Centre of mass with variable parameter

2 \includegraphics[max width=\textwidth, alt={}, center]{b8e52188-f9a6-46fc-90bf-97965c6dd324-04_606_376_260_881} A uniform object is made by joining together three solid cubes with edges \(3 \mathrm {~m} , 2 \mathrm {~m}\) and 1 m . The object has an axis of symmetry, with the cubes stacked vertically and the cube of edge 2 m between the other two cubes (see diagram).

1. Calculate the distance of the centre of mass of the object above the base of the largest cube.
   The smallest cube is now removed from the object. It is replaced by a heavier uniform cube with 1 m edges which is made of a different material. The centre of mass of the object is now at the base of the 2 m cube.
2. Find the ratio of the masses of the two cubes of edge 1 m .

6 A uniform solid consists of a hemisphere with centre \(O\) and radius 0.6 m joined to a cylinder of radius 0.6 m and height 0.6 m . The plane face of the hemisphere coincides with one of the plane faces of the cylinder.

1. Calculate the distance of the centre of mass of the solid from \(O\).
   [0pt] [The volume of a hemisphere of radius \(r\) is \(\frac { 2 } { 3 } \pi r ^ { 3 }\).]
2. \includegraphics[max width=\textwidth, alt={}, center]{a093cbad-3ba0-45ce-a617-d4ecc8cb1ec9-4_547_631_593_797} A cylindrical hole, of length 0.48 m , starting at the plane face of the solid, is made along the axis of symmetry (see diagram). The resulting solid has its centre of mass at \(O\). Show that the area of the cross-section of the hole is \(\frac { 3 } { 16 } \pi \mathrm {~m} ^ { 2 }\).
3. It is possible to increase the length of the cylindrical hole so that the solid still has its centre of mass at \(O\). State the increase in the length of the hole.

3. \begin{figure}[h] \end{figure} A uniform circular disc \(C\) has centre \(X\) and radius \(R\).
A disc with centre \(Y\) and radius \(r\), where \(0 < r < R\) and \(X Y = R - r\), is removed from \(C\) to form the template shown shaded in Figure 1. The centre of mass of the template is a distance \(k r\) from \(X\).

1. Show that \(r = \frac { k } { 1 - k } R\)
2. Hence find the range of possible values of \(k\). The point \(P\) is on the outer edge of the template and \(P X\) is perpendicular to \(X Y\).
   The template is freely suspended from \(P\) and hangs in equilibrium.
   Given that \(k = \frac { 4 } { 9 }\)
3. find the angle that \(X Y\) makes with the vertical. The mass of the template is \(M\).
4. Find, in terms of \(M\), the mass of the lightest particle that could be attached to the template so that it would hang in equilibrium from \(P\) with \(X Y\) horizontal.

6. \begin{figure}[h] \end{figure} The uniform lamina \(P Q R S T U V\) shown in Figure 2 is formed from two identical rectangles, \(P Q U V\) and \(Q R S T U\).
The rectangles have sides \(P Q = R S = 2 a\) and \(P V = Q R = k a\).

1. Show that the centre of mass of the lamina is \(\left( \frac { 6 + k } { 4 } \right) a\) from \(P V\) The lamina is freely suspended from \(P\) and hangs in equilibrium with \(P R\) at an angle of \(\alpha\) to the downward vertical. Given that \(\tan \alpha = \frac { 7 } { 15 }\)
2. find the value of \(k\).

5. \includegraphics[max width=\textwidth, alt={}, center]{9e1d8a2f-0c35-4398-98ff-083ec76653ec-1_367_529_2122_383} A sign-board consists of a rectangular sheet of metal, of mass \(M\), which is 3 metres wide and 1 metre high, attached to two thin metal supports, each of mass \(m\) and length 2 metres. The board stands on horizontal ground.

1. Calculate the height above the ground of the centre of mass of the sign-board, in terms of \(M\) and \(m\). Given now that the centre of mass of the sign-board is \(2 \cdot 2\) metres above the ground, (b) find the ratio \(M : m\), in its simplest form. \section*{MECHANICS 2 (A) TEST PAPER 9 Page 2}

4 Fig. 4.1 shows a hollow circular cylinder open at one end and closed at the other. The radius of the cylinder is 0.1 m and its height is \(h \mathrm {~m} . \mathrm { O }\) and C are points on the axis of symmetry at the centres of the open and closed ends, respectively. The thin material used for the closed end has four times the density of the thin material used for the curved surface. \begin{figure}[h] \end{figure} Cylinders of this type are made with different values of \(h\).

