6.04b Find centre of mass: using symmetry

4 A uniform rectangular lamina \(A B C D\) has a mass of 8 kg . The side \(A B\) has length 20 cm , the side \(B C\) has length 10 cm , and \(P\) is the mid-point of \(A B\). A uniform circular lamina, of mass 2 kg and radius 5 cm , is fixed to the rectangular lamina to form a sign. The centre of the circular lamina is 5 cm from each of \(A B\) and \(B C\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{851cb2a3-5bc8-4af9-b1fc-a143d37beebe-3_661_1200_589_406}

1. Find the distance of the centre of mass of the sign from \(A D\).
2. Write down the distance of the centre of mass of the sign from \(A B\).
3. The sign is freely suspended from \(P\). Find the angle between \(A D\) and the vertical when the sign is in equilibrium.
4. Explain how you have used the fact that each lamina is uniform in your solution to this question.

2 A piece of modern art is modelled as a uniform lamina and three particles. The diagram shows the lamina, the three particles \(A , B\) and \(C\), and the \(x\) - and \(y\)-axes. \includegraphics[max width=\textwidth, alt={}, center]{06b431ca-d3a8-46d6-b9f8-bac08d3fd51e-2_875_1004_1414_502} The lamina, which is fixed in the \(x - y\) plane, has mass 10 kg and its centre of mass is at the point (12, 9). The three particles are attached to the lamina.
Particle \(A\) has mass 3 kg and is at the point (15, 6).
Particle \(B\) has mass 1 kg and is at the point ( 7,14 ).
Particle \(C\) has mass 6 kg and is at the point ( 8,7 ).
Find the coordinates of the centre of mass of the piece of modern art.

3 Three particles are attached to a light rectangular lamina \(O A B C\), which is fixed in a horizontal plane. Take \(O A\) and \(O C\) as the \(x\) - and \(y\)-axes, as shown. Particle \(P\) has mass 1 kg and is attached at the point \(( 25,10 )\).
Particle \(Q\) has mass 4 kg and is attached at the point ( 12,7 ).
Particle \(R\) has mass 5 kg and is attached at the point \(( 4,18 )\). \includegraphics[max width=\textwidth, alt={}, center]{03994596-21ad-4201-8d64-ba2d7b7e0a77-3_782_1033_703_482} Find the coordinates of the centre of mass of the three particles.

3 Three particles are attached to a light rod, \(A B\), of length 40 cm The particles are attached at \(A , B\) and the midpoint of the rod.
The particle at \(A\) has mass 5 kg
The particle at \(B\) has mass 1 kg
The particle at the midpoint has mass 4 kg
Find the distance of the centre of mass of this system from the midpoint of the rod.
Circle your answer.
[0pt] [1 mark] \(4 \mathrm {~cm} \quad 8 \mathrm {~cm} \quad 12 \mathrm {~cm} \quad 28 \mathrm {~cm}\) Turn over for the next question

4 Fig. 4.1 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. 4.2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8859baf3-f8e8-4fbf-b54f-34f550b02c26-03_748_743_1594_251} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \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 \(m\). Total Marks for Question Set 3: 40

4 Particles \(A , B\) and \(C\) of masses \(2 \mathrm {~kg} , 3 \mathrm {~kg}\) and 5 kg respectively are joined by light rigid rods to form a triangular frame. The frame is placed at rest on a horizontal plane with \(A\) at the point \(( 0,0 )\), \(B\) at the point ( \(0.6,0\) ) and \(C\) at the point ( \(0.4,0.2\) ), where distances in the coordinate system are measured in metres (see Fig. 1). \begin{figure}[h] \end{figure} \(G\), which is the centre of mass of the frame, is at the point \(( \bar { x } , \bar { y } )\).

1. - Show that \(\bar { x } = 0.38\).
   A rough plane, \(\Pi\), is inclined at an angle \(\theta\) to the horizontal where \(\sin \theta = \frac { 3 } { 5 }\). The frame is placed on \(\Pi\) with \(A B\) vertical and \(B\) in contact with \(\Pi . C\) is in the same vertical plane as \(A B\) and a line of greatest slope of \(\Pi . C\) is on the down-slope side of \(A B\). The frame is kept in equilibrium by a horizontal light elastic string whose natural length is \(l \mathrm {~m}\) and whose modulus of elasticity is \(g \mathrm {~N}\). The string is attached to \(A\) at one end and to a fixed point on \(\Pi\) at the other end (see Fig. 2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{709f3a7a-d857-4813-98ab-de6b41a3a8dc-03_605_828_1525_248} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The coefficient of friction between \(B\) and \(\Pi\) is \(\mu\).
2. Show that \(l = 0.3\).
3. Show that \(\mu \geqslant \frac { 14 } { 27 }\).

