Toppling and sliding of solids

A question is this type if and only if it requires determining the critical angle at which a uniform solid (cylinder, cone, cube, prism) on an inclined plane will topple or slide, or comparing these conditions.

13 questions · Standard +0.6

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CAIE M2 2006 June Q2
5 marks Standard +0.3
2 A uniform solid cone has height 38 cm .
  1. Write down the distance of the centre of mass of the cone from its base. \includegraphics[max width=\textwidth, alt={}, center]{ece63d46-5e56-4668-939a-9dbbcfc1a77a-2_497_547_1224_840} The cone is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted, and the cone remains in equilibrium until the angle of inclination of the plane reaches \(31 ^ { \circ }\) (see diagram), when the cone topples.
  2. Find the radius of the cone.
  3. Show that \(\mu \geqslant 0.601\), correct to 3 significant figures, where \(\mu\) is the coefficient of friction between the cone and the plane.
CAIE M2 2003 November Q2
6 marks Standard +0.3
2 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{be83d46f-bf5b-4382-b424-bb5067626adc-2_376_569_559_466} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{be83d46f-bf5b-4382-b424-bb5067626adc-2_485_456_450_1226} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A uniform solid cone has height 20 cm and base radius 10 cm . It is placed with its axis vertical on a rough horizontal plane (see Fig. 1). The plane is slowly tilted and the cone remains in equilibrium until the angle of inclination of the plane reaches \(\theta ^ { \circ }\), when the cone begins to topple without sliding (see Fig. 2).
  1. Find the value of \(\theta\).
  2. What can you say about the value of the coefficient of friction between the cone and the plane?
CAIE M2 2008 November Q2
4 marks Standard +0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{5109244c-3062-4f5f-9277-fc6b5b28f2d4-2_485_863_495_641} A uniform solid cylinder has height 24 cm and radius \(r \mathrm {~cm}\). A uniform solid cone has base radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The cylinder and the cone are both placed with their axes vertical on a rough horizontal plane (see diagram, which shows cross-sections of the solids). The plane is slowly tilted and both solids remain in equilibrium until the angle of inclination of the plane reaches \(\alpha ^ { \circ }\), when both solids topple simultaneously.
  1. Find the value of \(h\).
  2. Given that \(r = 10\), find the value of \(\alpha\).
CAIE M2 2011 November Q4
9 marks Challenging +1.2
4 A uniform solid cylinder has radius 0.7 m and height \(h \mathrm {~m}\). 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, \(\theta ^ { \circ }\), is increased gradually until the cone is about to topple.
  1. Find the value of \(\theta\) at which the cone is about to topple.
  2. Given that the cylinder does not topple, find the greatest possible value of \(h\). 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[max width=\textwidth, alt={}, center]{d1f1f036-1676-443e-b733-ca1fe79972d4-3_476_1211_836_466}
  3. Given that the solid immediately topples, find the least possible value of \(h\).
OCR M2 2007 January Q1
3 marks Moderate -0.3
1 A uniform solid cylinder has height 20 cm and diameter 12 cm . It is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted until the cylinder topples when the angle of inclination is \(\alpha\). Find \(\alpha\).
OCR M2 2005 June Q1
5 marks Standard +0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-2_531_533_269_806} A uniform solid cone has vertical height 20 cm and base radius \(r \mathrm {~cm}\). 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 \(24 ^ { \circ }\) (see diagram).
  1. Find \(r\), correct to 1 decimal place. 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 \(24 ^ { \circ }\).
  2. State, with justification, whether this cone will topple.
OCR M2 2014 June Q2
5 marks Standard +0.3
2 A uniform solid cylinder of height 12 cm and radius \(r \mathrm {~cm}\) is in equilibrium on a rough inclined plane with one of its circular faces in contact with the plane.
