6.04c Composite bodies: centre of mass

\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]

\includegraphics{figure_1} The trapezium \(ABCD\) is a uniform lamina with \(AB = 4\) m and \(BC = CD = DA = 2\) m, as shown in Figure 1.

1. Show that the centre of mass of the lamina is \(\frac{4\sqrt{3}}{9}\) m from \(AB\). [5]

The lamina is freely suspended from \(D\) and hangs in equilibrium.

1. Find the angle between \(DC\) and the vertical through \(D\). [5]

\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]

\includegraphics{figure_1} A uniform lamina \(ABCD\) is made by taking a uniform sheet of metal in the form of a rectangle \(ABED\), with \(AB = 3a\) and \(AD = 2a\), and removing the triangle \(BCE\), where \(C\) lies on \(DE\) and \(CE = a\), as shown in Fig. 1.

1. Find the distance of the centre of mass of the lamina from \(AD\). [5]

The lamina has mass \(M\). A particle of mass \(m\) is attached to the lamina at \(B\). When the loaded lamina is freely suspended from the mid-point of \(AB\), it hangs in equilibrium with \(AB\) horizontal.

1. Find \(m\) in terms of \(M\). [4]

Figure 1 \includegraphics{figure_1} Figure 1 shows four uniform rods joined to form a rigid rectangular framework \(ABCD\), where \(AB = CD = 2a\), and \(BC = AD = 3a\). Each rod has mass \(m\). Particles, of mass \(6m\) and \(2m\), are attached to the framework at points \(C\) and \(D\) respectively.

1. Find the distance of the centre of mass of the loaded framework from
   1. \(AB\),
   2. \(AD\).
   [7]

The loaded framework is freely suspended from \(B\) and hangs in equilibrium.

1. Find the angle which \(BC\) makes with the vertical. [3]

\includegraphics{figure_1} A triangular frame is formed by cutting a uniform rod into 3 pieces which are then joined to form a triangle \(ABC\), where \(AB = AC = 10\) cm and \(BC = 12\) cm, as shown in Figure 1.

1. Find the distance of the centre of mass of the frame from \(BC\). (5)

The frame has total mass \(M\). A particle of mass \(M\) is attached to the frame at the mid-point of \(BC\). The frame is then freely suspended from \(B\) and hangs in equilibrium.

1. Find the size of the angle between \(BC\) and the vertical. (4)

\includegraphics{figure_1} Figure 1 shows a uniform lamina \(ABCDE\) such that \(ABDE\) is a rectangle, \(BC = CD\), \(AB = 4a\) and \(AE = 2a\). The point \(F\) is the midpoint of \(BD\) and \(FC = a\).

1. Find, in terms of \(a\), the distance of the centre of mass of the lamina from \(AE\). [4]

The lamina is freely suspended from \(A\) and hangs in equilibrium.

1. Find the angle between \(AB\) and the downward vertical. [3]

\includegraphics{figure_2} A uniform triangular lamina \(ABC\) of mass \(M\) is such that \(AB = AC\), \(BC = 2a\) and the distance of \(A\) from \(BC\) is \(h\). A line, parallel to \(BC\) and at a distance \(\frac{2h}{3}\) from \(A\), cuts \(AB\) at \(D\) and cuts \(AC\) at \(E\), as shown in Figure 2. It is given that the mass of the trapezium \(BCED\) is \(\frac{5M}{9}\).

1. Show that the centre of mass of the trapezium \(BCED\) is \(\frac{7h}{45}\) from \(BC\). [5]

\includegraphics{figure_3} The portion \(ADE\) of the lamina is folded through 180° about \(DE\) to form the folded lamina shown in Figure 3.

1. Find the distance of the centre of mass of the folded lamina from \(BC\). [4]

The folded lamina is freely suspended from \(D\) and hangs in equilibrium. The angle between \(DE\) and the downward vertical is \(\alpha\).