Answer only one of the following two alternatives. **EITHER** \includegraphics{figure_11a} A uniform plane object consists of three identical circular rings, \(X\), \(Y\) and \(Z\), enclosed in a larger circular ring \(W\). Each of the inner rings has mass \(m\) and radius \(r\). The outer ring has mass \(3m\) and radius \(R\). The centres of the inner rings lie at the vertices of an equilateral triangle of side \(2r\). The outer ring touches each of the inner rings and the rings are rigidly joined together. The fixed axis \(AB\) is the diameter of \(W\) that passes through the centre of \(X\) and the point of contact of \(Y\) and \(Z\) (see diagram). It is given that \(R = \left(1 + \frac{2}{3}\sqrt{3}\right)r\).

1. Show that the moment of inertia of the object about \(AB\) is \(\left(7 + 2\sqrt{3}\right)mr^2\). [8]

The line \(CD\) is the diameter of \(W\) that is perpendicular to \(AB\). A particle of mass \(9m\) is attached to \(D\). The object is now held with its plane horizontal. It is released from rest and rotates freely about the fixed horizontal axis \(AB\).

1. Find, in terms of \(g\) and \(r\), the angular speed of the object when it has rotated through \(60°\). [6]

**OR** Fish of a certain species live in two separate lakes, \(A\) and \(B\). A zoologist claims that the mean length of fish in \(A\) is greater than the mean length of fish in \(B\). To test his claim, he catches a random sample of 8 fish from \(A\) and a random sample of 6 fish from \(B\). The lengths of the 8 fish from \(A\), in appropriate units, are as follows. $$15.3 \quad 12.0 \quad 15.1 \quad 11.2 \quad 14.4 \quad 13.8 \quad 12.4 \quad 11.8$$ Assuming a normal distribution, find a 95% confidence interval for the mean length of fish in \(A\). [5] The lengths of the 6 fish from \(B\), in the same units, are as follows. $$15.0 \quad 10.7 \quad 13.6 \quad 12.4 \quad 11.6 \quad 12.6$$ Stating any assumptions that you make, test at the 5% significance level whether the mean length of fish in \(A\) is greater than the mean length of fish in \(B\). [7] Calculate a 95% confidence interval for the difference in the mean lengths of fish from \(A\) and from \(B\). [2]

Answer only one of the following two alternatives. EITHER \includegraphics{figure_10a} An object is formed by attaching a thin uniform rod \(PQ\) to a uniform rectangular lamina \(ABCD\). The lamina has mass \(m\), and \(AB = DC = 6a\), \(BC = AD = 3a\). The rod has mass \(M\) and length \(3a\). The end \(P\) of the rod is attached to the mid-point of \(AB\). The rod is perpendicular to \(AB\) and in the plane of the lamina (see diagram). Show that the moment of inertia of the object about a smooth horizontal axis \(l_1\), through \(Q\) and perpendicular to the plane of the lamina, is \(3(8m + M)a^2\). [4] Show that the moment of inertia of the object about a smooth horizontal axis \(l_2\), through the mid-point of \(PQ\) and perpendicular to the plane of the lamina, is \(\frac{3}{4}(17m + M)a^2\). [2] Find expressions for the periods of small oscillations of the object about the axes \(l_1\) and \(l_2\), and verify that these periods are equal when \(m = M\). [8] OR A farmer \(A\) grows two types of potato plants, Royal and Majestic. A random sample of 10 Royal plants is taken and the potatoes from each plant are weighed. The total mass of potatoes on a plant is \(x\) kg. The data are summarised as follows. $$\Sigma x = 42.0 \qquad \Sigma x^2 = 180.0$$ A random sample of 12 Majestic plants is taken. The total mass of potatoes on a plant is \(y\) kg. The data are summarised as follows. $$\Sigma y = 57.6 \qquad \Sigma y^2 = 281.5$$ Test, at the 5% significance level, whether the population mean mass of potatoes from Royal plants is the same as the population mean mass of potatoes from Majestic plants. You may assume that both distributions are normal and you should state any additional assumption that you make. [9] A neighbouring farmer \(B\) grows Crown potato plants. His plants produce 3.8 kg of potatoes per plant, on average. Farmer \(A\) claims that her Royal plants produce a higher mean mass of potatoes than Farmer \(B\)'s Crown plants. Test, at the 5% significance level, whether Farmer \(A\)'s claim is justified. [5]

\includegraphics{figure_3} A uniform disc, of radius \(a\) and mass \(2M\), is attached to a thin uniform rod \(AB\) of length \(6a\) and mass \(M\). The rod lies along a diameter of the disc, so that the centre of the disc is a distance \(x\) from \(A\) (see diagram).

1. Find the moment of inertia of the object, consisting of disc and rod, about a fixed horizontal axis \(l\) through \(A\) and perpendicular to the plane of the disc. [4]

The object is free to rotate about the axis \(l\). The object is held with \(AB\) horizontal and is released from rest. When \(AB\) makes an angle \(\theta\) with the vertical, where \(\cos \theta = \frac{3}{5}\), the angular speed of the object is \(\sqrt{\left(\frac{2g}{5a}\right)}\).

