Lamina with removed circle/semicircle

7 \includegraphics[max width=\textwidth, alt={}, center]{9d377c95-09b8-4893-b29f-8517a5016e8b-4_597_1011_251_566} \(A B C D E\) is the cross-section through the centre of mass of a uniform prism resting in equilibrium with \(D E\) on a horizontal surface. The cross-section has the shape of a square \(O B C D\) with sides of length \(a \mathrm {~m}\), from which a quadrant \(O A E\) of a circle of radius 1 m has been removed (see diagram).

1. Find the distance of the centre of mass of the prism from \(O\), giving the answer in terms of \(a , \pi\) and \(\sqrt { } 2\).
2. Hence show that $$3 a ^ { 2 } ( 2 - a ) < \frac { 3 } { 2 } \pi - 2$$ and verify that this inequality is satisfied by \(a = 1.68\) but not by \(a = 1.67\).

A uniform circular disc has centre \(O\) and radius \(1.2\text{ m}\). The centre of the disc is at the origin of \(x\)- and \(y\)-axes. Two circular holes with centres at \(A\) and \(B\) are made in the disc (see diagram). The point \(A\) is on the negative \(x\)-axis with \(OA = 0.5\text{ m}\). The point \(B\) is on the negative \(y\)-axis with \(OB = 0.7\text{ m}\). The hole with centre \(A\) has radius \(0.3\text{ m}\) and the hole with centre \(B\) has radius \(0.4\text{ m}\). Find the distance of the centre of mass of the object from

1. the \(x\)-axis, [4]
2. the \(y\)-axis. [3]

The object can rotate freely in a vertical plane about a horizontal axis through \(O\).

1. Calculate the angle which \(OA\) makes with the vertical when the object rests in equilibrium. [2]

\includegraphics{figure_6} A uniform circular disc has centre \(O\) and radius \(1.2\,\text{m}\). The centre of the disc is at the origin of \(x\)- and \(y\)-axes. Two circular holes with centres at \(A\) and \(B\) are made in the disc (see diagram). The point \(A\) is on the negative \(x\)-axis with \(OA = 0.5\,\text{m}\). The point \(B\) is on the negative \(y\)-axis with \(OB = 0.7\,\text{m}\). The hole with centre \(A\) has radius \(0.3\,\text{m}\) and the hole with centre \(B\) has radius \(0.4\,\text{m}\). Find the distance of the centre of mass of the object from

1. the \(x\)-axis, [4]
2. the \(y\)-axis. [3]

The object can rotate freely in a vertical plane about a horizontal axis through \(O\).

1. Calculate the angle which \(OA\) makes with the vertical when the object rests in equilibrium. [2]

\includegraphics{figure_1} Figure 1 shows a decoration which is made by cutting 2 circular discs from a sheet of uniform card. The discs are joined so that they touch at a point \(D\) on the circumference of both discs. The discs are coplanar and have centres \(A\) and \(B\) with radii 10 cm and 20 cm respectively.

1. Find the distance of the centre of mass of the decoration from \(B\). [5]

The point \(C\) lies on the circumference of the smaller disc and \(\angle CAB\) is a right angle. The decoration is freely suspended from \(C\) and hangs in equilibrium.

1. Find, in degrees to one decimal place, the angle between \(AB\) and the vertical. [4]

\includegraphics{figure_1} A uniform lamina \(L\) is formed by taking a uniform square sheet of material \(ABCD\), of side 10 cm, and removing the semi-circle with diameter \(AB\) from the square, as shown in Fig. 2.

1. Find, in cm to 2 decimal places, the distance of the centre of mass of the lamina \(L\) from the mid-point of \(AB\). [7]

[The centre of mass of a uniform semi-circular lamina, radius \(a\), is at a distance \(\frac{4a}{3π}\) from the centre of the bounding diameter.] The lamina is freely suspended from \(D\) and hangs at rest.

1. Find, in degrees to one decimal place, the angle between \(CD\) and the vertical. [4]

\includegraphics{figure_1} Figure 1 shows a template \(T\) made by removing a circular disc, of centre \(X\) and radius 8 cm, from a uniform circular lamina, of centre \(O\) and radius 24 cm. The point \(X\) lies on the diameter \(AOB\) of the lamina and \(AX = 16\) cm. The centre of mass of \(T\) is at the point \(G\).