1. Find tan \(\alpha\) in terms of \(a\) and \(h\). [4]

\includegraphics{figure_1} A child's toy consists of a uniform solid hemisphere, of mass \(M\) and base radius \(r\), joined to a uniform solid right circular cone of mass \(m\), where \(2m < M\). The cone has vertex \(O\), base radius \(r\) and height \(3r\). Its plane face, with diameter \(AB\), coincides with the plane face of the hemisphere, as shown in Figure 1.

1. Show that the distance of the centre of mass of the toy from \(AB\) is $$\frac{3(M - 2m)}{8(M + m)}r.$$ [5]

The toy is placed with \(OA\) on a horizontal surface. The toy is released from rest and does not remain in equilibrium.

1. Show that \(M > 26m\). [4]

\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]

\includegraphics{figure_3} An ornament \(S\) is formed by removing a solid right circular cone, of radius \(r\) and height \(\frac{1}{2}h\), from a solid uniform cylinder, of radius \(r\) and height \(h\), as shown in Fig. 3.

1. Show that the distance of the centre of mass \(S\) from its plane face is \(\frac{19}{30}h\). [7]

The ornament is suspended from a point on the circular rim of its open end. It hangs in equilibrium with its axis of symmetry inclined at an angle \(\alpha\) to the horizontal. Given that \(h = 4r\),

1. find, in degrees to one decimal place, the value of \(\alpha\). [4]

A bowl consists of a uniform solid metal hemisphere, of radius \(a\) and centre \(O\), from which is removed the solid hemisphere of radius \(\frac{1}{4}a\) with the same centre \(O\).

1. Show that the distance of the centre of mass of the bowl from \(O\) is \(\frac{45}{112}a\). [5]

The bowl is fixed with its plane face uppermost and horizontal. It is now filled with liquid. The mass of the bowl is \(M\) and the mass of the liquid is \(kM\), where \(k\) is a constant. Given that the distance of the centre of mass of the bowl and liquid together from \(O\) is \(\frac{17}{48}a\),

1. Find the value of \(k\). [5]

An open container \(C\) is modelled as a thin uniform hollow cylinder of radius \(h\) and height \(h\) with a base but no lid. The centre of the base is \(O\).

1. Show that the distance of the centre of mass of \(C\) from \(O\) is \(\frac{1}{4}h\). [5]

The container is filled with uniform liquid. Given that the mass of the container is \(M\) and the mass of the liquid is \(M\),

1. find the distance of the centre of mass of the filled container from \(O\). [5]

[The centre of mass of a uniform hollow cone of height \(h\) is \(\frac{1}{3}h\) above the base on the line from the centre of the base to the vertex.] \includegraphics{figure_1} A marker for the route of a charity walk consists of a uniform hollow cone fixed on to a uniform solid cylindrical ring, as shown in Figure 1. The hollow cone has base radius \(r\), height \(9h\) and mass \(m\). The solid cylindrical ring has outer radius \(r\), height \(2h\) and mass \(3m\). The marker stands with its base on a horizontal surface.

1. Find, in terms of \(h\), the distance of the centre of mass of the marker from the horizontal surface. [5]

When the marker stands on a plane inclined at arctan \(\frac{1}{12}\) to the horizontal it is on the point of toppling over. The coefficient of friction between the marker and the plane is large enough to be certain that the marker will not slip.

1. Find \(h\) in terms of \(r\). [3]

\includegraphics{figure_2} Figure 2 shows the cross-section \(AVBC\) of the solid \(S\) formed when a uniform right circular cone of base radius \(a\) and height \(a\), is removed from a uniform right circular cone of base radius \(a\) and height \(2a\). Both cones have the same axis \(VCO\), where \(O\) is the centre of the base of each cone.

1. Show that the distance of the centre of mass of \(S\) from the vertex \(V\) is \(\frac{5}{4}a\). [5]

The mass of \(S\) is \(M\). A particle of mass \(kM\) is attached to \(S\) at \(B\). The system is suspended by a string attached to the vertex \(V\), and hangs freely in equilibrium. Given that \(VA\) is at an angle \(45°\) to the vertical through \(V\),

1. find the value of \(k\). [5]

\includegraphics{figure_1} Figure 1 shows a plank \(AB\) of mass 40 kg and length 6 m, which rests on supports at each of its ends. The plank is wedge-shaped, being thicker at end \(A\) than at end \(B\). A woman of mass 60 kg stands on the plank at a distance of 2 m from \(B\).