1. Find the possible values of \(x\). [5]

\includegraphics{figure_1} A frame consists of a uniform semicircular wire of radius 20 cm and mass 2 kg, and a uniform straight wire of length 40 cm and mass 0.9 kg. The ends of the semicircular wire are attached to the ends of the straight wire (see diagram). Find the distance of the centre of mass of the frame from the straight wire. [4]

\includegraphics{figure_7} A uniform solid cone has height \(1.2 \text{ m}\) and base radius \(0.5 \text{ m}\). A uniform object is made by drilling a cylindrical hole of radius \(0.2 \text{ m}\) through the cone along the axis of symmetry (see diagram).

1. Show that the height of the object is \(0.72 \text{ m}\) and that the volume of the cone removed by the drilling is \(0.0352\pi \text{ m}^3\). [4]

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

1. Find the distance of the centre of mass of the object from its base. [6]

\includegraphics{figure_3} \(ABC\) is an object made from a uniform wire consisting of two straight portions \(AB\) and \(BC\), in which \(AB = a\), \(BC = x\) and angle \(ABC = 90°\). When the object is freely suspended from \(A\) and in equilibrium, the angle between \(AB\) and the horizontal is \(\theta\) (see diagram).

1. Show that \(x^2 \tan \theta - 2ax - a^2 = 0\). [3]
2. Given that \(\tan \theta = 1.25\), calculate the length of the wire in terms of \(a\). [2]

\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 \(\frac{1}{3}\pi r^2 h\).]

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]

\includegraphics{figure_6} A uniform lamina \(OABCD\) consists of a semicircle \(BCD\) with centre \(O\) and radius \(0.6\) m and an isosceles triangle \(OAB\), joined along \(OB\) (see diagram). The triangle has area \(0.36\) m\(^2\) and \(AB = AO\).

1. Show that the centre of mass of the lamina lies on \(OB\). [4]
2. Calculate the distance of the centre of mass of the lamina from \(O\). [4]

\includegraphics{figure_1} A seesaw in a playground consists of a beam \(AB\) of length \(4\) m which is supported by a smooth pivot at its centre \(C\). Jill has mass \(25\) kg and sits on the end \(A\). David has mass \(40\) kg and sits at a distance \(x\) metres from \(C\), as shown in Figure 1. The beam is initially modelled as a uniform rod. Using this model,

1. find the value of \(x\) for which the seesaw can rest in equilibrium in a horizontal position. [3]
2. State what is implied by the modelling assumption that the beam is uniform. [1]

David realises that the beam is not uniform as he finds that he must sit at a distance \(1.4\) m from \(C\) for the seesaw to rest horizontally in equilibrium. The beam is now modelled as a non-uniform rod of mass \(15\) kg. Using this model,

1. find the distance of the centre of mass of the beam from \(C\). [4]

\includegraphics{figure_2} A uniform plank \(AB\) has weight 120 N and length 3 m. The plank rests horizontally in equilibrium on two smooth supports \(C\) and \(D\), where \(AC = 1\) m and \(CD = x\) m, as shown in Figure 2. The reaction of the support on the plank at \(D\) has magnitude 80 N. Modelling the plank as a rod,

1. show that \(x = 0.75\) [3]

A rock is now placed at \(B\) and the plank is on the point of tilting about \(D\). Modelling the rock as a particle, find

1. the weight of the rock, [4]
2. the magnitude of the reaction of the support on the plank at \(D\). [2]
3. State how you have used the model of the rock as a particle. [1]

Three particles of mass \(3m\), \(5m\) and \(2m\) are placed at points with coordinates \((4, 0)\), \((0, -3)\) and \((4, 2)\) respectively. The centre of mass of the system of three particles is at \((2, k)\).

1. Show that \(λ = 2\). [4]

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

Three particles of mass \(3m\), \(5m\) and \(\lambda m\) are placed at points with coordinates \((4, 0)\), \((0, -3)\) and \((4, 2)\) respectively. The centre of mass of the system of three particles is at \((2, k)\).

1. Show that \(\lambda = 2\). [4]
2. Calculate the value of \(k\). [3]

\includegraphics{figure_2} Figure 2 shows a uniform lamina \(ABCDE\) such that \(ABDE\) is a rectangle, \(BC = CD\), \(AB = 8a\) and \(AE = 6a\). The point \(X\) is the mid-point of \(BD\) and \(XC = 4a\). The centre of mass of the lamina is at \(G\).

1. Show that \(GX = \frac{14}{15}a\). [6]

The mass of the lamina is \(M\). A particle of mass \(\lambda M\) is attached to the lamina at \(C\). The lamina is suspended from \(B\) and hangs freely under gravity with \(AB\) horizontal.

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

\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]