A B\),
   2. side \(B C\). The lamina is freely suspended from \(A\) and hangs in equilibrium.
2. Find, to the nearest degree, the size of the angle between \(A B\) and the vertical.

4 The diagram shows a uniform lamina \(A B C D E F G H\). \includegraphics[max width=\textwidth, alt={}, center]{6a49fdd7-f180-451c-8f37-ad764fe13dfd-3_346_933_1123_577}

1. Explain why the centre of mass is 25 cm from \(A H\).
2. Show that the centre of mass is 4.375 cm from \(H G\).
3. The lamina is freely suspended from \(A\). Find the angle between \(A B\) and the vertical when the lamina is in equilibrium.
4. Explain, briefly, how you have used the fact that the lamina is uniform.

4 The diagram shows a uniform lamina which is in the shape of two identical rectangles \(A X G H\) and \(Y B C D\) and a square \(X Y E F\), arranged as shown. The length of \(A X\) is 10 cm , the length of \(X Y\) is 10 cm and the length of \(A H\) is 30 cm . \includegraphics[max width=\textwidth, alt={}, center]{85514b55-3f13-4746-a3ef-747239b64cca-3_1183_1278_513_374}

1. Explain why the centre of mass of the lamina is 15 cm from \(A H\).
2. Find the distance of the centre of mass of the lamina from \(A B\).
3. The lamina is freely suspended from the point \(H\). Find, to the nearest degree, the angle between \(H G\) and the horizontal when the lamina is in equilibrium.

2 A uniform lamina is in the shape of a rectangle \(A B C D\) and a square \(E F G H\), as shown in the diagram. The length \(A B\) is 20 cm , the length \(B C\) is 30 cm , the length \(D E\) is 5 cm and the length \(E F\) is 10 cm . The point \(P\) is the midpoint of \(A B\) and the point \(Q\) is the midpoint of \(H G\). \includegraphics[max width=\textwidth, alt={}, center]{676e753d-1b80-413c-a4b9-21861db8dde5-2_615_1221_1585_429}

1. Explain why the centre of mass of the lamina lies on \(P Q\).
2. Find the distance of the centre of mass of the lamina from \(A B\).
3. The lamina is freely suspended from \(A\). Find, to the nearest degree, the angle between \(A D\) and the vertical when the lamina is in equilibrium.

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.

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.

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 \(B\).

3 Two identical uniform rectangular laminas, P and Q , each having length \(k a\) and width \(a\) are fixed together, in the same plane, to form a lamina R.
With reference to coordinate axes, the corners of P are at ( 0,0 ), ( \(k a , 0\) ), ( \(k a , a\) ) and ( \(0 , a\) ) and the corners of Q are at \(( k a , 0 ) , ( k a + a , 0 ) , ( k a + a , k a )\) and \(( k a , k a )\), as shown in Fig. 3. \begin{figure}[h] \end{figure} Determine the range of values of \(k\) for which the centre of mass of R lies outside the boundary of R.

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

\includegraphics{figure_1}

\includegraphics{figure_2}

\includegraphics{figure_3}

\includegraphics{figure_4}

\includegraphics{figure_5}

\includegraphics{figure_6}

\includegraphics{figure_2} The uniform L-shaped lamina \(OABCDE\), shown in Figure 2, is made from two identical rectangles. Each rectangle is 4 metres long and \(a\) metres wide. Giving each answer in terms of \(a\), find the distance of the centre of mass of the lamina from

1. \(OE\), [4]
2. \(OA\). [4]

The lamina is freely suspended from \(O\) and hangs in equilibrium with \(OE\) at an angle \(\theta\) to the downward vertical through \(O\), where \(\tan \theta = \frac{4}{3}\).

1. Find the value of \(a\). [4]

The diagram shows a uniform lamina \(ABCDEFGHIJKL\). \includegraphics{figure_3}

1. Explain why the centre of mass of the lamina is \(35\) cm from \(AL\). [1 mark]
2. Find the distance of the centre of mass from \(AF\). [4 marks]
3. The lamina is freely suspended from \(A\). Find the angle between \(AB\) and the vertical when the lamina is in equilibrium. [3 marks]
4. Explain, briefly, how you have used the fact that the lamina is uniform. [1 mark]

\includegraphics{figure_2} A key is modelled as a lamina which consists of a circle of radius 3 cm, with a circle of radius 1 cm removed from its centre, attached to a rectangle of length 8 cm and width 1 cm, with a rectangle measuring 3 cm by 1 cm fixed to its end as shown. Calculate the distance of the centre of mass of the key from the line marked \(AB\). [7 marks]