1. Show that the centres of mass of these cylinders are on the line OC at a distance \(\frac { 5 h ^ { 2 } + 2 h } { 2 + 10 h } \mathrm {~m}\) from O . Fig. 4.2 shows one of the cylinders placed with its open end on a slope inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 2 } { 3 }\). The cylinder does not slip but is on the point of tipping.
2. Show that \(50 h ^ { 2 } + 5 h - 3 = 0\) and hence that \(h = 0.2\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-5_383_497_1178_1402} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure} Fig. 4.3 shows another of the cylinders that has weight 42 N and \(h = 0.5\). This cylinder has its open end on a rough horizontal plane. A force of magnitude \(T \mathrm {~N}\) is applied to a point P on the circumference of the closed end. This force is at an angle \(\beta\) with the horizontal such that \(\tan \beta = \frac { 3 } { 4 }\) and the force is in the vertical plane containing \(\mathrm { O } , \mathrm { C }\) and P . The cylinder does not slip but is on the point of tipping. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-5_451_679_1955_685} \captionsetup{labelformat=empty} \caption{Fig. 4.3}
   \end{figure}
3. Calculate \(T\).

3. \begin{figure}[h] \end{figure} Figure 1 shows the shape and dimensions of a template \(O P Q R S T U V\) made from thin uniform metal. \(O P = 5 \mathrm {~m} , P Q = 2 \mathrm {~m} , Q R = 1 \mathrm {~m} , R S = 1 \mathrm {~m} , T U = 2 \mathrm {~m} , U V = 1 \mathrm {~m} , V O = 3 \mathrm {~m}\).
Figure 1 also shows a coordinate system with \(O\) as origin and the \(x\)-axis and \(y\)-axis along \(O P\) and \(O V\) respectively. The unit of length on both axes is the metre. The centre of mass of the template has coordinates \(( \bar { x } , \bar { y } )\).

1. 1. Show that \(\bar { y } = 1\)
   2. Find the value of \(\bar { x }\). A new design requires the template to have its centre of mass at the point (2.5,1). In order to achieve this, two circular discs, each of radius \(r\) metres, are removed from the template which is shown in Figure 1, to form a new template \(L\). The centre of the first disc is ( \(0.5,0.5\) ) and the centre of the second disc is ( \(0.5 , a\) ) where \(a\) is a constant.
2. Find the value of \(r\).
3. 1. Explain how symmetry can be used to find the value of \(a\).
   2. Find the value of \(a\). The template \(L\) is now freely suspended from the point \(U\) and hangs in equilibrium.
4. Find the size of the angle between the line \(T U\) and the horizontal.

6. \begin{figure}[h] \end{figure} The shaded region shown in Figure 4 is bounded by the \(x\)-axis, the line with equation \(x = 9\) and the line with equation \(y = \frac { 1 } { 3 } x\). This shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution. This solid of revolution is used to model a solid right circular cone of height 9 cm and base radius 3 cm . The cone is non-uniform and the mass per unit volume of the cone at the point ( \(x , y , z\) ) is \(\lambda x \mathrm {~kg} \mathrm {~cm} ^ { - 3 }\), where \(0 \leqslant x \leqslant 9\) and \(\lambda\) is constant.

1. Find the distance of the centre of mass of the cone from its vertex. A toy is made by joining the circular plane face of the cone to the circular plane face of a uniform solid hemisphere of radius 3 cm , so that the centres of the two plane surfaces coincide. The weight of the cone is \(W\) newtons and the weight of the hemisphere is \(k W\) newtons.
   When the toy is placed on a smooth horizontal plane with any point of the curved surface of the hemisphere in contact with the plane, the toy will remain at rest.
2. Find the value of \(k\)

1. A flag pole is 15 m long.

The flag pole is non-uniform so that, at a distance \(x\) metres from its base, the mass per unit length of the flag pole, \(m \mathrm {~kg} \mathrm {~m} ^ { - 1 }\) is given by the formula \(m = 10 \left( 1 - \frac { x } { 25 } \right)\). The flag pole is modelled as a rod.