1. Find \(AG\). [6]

The template \(T\) is free to rotate about a smooth fixed horizontal axis, perpendicular to the plane of \(T\), which passes through the mid-point of \(OB\). A small stud of mass \(\frac{1}{4}m\) is fixed at \(B\), and \(T\) and the stud are in equilibrium with \(AB\) horizontal. Modelling the stud as a particle,

1. find the mass of \(T\) in terms of \(m\). [4]

\includegraphics{figure_1} A set square \(S\) is made by removing a circle of centre \(O\) and radius 3 cm from a triangular piece of wood. The piece of wood is modelled as a uniform triangular lamina \(ABC\), with \(\angle ABC = 90°\), \(AB = 12\) cm and \(BC = 21\) cm. The point \(O\) is 5 cm from \(AB\) and 5 cm from \(BC\), as shown in Figure 1.

1. Find the distance of the centre of mass of \(S\) from
   1. \(AB\),
   2. \(BC\). [9]

The set square is freely suspended from \(C\) and hangs in equilibrium.

1. Find, to the nearest degree, the angle between \(CB\) and the vertical. [3]

[The centre of mass of a semi-circular lamina of radius \(r\) is \(\frac{4r}{3\pi}\) from the centre] \includegraphics{figure_3} A template \(T\) consists of a uniform plane lamina \(PQRQS\), as shown in Figure 3. The lamina is bounded by two semicircles, with diameters \(SQ\) and \(QR\), and by the sides \(SP\), \(PQ\) and \(QR\) of the rectangle \(PQRS\). The point \(O\) is the mid-point of \(SR\), \(PQ = 12\) cm and \(QR = 2\) cm.

1. Show that the centre of mass of \(T\) is a distance \(\frac{4|2x^2 - 3|}{8x + 3\pi}\) cm from \(SR\). [7]

The template \(T\) is freely suspended from the point \(P\) and hangs in equilibrium. Given that \(x = 2\) and that \(\theta\) is the angle that \(PQ\) makes with the horizontal,

1. show that \(\tan \theta = \frac{48 + 9\pi}{22 + 6\pi}\). [4]

\includegraphics{figure_1} A uniform lamina \(L\) is formed by taking a uniform square sheet of material \(ABCD\), of side 10 cm, and removing the semi-circle with diameter \(AB\) from the square, as shown in Fig. 2.

1. Find, in cm to 2 decimal places, the distance of the centre of mass of the lamina \(L\) from the mid-point of \(AB\). [7]

[The centre of mass of a uniform semi-circular lamina, radius \(a\), is at a distance \(\frac{4a}{3\pi}\) from the centre of the bounding diameter.] The lamina is freely suspended from \(D\) and hangs at rest.

1. Find, in degrees to one decimal place, the angle between \(CD\) and the vertical. [4]

A rectangular piece of cardboard \(ABCD\), measuring \(30\) cm by \(12\) cm, has a semicircle of radius \(5\) cm removed from it as shown. \includegraphics{figure_6}

1. Calculate the distances of the centre of mass of the remaining piece of cardboard from \(AB\) and from \(BC\). [7 marks]

The remaining cardboard is suspended from \(A\) and hangs in equilibrium.

1. Find the angle made by \(AB\) with the vertical. [4 marks]

\includegraphics{figure_4} A uniform square lamina \(ABCD\) of side 6 cm has a semicircular piece, with \(AB\) as diameter, removed (see diagram).

1. Find the distance of the centre of mass of the remaining shape from \(CD\). [6]

The remaining shape is suspended from a fixed point by a string attached at \(C\) and hangs in equilibrium.

1. Find the angle between \(CD\) and the vertical. [2]

\includegraphics{figure_1} A uniform lamina \(ABDC\) is bounded by two semicircular arcs \(AB\) and \(CD\), each with centre \(O\) and of radii \(3a\) and \(a\) respectively, and two straight edges, \(AC\) and \(DB\), which lie on the line \(AOB\) (see Fig. 1).

1. Show that the distance of the centre of mass of the lamina from \(O\) is \(\frac{13a}{3\pi}\). [5]

\includegraphics{figure_2} The lamina has mass 3 kg and is freely pivoted to a fixed point at \(A\). The lamina is held in equilibrium with \(AB\) vertical by means of a light string attached to \(B\). The string lies in the same plane as the lamina and is at an angle of \(40°\) below the horizontal (see Fig. 2).

1. Calculate the tension in the string. [3]
2. Find the direction of the force acting on the lamina at \(A\). [4]