A uniform disc, of radius \(a\) and mass \(2 M\), is attached to a thin uniform rod \(A B\) of length \(6 a\) and mass \(M\). The rod lies along a diameter of the disc, so that the centre of the disc is a distance \(x\) from \(A\) (see diagram).
(i) Find the moment of inertia of the object, consisting of disc and rod, about a fixed horizontal axis \(l\) through \(A\) and perpendicular to the plane of the disc.

The object is free to rotate about the axis \(l\). The object is held with \(A B\) horizontal and is released from rest. When \(A B\) makes an angle \(\theta\) with the vertical, where \(\cos \theta=\frac{3}{5}\), the angular speed of the object is \(\sqrt{ }\left(\frac{2 g}{5 a}\right)\).
(ii) Find the possible values of \(x\).

An object is formed from a uniform circular disc, of radius \(2 a\) and mass \(3 M\), and a uniform \(\operatorname{rod} A B\), of length \(3 a\) and mass \(k M\), where \(k\) is a constant. The centre of the disc is \(O\). The end \(B\) of the rod is rigidly joined to a point on the circumference of the disc so that \(O B A\) is a straight line. The fixed horizontal axis \(l\) is in the plane of the object, passes through \(A\) and is perpendicular to \(A B\).
(i) Show that the moment of inertia of the object about the axis \(l\) is \(3 M a^{2}(26+k)\).

The object is free to rotate about \(l\).
(ii) Show that small oscillations of the object about \(l\) are approximately simple harmonic. Given that the period of these oscillations is \(4 \pi \sqrt{ }\left(\frac{a}{g}\right)\), find the value of \(k\).

Three identical uniform discs, \(A, B\) and \(C\), each have mass \(m\) and radius \(a\). They are joined together by uniform rods, each of which has mass \(\frac{1}{3} m\) and length \(2 a\). The discs lie in the same plane and their centres form the vertices of an equilateral triangle of side \(4 a\). Each rod has one end rigidly attached to the circumference of a disc and the other end rigidly attached to the circumference of an adjacent disc, so that the rod lies along the line joining the centres of the two discs (see diagram).
(i) Find the moment of inertia of this object about an axis \(l\), which is perpendicular to the plane of the object and through the centre of disc \(A\).

The object is free to rotate about the horizontal axis \(l\). It is released from rest in the position shown, with the centre of disc \(B\) vertically above the centre of \(\operatorname{disc} A\).
(ii) Write down the change in the vertical position of the centre of mass of the object when the centre of disc \(B\) is vertically below the centre of disc \(A\). Hence find the angular velocity of the object when the centre of disc \(B\) is vertically below the centre of disc \(A\).

A uniform picture frame of mass \(m\) is made by removing a rectangular lamina \(E F G H\) in which \(E F=4 a\) and \(F G=2 a\) from a larger rectangular lamina \(A B C D\) in which \(A B=6 a\) and \(B C=4 a\). The side \(E F\) is parallel to the side \(A B\). The point of intersection of the diagonals \(A C\) and \(B D\) coincides with the point of intersection of the diagonals \(E G\) and \(F H\). One end of a light inextensible string of length \(10 a\) is attached to \(A\) and the other end is attached to \(B\). The frame is suspended from the mid-point \(O\) of the string. A small object of mass \(\frac{11}{12} m\) is fixed to the mid-point of \(A B\) (see diagram).
(i) Show that the moment of inertia of the system, consisting of frame and small object, about an axis through \(O\) perpendicular to the plane of the frame, is \(\frac{169}{3} m a^{2}\).
(ii) Show that small oscillations of the system about this axis are approximately simple harmonic and state their period.

Axis \(l\)

Three thin uniform rings \(A, B\) and \(C\) are joined together, so that each ring is in contact with each of the other two rings. Ring \(A\) has radius \(2 a\) and mass \(3 M\); rings \(B\) and \(C\) each have radius \(3 a\) and mass \(2 M\). The rings lie in the same plane and the centres of the rings are at the vertices of an isosceles triangle. The object consisting of the three rings is free to rotate about the horizontal axis \(l\) which is tangential to ring \(A\), in the plane of the object and perpendicular to the line of symmetry of the object (see diagram).
(i) Show that the moment of inertia of the object about the axis \(l\) is \(180 M a^{2}\).

(ii) Show that small oscillations of the object about the axis \(l\) are approximately simple harmonic, and state the period.

Question 11 EITHER alternative.