1. Suggest suitable modelling assumptions which can be made about
   1. the plank,
   2. the woman. [3 marks]
   Given that the reactions at each support are of equal magnitude,
2. find the magnitude of the reaction on the support at \(A\), [2 marks]
3. calculate the distance of the centre of mass of the plank from \(A\). [4 marks]

Four tools are attached to a board. The board is to be modelled as a uniform lamina and the four tools as four particles. The diagram shows the lamina, the four particles \(A\), \(B\), \(C\) and \(D\), and the \(x\) and \(y\) axes. \includegraphics{figure_3} The lamina has mass 5 kg and its centre of mass is at the point \((7, 6)\). Particle \(A\) has mass 4 kg and is at the point \((11, 2)\). Particle \(B\) has mass 3 kg and is at the point \((3, 6)\). Particle \(C\) has mass 7 kg and is at the point \((5, 9)\). Particle \(D\) has mass 1 kg and is at the point \((1, 4)\). Find the coordinates of the centre of mass of the system of board and tools. [5 marks]

A uniform wire \(ABCD\) is bent into the shape shown, where the sections \(AB\), \(BC\) and \(CD\) are straight and of length \(3a\), \(10a\) and \(5a\) respectively and \(AD\) is parallel to \(BC\). \includegraphics{figure_6}

1. Show that the cosine of angle \(BCD\) is \(\frac{3}{5}\). [2 marks]
2. Find the distances of the centre of mass of the bent wire from (i) \(AB\), (ii) \(BC\). [6 marks]

The wire is hung over a smooth peg at \(B\) and rests in equilibrium.

1. Find, to the nearest 0.1°, the angle between \(BC\) and the vertical in this position. [4 marks]

A uniform lamina is in the form of a trapezium \(ABCD\), as shown. \(AB\) and \(DC\) are perpendicular to \(BC\). \(AB = 17\) cm, \(BC = 21\) cm and \(CD = 8\) cm. \includegraphics{figure_7}

1. Find the distances of the centre of mass of the lamina from
   1. \(AB\),
   2. \(BC\). [8 marks]

The lamina is freely suspended from \(C\) and rests in equilibrium.

1. Find the angle between \(CD\) and the vertical. [3 marks]

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} The diagram shows a uniform lamina \(ABCDE\) formed by removing a symmetrical triangular section from a rectangular sheet of metal measuring 30 cm by 25 cm.

1. Find the distance of the centre of mass of the lamina from \(ED\). [4 marks]

The lamina has mass \(m\). A particle \(P\) is attached to the lamina at \(B\). The lamina is then suspended freely from \(A\) and hangs in equilibrium with \(AD\) vertical.

1. Find, in terms of \(m\), the mass of \(P\). [5 marks]

\includegraphics{figure_2} A key is modelled as a lamina which consists of a circle of radius 3 cm, with a circle of radius 1 cm removed from its centre, attached to a rectangle of length 8 cm and width 1 cm, with a rectangle measuring 3 cm by 1 cm fixed to its end as shown. Calculate the distance of the centre of mass of the key from the line marked \(AB\). [7 marks]

\includegraphics{figure_7} A barrier is modelled as a uniform rectangular plank of wood, \(ABCD\), rigidly joined to a uniform square metal plate, \(DEFG\). The plank of wood has mass 50 kg and dimensions 4.0 m by 0.25 m. The metal plate has mass 80 kg and side 0.5 m. The plank and plate are joined in such a way that \(CDE\) is a straight line (see diagram). The barrier is smoothly pivoted at the point \(D\). In the closed position, the barrier rests on a thin post at \(H\). The distance \(CH\) is 0.25 m.

1. Calculate the contact force at \(H\) when the barrier is in the closed position. [3]

In the open position, the centre of mass of the barrier is vertically above \(D\).

1. Calculate the angle between \(AB\) and the horizontal when the barrier is in the open position. [8]