Centre of Mass 2

2 The region enclosed by the curve \(y = \sqrt { } x\) for \(0 \leqslant x \leqslant 9\), the \(x\)-axis and the line \(x = 9\) is occupied by a uniform lamina. Find the coordinates of the centre of mass of this lamina.

2. \begin{figure}[h] \end{figure} A uniform lamina is in the shape of the region \(R\) which is bounded by the curve with equation \(y = \frac { 3 } { x ^ { 2 } }\), the lines \(x = 1\) and \(x = 3\), and the \(x\)-axis, as shown in Figure 1. The centre of mass of the lamina has coordinates \(( \bar { x } , \bar { y } )\).
Use algebraic integration to find

1. the value of \(\bar { x }\),
2. the value of \(\bar { y }\).

\includegraphics{figure_8} The diagram shows the shaded region R bounded by the curve \(y = \sqrt{3x + 4}\), the \(x\)-axis, the \(y\)-axis, and the straight line that passes through the points \((k, 0)\) and \((4, 4)\), where \(0 < k < 4\). Region R is occupied by a uniform lamina.

1. Determine, in terms of \(k\), an expression for the \(y\)-coordinate of the centre of mass of the lamina. Give your answer in the form \(\frac{a + bk}{c + dk}\), where \(a\), \(b\), \(c\) and \(d\) are integers to be determined. [6]
2. Show that the \(y\)-coordinate of the centre of mass of the lamina cannot be \(\frac{3}{2}\). [2]

3 The region between the curve \(y = x \sqrt { } ( 3 - x )\) and the \(x\)-axis for \(0 \leqslant x \leqslant 3\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution. Find the \(x\)-coordinate of the centre of mass of this solid.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the region \(R\) bounded by the curve with equation \(y = \mathrm { e } ^ { - x }\), the line \(x = 1\), the \(x\)-axis and the \(y\)-axis. A uniform solid \(S\) is formed by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis.

1. Show that the volume of \(S\) is \(\frac { \pi } { 2 } \left( 1 - \mathrm { e } ^ { - 2 } \right)\).
2. Find, in terms of e, the distance of the centre of mass of \(S\) from \(O\).

6. \begin{figure}[h] \end{figure} The shaded region \(R\) is bounded by part of the curve with equation \(y = x ^ { 2 } + 3\), the \(x\)-axis, the \(y\)-axis and the line with equation \(x = 2\), as shown in Figure 4. The unit of length on each axis is one centimetre. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid \(S\).
Using algebraic integration,

1. show that the volume of \(S\) is \(\frac { 202 } { 5 } \pi \mathrm {~cm} ^ { 3 }\),
2. show that, to 2 decimal places, the centre of mass of \(S\) is 1.30 cm from \(O\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b7cfcf0a-8f54-4350-8e07-a3b51d94d0f2-11_478_472_1407_762} \captionsetup{labelformat=empty} \caption{Figure 5}
   \end{figure} A uniform right circular solid cone, of base radius 7 cm and height 6 cm , is joined to \(S\) to form a solid \(T\). The base of the cone coincides with the larger plane face of \(S\), as shown in Figure 5. The vertex of the cone is \(V\).
   The mass per unit volume of \(S\) is twice the mass per unit volume of the cone.
3. Find the distance from \(V\) to the centre of mass of \(T\). The point \(A\) lies on the circumference of the base of the cone. The solid \(T\) is suspended from \(A\) and hangs freely in equilibrium.
4. Find the size of the angle between \(V A\) and the vertical.

2 \includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-2_406_483_431_829} \(A O B\) is a uniform lamina in the shape of a quadrant of a circle with centre \(O\) and radius 0.6 m (see diagram).

1. Calculate the distance of the centre of mass of the lamina from \(A\). The lamina is freely suspended at \(A\) and hangs in equilibrium.
2. Find the angle between the vertical and the side \(A O\) of the lamina.

2 \includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-2_406_483_431_829} \(A O B\) is a uniform lamina in the shape of a quadrant of a circle with centre \(O\) and radius 0.6 m (see diagram).

1. Calculate the distance of the centre of mass of the lamina from \(A\). The lamina is freely suspended at \(A\) and hangs in equilibrium.
2. Find the angle between the vertical and the side \(A O\) of the lamina.