An object is formed from a square frame \(A B C D\) with a square lamina attached in one corner of the frame. The frame consists of four identical thin rods, each of mass \(M\) and length \(2a\). The lamina has mass \(kM\) and edges of length \(a\). It has one vertex at \(C\) and adjacent sides in contact with \(C B\) and \(C D\).

(i) Show that the moment of inertia of the object about an axis \(l\) through \(A\) perpendicular to the plane of the object is

\[\frac23Ma^2(7k+20).\]

The object is released from rest with the edge \(AB\) horizontal and \(D\) vertically above \(A\). The object rotates freely about the fixed axis \(l\). The angular speed of the object is \(\frac12\sqrt{\frac{5g}{a}}\) when \(D\) is first vertically below \(A\).

(ii) Find the value of \(k\).

A uniform lamina \(O A B C D\) consists of a rectangle \(O A C D\) and a triangle \(A B C\). The length of \(O A\) is \(k a\), the length of \(O D\) is \(2 a\), the height of triangle \(A B C\) is \(h\) and angle \(C A B\) is \(45^{\circ}\) (see diagram). Relative to axes through \(O\), parallel and perpendicular to \(O A\) as shown, the centre of mass of triangle \(A B C\) is \((\bar{x}, \bar{y})\). (a) Show that \(\bar{x}\) is \(\frac{1}{3}(3 k a+h)\), and find an expression for \(\bar{y}\).

The lamina \(O A B C D\) is placed vertically on its edge \(O A\) on a horizontal plane. (b) Find, in terms of \(a\) and \(k\), the set of values of \(h\) for which the lamina is in equilibrium.

It is now given that \(k=\frac{\sqrt{3}}{3}\) and that the lamina is on the point of toppling. (c) Find, in terms of \(a\), the coordinates of the centre of mass of the triangle \(A B C\).

A uniform lamina \(O A B C D\) is in the form of a rectangle, \(O B C D\), joined along the edge \(O B\) to a quarter circle \(O A B\). The length of \(D O\) is \(k a\) and the length of \(O B\) is \(a\). The lamina rests in a vertical plane with its edge \(C B\) on a horizontal surface (see diagram). (a) Find, in terms of \(k, a\) and \(\pi\), an expression for the distance of the centre of mass above the horizontal surface. [You may use without proof the result for the centre of mass of a circular sector in the list of formulae (MF19).]

The lamina is on the point of toppling about \(B\). (b) Find the value of \(k\).

The lamina \(B F D E\) is obtained by removing triangles \(A E D\) and \(B C F\) from a uniform square lamina \(A B C D\) of side \(2 a\). The length of side \(A E\) is \(a\) and the length of side \(F C\) is \(h\) (see diagram). The centre of mass of \(B F D E\) is at a distance \(\bar{x}\) from \(A D\), and at a distance \(\bar{y}\) from \(A B\). (a) Show that \(\bar{x}=\frac{h^{2}-6 a h+11 a^{2}}{3(3 a-h)}\) and find a corresponding expression for \(\bar{y}\).

(b) The lamina \(B F D E\) is placed vertically on its edge \(E B\) on a smooth horizontal surface.

Find, in terms of \(a\), the set of possible values of \(h\) for which the lamina remains in equilibrium.

An object consists of a uniform lamina with a particle attached. The lamina \(ABCEFD\), of mass \(m\), is formed from a rectangle \(ABCD\) and an isosceles triangle \(CEF\), where \(F\) is the midpoint of \(CD\). The rectangle has sides \(AB=2a\) and \(AD=a\). The triangle has base \(a\) and height \(2a\). A particle of mass \(km\) is attached at \(E\). The object rests with edge \(AD\) on horizontal ground and is on the point of toppling about \(D\). Find \(k\).

\(ABCD\) is a uniform square lamina of side \(6a\). Points \(E\) and \(F\) are on \(DC\) and \(AB\), respectively, and are such that \(DE=FB=h\). The quadrilateral \(BCEF\) is removed from the square lamina.

(a) Show that the distance of the centre of mass of the resulting lamina \(AFED\) from \(AD\) is \(\dfrac{h^2-6ah+36a^2}{18a}\), and find a corresponding expression for the distance of the centre of mass from \(AB\).

When the lamina \(AFED\) is suspended from the point \(D\), the edge \(DA\) makes an angle \(\theta\) with the downward vertical, where \(\tan\theta=\dfrac{7}{15}\).

(b) Find, in terms of \(a\), the two possible values of \(h\).

A uniform lamina is in the form of a triangle \(OBC\), with \(OC=18a\), \(OB=24a\), and \(\angle COB=90^\circ\). The point \(A\) on \(OB\) is such that \(OA=x\). The triangle \(OAC\) is removed from the lamina.

