Cone stability and toppling conditions

2 \includegraphics[max width=\textwidth, alt={}, center]{7f8646df-a7d8-4ca1-a6ee-3ceab6bb83af-2_439_608_1181_772} A uniform solid hemisphere, with centre \(O\) and radius 4 cm , is held so that a point \(P\) of its rim is in contact with a horizontal surface. The plane face of the hemisphere makes an angle of \(70 ^ { \circ }\) with the horizontal. \(Q\) is the point on the axis of symmetry of the hemisphere which is vertically above \(P\). The diagram shows the vertical cross-section of the hemisphere which contains \(O , P\) and \(Q\).

1. Determine whether or not the centre of mass of the hemisphere is between \(O\) and \(Q\). The hemisphere is now released.
2. State whether or not the hemisphere falls on to its plane face, giving a reason for your answer.

1 \includegraphics[max width=\textwidth, alt={}, center]{a20a6641-d771-4c89-b40f-168a0c61f99d-2_552_604_264_772} A uniform solid cone has vertical height 28 cm and base radius 6 cm . The cone is held with a point of the circumference of its base in contact with a horizontal table, and with the base making an angle of \(\theta ^ { \circ }\) with the horizontal (see diagram). When the cone is released, it moves towards the equilibrium position in which its base is in contact with the table. Show that \(\theta < 40.6\), correct to 1 decimal place.

5 The triangular region shown below is rotated through \(360 ^ { \circ }\) around the \(x\)-axis, to form a solid cone. \includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-06_328_755_415_644} The coordinates of the vertices of the triangle are \(( 0,0 ) , ( 8,0 )\) and \(( 0,4 )\).
All units are in centimetres. 5

1. State an assumption that you should make about the cone in order to find the position of its centre of mass. 5
2. Using integration, prove that the centre of mass of the cone is 2 cm from its plane face.
   5
3. The cone is placed with its plane face on a rough board. One end of the board is lifted so that the angle between the board and the horizontal is gradually increased. Eventually the cone topples without sliding. 5 (c) (i) Find the angle between the board and the horizontal when the cone topples, giving your answer to the nearest degree. 5 (c) (ii) Find the range of possible values for the coefficient of friction between the cone and the board.

A uniform solid cylinder has radius 0.7 m and height \(h\) m. A uniform solid cone has base radius 0.7 m and height 2.4 m. The cylinder and the cone both rest in equilibrium each with a circular face in contact with a horizontal plane. The plane is now tilted so that its inclination to the horizontal, \(θ°\), is increased gradually until the cone is about to topple.

1. Find the value of \(θ\) at which the cone is about to topple. [2]
2. Given that the cylinder does not topple, find the greatest possible value of \(h\). [2]

The plane is returned to a horizontal position, and the cone is fixed to one end of the cylinder so that the plane faces coincide. It is given that the weight of the cylinder is three times the weight of the cone. The curved surface of the cone is placed on the horizontal plane (see diagram). \includegraphics{figure_4}

1. Given that the solid immediately topples, find the least possible value of \(h\). [5]

A uniform solid cone has vertical height 20 cm and base radius \(r\) cm. It is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted until the cone topples when the angle of inclination is \(24°\) (see diagram). \includegraphics{figure_1}

1. Find \(r\), correct to 1 decimal place. [4]

A uniform solid cone of vertical height 20 cm and base radius 2.5 cm is placed on the plane which is inclined at an angle of \(24°\).

1. State, with justification, whether this cone will topple. [1]

1. Show that the centre of mass of a uniform solid hemisphere of radius \(r\) is at a distance \(\frac{3r}{8}\) from the centre \(O\) of the plane face. [7 marks]

The figure shows the vertical cross-section of a rough solid hemisphere at rest on a rough inclined plane inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{10}\). \includegraphics{figure_7} \begin{enumerate}[label=(\alph*)] \setcounter{enumi}{1} \item Indicate on a copy of the figure the three forces acting on the hemisphere, clearly stating what they are, and paying special attention to their lines of action. [3 marks] \item Given that the plane face containing the diameter \(AB\) makes an angle \(\alpha\) with the vertical, show that \(\cos \alpha = \frac{4}{5}\). [6 marks] \end{enumerate]

A uniform solid sphere, of radius \(a\), is divided into two sections by a plane at a distance \(\frac{a}{2}\) from the centre and parallel to a diameter.

1. Show that the centre of gravity of the smaller cap from its plane face is \(\frac{7a}{40}\). [9 marks]

This smaller cap is now placed on an inclined plane whose angle of inclination to the horizontal is \(\theta\). The plane is rough enough to prevent slipping and the cap rests with its curved surface in contact with the plane.

1. If the maximum value of \(\theta\) for which this is possible without the cap turning over is 30°, find the corresponding maximum inclination of the axis of symmetry of the cap to the vertical. [6 marks]

1. Prove that the centre of mass of a uniform solid right circular cone of height \(h\) and base radius \(r\) is at a distance \(\frac{3h}{4}\) from the vertex. [7 marks]

An item of confectionery consists of a thin wafer in the form of a hollow right circular cone of height \(h\) and mass \(m\), filled with solid chocolate, also of mass \(m\), to a depth of \(kh\) as shown. The centre of mass of the item is at \(O\), the centre of the horizontal plane face of the chocolate. \includegraphics{figure_3}

1. Show that \(k = \frac{8h}{15}\). [3 marks]

In the packaging process, the cone has to move on a conveyor belt inclined at an angle \(\alpha\) to the horizontal as shown. If the belt is rough enough to prevent sliding, and the maximum value of \(\alpha\) for which the cone does not topple is \(45°\), \includegraphics{figure_4}

1. find the radius of the base of the cone in terms of \(h\). [4 marks]

The finite region enclosed by the line \(y = kx\), the \(x\)-axis and the line \(x = 5\) is rotated through 360° around the \(x\) axis to form a solid cone.

1. 1. Use integration to show that the position of the centre of mass of the cone is independent of \(k\) [4 marks]
   2. State the distance between the base of the cone and its centre of mass. [1 mark]
2. State one assumption that you have made about the cone. [1 mark]
3. The plane face of the cone is placed on a rough inclined plane. The coefficient of friction between the cone and the plane is 0.8 The angle between the plane and the horizontal is gradually increased from 0° Find the range of values of \(k\) for which the cone slides before it topples. [4 marks]