5 In this question you must show detailed reasoning. The region bounded by the \(x\)-axis, the \(y\)-axis, the line \(x = 4\) and the curve with equation \(\mathrm { y } = \frac { 15 } { \sqrt { \mathrm { x } ^ { 2 } + 9 } }\) is occupied by a uniform lamina. The centre of mass of the lamina is at the point \(G ( \bar { x } , \bar { y } )\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{857eca7f-c42d-49a9-ac39-a2fb5bcb9cd5-4_944_954_598_228}

1. Show that \(\bar { x } = \frac { 2 } { \ln 3 }\).
2. Determine the value of \(\bar { y }\). Give your answer correct to \(\mathbf { 3 }\) significant figures. \(P\) is the point on the curved edge of the lamina where \(x = 3\). The lamina is freely suspended from \(P\) and hangs in equilibrium in a vertical plane.
3. Determine the acute angle that the longest straight edge of the lamina makes with the vertical.

6 A uniform solid is formed by rotating the region enclosed by the positive \(x\)-axis, the line \(x = 2\) and the curve \(y = \frac { 1 } { 2 } x ^ { 2 }\) through \(360 ^ { \circ }\) around the \(x\)-axis. 6

1. Find the centre of mass of this solid.
   6
2. The solid is placed with its plane face on a rough inclined plane and does not slide. The angle between the inclined plane and the horizontal is gradually increased. When the angle between the inclined plane and the horizontal is \(\alpha\), the solid is on the point of toppling. Find \(\alpha\), giving your answer to the nearest \(0.1 ^ { \circ }\)

3. \begin{figure}[h] \end{figure} The shaded region in Figure 2 is bounded by the \(x\)-axis, the line with equation \(x = 2\) and the curve with equation \(y = \frac { 1 } { 4 } x ( 3 - x )\).
This region is rotated through \(2 \pi\) radians about the \(x\)-axis, to form a solid of revolution which is used to model a uniform solid \(S\). The volume of \(S\) is \(\frac { 2 } { 5 } \pi\)

1. Use the model and algebraic integration to show that the \(x\) coordinate of the centre of mass of \(S\) is \(\frac { 31 } { 24 }\) The solid \(S\) is placed with its circular face on a rough plane which is inclined at \(\alpha ^ { \circ }\) to the horizontal. The plane is sufficiently rough to prevent \(S\) from sliding. The solid \(S\) is on the point of toppling.
2. Find the value of \(\alpha\)

6 A uniform solid is formed by rotating the region enclosed by the positive \(x\)-axis, the line \(x = 2\) and the curve \(y = \frac { 1 } { 2 } x ^ { 2 }\) through \(360 ^ { \circ }\) around the \(x\)-axis. 6

1. Find the centre of mass of this solid.
   6
2. The solid is placed with its plane face on a rough inclined plane and does not slide. The angle between the inclined plane and the horizontal is gradually increased. When the angle between the inclined plane and the horizontal is \(\alpha\), the solid is on the point of toppling. Find \(\alpha\), giving your answer to the nearest \(0.1 ^ { \circ }\)

2 \includegraphics[max width=\textwidth, alt={}, center]{f3a35846-075d-4e03-ba6b-82774ef0e4f8-04_442_554_260_794} A uniform lamina \(A B C E F G\) is formed from a square \(A B D G\) by removing a smaller square \(C D F E\) from one corner. \(A B = 0.7 \mathrm {~m}\) and \(D F = 0.3 \mathrm {~m}\) (see diagram). Find the distance of the centre of mass of the lamina from \(A\).

2 \includegraphics[max width=\textwidth, alt={}, center]{f3a35846-075d-4e03-ba6b-82774ef0e4f8-04_442_554_260_794} A uniform lamina \(A B C E F G\) is formed from a square \(A B D G\) by removing a smaller square \(C D F E\) from one corner. \(A B = 0.7 \mathrm {~m}\) and \(D F = 0.3 \mathrm {~m}\) (see diagram). Find the distance of the centre of mass of the lamina from \(A\).