  1. The cylinder is on the point of toppling when the angle of inclination of the plane to the horizontal is \(21 ^ { \circ }\). Find \(r\). The cylinder is now placed on a different inclined plane with one of its circular faces in contact with the plane. This plane is also inclined at \(21 ^ { \circ }\) to the horizontal. The coefficient of friction between this plane and the cylinder is \(\mu\).
  2. The cylinder slides down this plane but does not topple. Find an inequality for \(\mu\).
OCR M2 Specimen Q2
7 marks Standard +0.3
2 A uniform circular cylinder, of radius 6 cm and height 15 cm , is in equilibrium on a fixed inclined plane with one of its ends in contact with the plane.
  1. Given that the cylinder is on the point of toppling, find the angle the plane makes with the horizontal. The cylinder is now placed on a horizontal board with one of its ends in contact with the board. The board is then tilted so that the angle it makes with the horizontal gradually increases.
  2. Given that the coefficient of friction between the cylinder and the board is \(\frac { 3 } { 4 }\), determine whether or not the cylinder will slide before it topples, justifying your answer.
CAIE M2 2019 March Q6
8 marks Challenging +1.2
  1. Find, in terms of \(r\), the distance of the centre of mass of the prism from the centre of the cylinder.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b8e52188-f9a6-46fc-90bf-97965c6dd324-11_633_729_258_708} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The prism has weight \(W \mathrm {~N}\) and is placed with its curved surface on a rough horizontal plane. The axis of symmetry of the cross-section makes an angle of \(30 ^ { \circ }\) with the vertical. A horizontal force of magnitude \(P \mathrm {~N}\) acting in the plane of the cross-section through the centre of mass is applied to the cylinder at the highest point of this cross-section (see Fig. 2). The prism rests in limiting equilibrium.
  2. Find the coefficient of friction between the prism and the plane.
CAIE M2 2012 November Q4
8 marks Challenging +1.2
  1. Find \(r\). The upper cylinder is now fixed to the lower cylinder to create a uniform object.
  2. Show that the centre of mass of the object is $$\frac { 25 h ^ { 2 } + 180 h + 81 } { 50 h + 180 } \mathrm {~m}$$ from \(A\). The object is placed with the plane face containing \(A\) in contact with a rough plane inclined at \(\alpha ^ { \circ }\) to the horizontal, where \(\tan \alpha = 0.5\). The object is on the point of toppling without sliding.
  3. Calculate \(h\).
CAIE Further Paper 3 2022 November Q3
7 marks Challenging +1.2
  1. Show that \(\mathrm { N } = \frac { 8 } { 15 } \mathrm {~W} ( 1 + 2 \mathrm { k } )\).
  2. Find the value of \(k\).
CAIE M2 2010 June Q2
5 marks Standard +0.3
\includegraphics{figure_2} A uniform solid cone has height 30 cm and base radius \(r\) cm. The cone is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted and the cone remains in equilibrium until the angle of inclination of the plane reaches \(35°\), when the cone topples. The diagram shows a cross-section of the cone.
  1. Find the value of \(r\). [3]
  2. Show that the coefficient of friction between the cone and the plane is greater than 0.7. [2]
CAIE Further Paper 3 2021 June Q4
7 marks Challenging +1.2
\includegraphics{figure_4} A uniform solid circular cone has vertical height \(kh\) and radius \(r\). A uniform solid cylinder has height \(h\) and radius \(r\). The base of the cone is joined to one of the circular faces of the cylinder so that the axes of symmetry of the two solids coincide (see diagram, which shows a cross-section). The cone and the cylinder are made of the same material.
  1. Show that the distance of the centre of mass of the combined solid from the base of the cylinder is \(\frac{h(k^2 + 4k + 6)}{4(3 + k)}\). [4]
The solid is placed on a plane that is inclined to the horizontal at an angle \(\theta\). The base of the cylinder is in contact with the plane. The plane is sufficiently rough to prevent sliding. It is given that \(3h = 2r\) and that the solid is on the point of toppling when \(\tan \theta = \frac{4}{3}\).
  1. Find the value of \(k\). [3]