1. Show that the mass of the flag pole is 105 kg .
2. Find the distance of the centre of mass of the flag pole from its base.

\includegraphics{figure_5} A uniform object is made by joining a solid cone of height 0.8 m and base radius 0.6 m and a cylinder. The cylinder has length 0.4 m and radius 0.5 m. The cylinder has a cylindrical hole of length 0.4 m and radius \(x\) m drilled through it along the axis of symmetry. A plane face of the cylinder is attached to the base of the cone so that the object has an axis of symmetry perpendicular to its base and passing through the vertex of the cone. The object is placed with points on the base of the cone and the base of the cylinder in contact with a horizontal surface (see diagram). The object is on the point of toppling.

1. Show that the centre of mass of the object is 0.15 m from the base of the cone. [3]
2. Find \(x\). [4]

[The volume of a cone is \(\frac{1}{3}\pi r^2 h\).]

\includegraphics{figure_2} A bow consists of a uniform curved portion \(AB\) of mass \(1.4 \text{ kg}\), and a uniform taut string of mass \(m \text{ kg}\) which joins \(A\) and \(B\). The curved portion \(AB\) is an arc of a circle centre \(O\) and radius \(0.8 \text{ m}\). Angle \(AOB\) is \(\frac{2}{3}\pi\) radians (see diagram). The centre of mass of the bow (including the string) is \(0.65 \text{ m}\) from \(O\). Calculate \(m\). [6]

\includegraphics{figure_2} A uniform object is made by attaching the base of a solid hemisphere to the base of a solid cone so that the object has an axis of symmetry. The base of the cone has radius \(0.3\text{ m}\), and the hemisphere has radius \(0.2\text{ m}\). The object is placed on a horizontal plane with a point \(A\) on the curved surface of the hemisphere and a point \(B\) on the circumference of the cone in contact with the plane (see diagram).

1. Given that the object is on the point of toppling about \(B\), find the distance of the centre of mass of the object from the base of the cone. [3]
2. Given instead that the object is on the point of toppling about \(A\), calculate the height of the cone. [3]

[The volume of a cone is \(\frac{1}{3}\pi r^2 h\). The volume of a hemisphere is \(\frac{2}{3}\pi r^3\).]

\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 \(\tan \theta = \frac{1}{3}\).

1. Find the value of \(k\). [3]

When the object is suspended from \(A\), the angle between \(AB\) and the vertical is \(\theta\), where \(\tan\theta = \frac{1}{2}\).

1. Given that \(h = \frac{8}{3}a\), find the possible values of \(k\). [3]

\includegraphics{figure_1} A toy is formed by joining a uniform solid hemisphere, of radius \(r\) and mass \(4m\), to a uniform right circular solid cone of mass \(km\). The cone has vertex \(A\), base radius \(r\) and height \(2r\). The plane face of the cone coincides with the plane face of the hemisphere. The centre of the plane face of the hemisphere is \(O\) and \(OB\) is a radius of its plane face as shown in Figure 1. The centre of mass of the toy is at \(O\).

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

A metal stud of mass \(2m\) is attached to the toy at \(A\). The toy is now suspended by a light string attached to \(B\) and hangs freely at rest. The angle between \(OB\) and the vertical is \(30°\).

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

The diagram shows the cross-section through the centre of mass of a uniform solid prism. The cross-section is a trapezium ABCD with AB and CD perpendicular to AD. The lengths of AB and AD are each 5 cm and the length of CD is \((a + 5)\) cm. \includegraphics{figure_7}

1. Show the distance of the centre of mass of the prism from AD is $$\frac{a^2 + 15a + 75}{3(a + 10)} \text{ cm.}$$ [5]

The prism is placed with the face containing AB in contact with a horizontal surface.

1. Find the greatest value of \(a\) for which the prism does not topple. [3]

The prism is now placed on an inclined plane which makes an angle \(\theta^o\) with the horizontal. AB lies along a line of greatest slope with B higher than A.

1. Using the value for \(a\) found in part (ii), and assuming the prism does not slip down the plane, find the great value of \(\theta\) for which the prism does not topple. [6]