7 Fig. 7.1 shows a uniform lamina in the shape of a sector of a circle of radius \(r\) and angle \(2 \theta\) where \(\theta\) is in radians. The sector consists of a triangle \(O A B\) and a segment bounded by the chord \(A B\). \begin{figure}[h] \end{figure}

1. Explain why the centre of mass of the segment lies on the radius through the midpoint of \(A B\).
2. Show that the distance of the centre of mass of the segment from \(O\) is \(\frac { 2 r \sin ^ { 3 } \theta } { 3 ( \theta - \sin \theta \cos \theta ) }\). A uniform circular lamina of radius 5 units is placed with its centre at the origin, \(O\), of an \(x - y\) coordinate system. A component for a machine is made by removing and discarding a segment from the lamina. The radius of the circle from which the segment is formed is 3 units and the centre of this circle is \(O\). The centre of the straight edge of the segment has coordinates ( 0,2 ) and this edge is perpendicular to the \(y\)-axis (see Fig. 7.2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{831ba5da-df19-43bb-b163-02bbddb4e2b8-6_766_757_1400_244} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
   \end{figure}
3. Find the \(y\)-coordinate of the centre of mass of the component, giving your answer correct to 3 significant figures. \(C\) is the point on the component with coordinates ( 0,5 ). The component is now placed horizontally and supported only at \(O\). A particle of mass \(m \mathrm {~kg}\) is placed on the component at \(C\) and the component and particle are in equilibrium.
4. Find the mass of the component in terms of \(m\).

2 A uniform semicircular lamina of radius 0.25 m has diameter \(A B\). It is freely suspended at \(A\) from a fixed point and hangs in equilibrium.

1. Find the distance of the centre of mass of the lamina from the diameter \(A B\).
2. Calculate the angle which the diameter \(A B\) makes with the vertical. The lamina is now held in equilibrium with the diameter \(A B\) vertical by means of a force applied at \(B\). This force has magnitude 6 N and acts at \(45 ^ { \circ }\) to the upward vertical in the plane of the lamina.
3. Calculate the weight of the lamina.

5. (a) Show, by integration, that the centre of mass of a uniform solid hemisphere of radius \(r\) is at a distance of \(\frac { 3 r } { 8 }\) from the plane face.
(b) The diagram shows a composite solid body which consists of a uniform right circular cylinder capped by a uniform hemisphere. The total height of the solid is \(3 r \mathrm {~cm}\), where \(r\) represents the common radius of the hemisphere and the cylinder. \includegraphics[max width=\textwidth, alt={}, center]{3578a810-46da-4d9e-a98f-248be72a517a-6_397_340_762_858} Given that the density of the hemisphere is \(50 \%\) more than that of the cylinder, find the distance of the centre of mass of the solid from its base along the axis of symmetry.

1 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.

[In this question, you may use the fact that the volume of a right circular cone of base radius \(r\) and height \(h\) is \(\frac{1}{3}\pi r^2 h\).]

1. By using integration, show that the centre of mass of a uniform solid right circular cone of height \(h\) and base radius \(r\) is at a distance \(\frac{3}{4}h\) from the vertex. [5]

\includegraphics{figure_8} Fig. 8 shows the side view of a toy formed by joining a uniform solid circular cylinder of radius \(r\) and height \(2r\) to a uniform solid right circular cone, made of the same material as the cylinder, of radius \(r\) and height \(r\). The toy is placed on a horizontal floor with the curved surface of the cone in contact with the floor.

1. Determine whether the toy will topple. [7]
2. Explain why it is not necessary to know whether the floor is rough or smooth in answering part (b). [1]

2. A metal sign is formed by removing triangle \(B C D\) from a rectangular lamina \(A C E F\) made of uniform material, and adding a quarter circle XYZ, made of the same uniform material, with a particle attached to its vertex at \(Y\). The sign is supported by two light chains fixed at \(E\) and \(F\). The quarter circle has radius 24 cm and the particle at \(Y\) has a mass equal to half of that of the removed triangle. \(X D\) is parallel to \(A C\) and \(B Z\) is parallel to \(A F\). The dimensions, in cm , are as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{3578a810-46da-4d9e-a98f-248be72a517a-3_885_636_712_715}

1. Calculate the distance of the centre of mass of the sign from
   1. \(A F\),
   2. \(A C\).
2. The support at \(F\) comes loose so that the sign is freely suspended at \(E\) by one chain alone. Given that it then hangs in equilibrium, calculate the angle that \(E F\) makes with the vertical.