(a) Find, in terms of \(a\) and \(x\), the distance of the centre of mass of the remaining object \(ABC\) from \(OC\).

The object \(ABC\) is suspended from \(C\). In its equilibrium position, the side \(AB\) makes an angle \(\theta\) with the vertical, where \(\tan\theta=\dfrac65\).

(b) Find \(x\) in terms of \(a\).

A uniform lamina is in the form of an isosceles triangle \(ABC\), in which \(AC=2a\) and \(\angle ABC=90^\circ\). The point \(D\) on \(AB\) is such that the ratio \(DB:AB=1:k\). The point \(E\) on \(CB\) is such that \(DE\) is parallel to \(AC\). The triangle \(DBE\) is removed from the lamina.

(a) Find, in terms of \(k\), the distance of the centre of mass of the remaining lamina \(ADEC\) from the midpoint of \(AC\).

When the lamina \(ADEC\) is freely suspended from the vertex \(A\), the edge \(AC\) makes an angle \(\theta\) with the downward vertical, where \(\tan\theta=\dfrac5{18}\).

(b) Find the value of \(k\).

A uniform lamina is in the form of a triangle \(A B C\), with \(A C=8 a, B C=6 a\) and angle \(A C B=90^{\circ}\). The point \(D\) on \(A C\) is such that \(A D=3 a\). The point \(E\) on \(C B\) is such that \(C E=x\) (see diagram). The triangle \(C D E\) is removed from the lamina.
(a) Find, in terms of \(a\) and \(x\), the distance of the centre of mass of the remaining object \(A D E B\) from \(A C\).

The object \(A D E B\) is on the point of toppling about the point \(E\) when the object is in the vertical plane with its edge \(E B\) on a smooth horizontal surface.
(b) Find \(x\) in terms of \(a\).

A uniform lamina \(O A B C\) is a trapezium whose vertices can be represented by coordinates in the \(x-y\) plane. The coordinates of the vertices are \(O(0,0), A(15,0), B(9,4)\) and \(C(3,4)\).

Find the \(x\)-coordinate of the centre of mass of the lamina.

A uniform lamina is in the form of a triangle \(A B C\) in which angle \(B\) is a right angle, \(A B=9 a\) and \(B C=6 a\). The point \(D\) is on \(B C\) such that \(B D=x\) (see diagram). The region \(A B D\) is removed from the lamina. The resulting shape \(A D C\) is placed with the edge \(D C\) on a horizontal surface and the plane \(A D C\) is vertical.

Find the set of values of \(x\), in terms of \(a\), for which the shape is in equilibrium.

A uniform lamina \(A B C D\) consists of two isosceles triangles \(A B D\) and \(B C D\). The diagonals of \(A B C D\) meet at the point \(O\). The length of \(A O\) is \(3 a\), the length of \(O C\) is \(6 a\) and the length of \(B D\) is \(16 a\) (see diagram).

Find the distance of the centre of mass of the lamina from \(D B\).

A uniform lamina \(A E C F\) is formed by removing two identical triangles \(B C E\) and \(C D F\) from a square lamina \(A B C D\). The square has side \(3 a\) and \(E B=D F=h\) (see diagram).
(a) Find the distance of the centre of mass of the lamina \(A E C F\) from \(A D\) and from \(A B\), giving your answers in terms of \(a\) and \(h\).

The lamina \(A E C F\) is placed vertically on its edge \(A E\) on a horizontal plane.
(b) Find, in terms of \(a\), the set of values of \(h\) for which the lamina remains in equilibrium.

A uniform square lamina \(A B C D\) has sides of length 10 cm . The point \(E\) is on \(B C\) with \(E C=7.5 \mathrm{~cm}\), and the point \(F\) is on \(D C\) with \(C F=x \mathrm{~cm}\). The triangle \(E F C\) is removed from \(A B C D\) (see diagram). The centre of mass of the resulting shape \(A B E F D\) is a distance \(\bar{x} \mathrm{~cm}\) from \(C B\) and a distance \(\bar{y} \mathrm{~cm}\) from \(C D\).
(a) Show that \(\bar{x}=\frac{400-x^{2}}{80-3 x}\) and find a corresponding expression for \(\bar{y}\).

The shape \(A B E F D\) is in equilibrium in a vertical plane with the edge \(D F\) resting on a smooth horizontal surface.
(b) Find the greatest possible value of \(x\), giving your answer in the form \(a+b \sqrt{2}\), where \(a\) and \(b\) are constants to be determined.