2 A rod \(A B\) of variable density has length 2 m . At a distance \(x\) metres from \(A\), the rod has mass per unit length ( \(0.7 - 0.3 x ) \mathrm { kg } \mathrm { m } ^ { - 1 }\). Find the distance of the centre of mass of the rod from \(A\).

3 The equation of a curve is \(y = \lambda x ^ { 2 }\), where \(\lambda > 0\). The region bounded by the curve, the \(x\)-axis and the line \(x = a\), where \(a > 0\), is denoted by \(R\). The \(y\)-coordinate of the centroid of \(R\) is \(a\). Show that \(\lambda = \frac { 10 } { 3 a }\).

1 The finite region enclosed by the line \(y = k x\), where \(k\) is a positive constant, the \(x\)-axis for \(0 \leqslant x \leqslant h\), and the line \(x = h\) is rotated through 1 complete revolution about the \(x\)-axis. Prove by integration that the centroid of the resulting cone is at a distance \(\frac { 3 } { 4 } h\) from the origin \(O\).
[0pt] [The volume of a cone of height \(h\) and base radius \(r\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]

5. \begin{figure}[h] \end{figure} A uniform solid right circular cylinder has height \(h\) and radius \(r\). The centre of one plane face is \(O\) and the centre of the other plane face is \(Y\). A cylindrical hole is made by removing a solid cylinder of radius \(\frac { 1 } { 4 } r\) and height \(\frac { 1 } { 4 } h\) from the end with centre \(O\). The axis of the cylinder removed is parallel to \(O Y\) and meets the end with centre \(O\) at \(X\), where \(O X = \frac { 1 } { 4 } r\). One plane face of the cylinder removed coincides with the plane face through \(O\) of the original cylinder. The resulting solid \(S\) is shown in Figure 3.

1. Show that the centre of mass of \(S\) is at a distance \(\frac { 85 h } { 168 }\) from the plane face
   containing \(O\). containing \(O\). The solid \(S\) is freely suspended from \(O\). In equilibrium the line \(O Y\) is inclined at an angle \(\arctan ( 17 )\) to the horizontal.
2. Find \(r\) in terms of \(h\).
   [0pt] [Question 5 Continued]
   [0pt] [Question 5 Continued]

3 An open box in the shape of a cube with edges of length 0.2 m is placed with its base horizontal and its four sides vertical. The four sides and base are uniform laminas, each with weight 3 N .

1. Calculate the height of the centre of mass of the box above its base.
   The box is now fitted with a thin uniform square lid of weight 3 N and with edges of length 0.2 m . The lid is attached to the box by a hinge of length 0.2 m and weight 2 N . The lid of the box is held partly open.
2. Find the angle which the lid makes with the horizontal when the centre of mass of the box (including the lid and hinge) is 0.12 m above the base of the box.

1 \includegraphics[max width=\textwidth, alt={}, center]{57f7ca89-f028-447a-9ac9-55f931201e6b-2_467_645_274_749} A uniform semicircular lamina has radius 5 m . The lamina rotates in a horizontal plane about a vertical axis through \(O\), the mid-point of its diameter. The angular speed of the lamina is \(4 \mathrm { rad } \mathrm { s } ^ { - 1 }\) (see diagram). Find

1. the distance of the centre of mass of the lamina from \(O\),
2. the speed with which the centre of mass of the lamina is moving.

7. A uniform square lamina \(P Q R S\), of mass \(m\) and side \(2 a\), is free to rotate about a fixed smooth horizontal axis which passes through \(P\) and \(Q\). The lamina hangs at rest in a vertical plane with \(S R\) below \(P Q\) and is given a horizontal impulse of magnitude \(J\) at the midpoint of \(S R\). The impulse is perpendicular to \(S R\).

1. Find the initial angular speed of the lamina.
2. Find the magnitude of the angular deceleration of the lamina at the instant when the lamina has turned through \(\frac { \pi } { 6 }\) radians.
3. Find the magnitude of the component of the force exerted on the lamina by the axis, in a direction perpendicular to the lamina, at the instant when the lamina has turned through \(\frac { \pi } { 6 }\) radians. \includegraphics[max width=\textwidth, alt={}, center]{f932d7cb-1299-41d1-8248-cfbf639795ed-12_2255_50_315_1978}

4. A uniform plane lamina of mass \(m\) is in the shape of an equilateral triangle of side \(2 a\). Find, using integration, the moment of inertia of the lamina about one of its edges.