6.04b Find centre of mass: using symmetry

4. The diagram shows a uniform lamina consisting of a rectangular section \(G P Q E\) with a semi-circular section EFG of radius 4 cm . Quadrants \(A P B\) and \(C Q D\) each with radius 2 cm are removed. Dimensions in cm are as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{3efc4ef6-8a80-4267-8e95-733200e875c5-3_758_604_497_651}

1. Write down the distance of the centre of mass of the lamina \(A B C D E F G\) from \(A G\).
2. Determine the distance of the centre of mass of the lamina \(A B C D E F G\) from \(B C\).
3. The lamina \(A B C D E F G\) is suspended freely from the point \(E\) and hangs in equilibrium. Calculate the angle EG makes with the vertical.

3 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
A composite body consists of a uniform rod, \(A B\), and a particle.
The rod has length 4 metres and mass 22.5 kilograms.
The particle, \(P\), has mass 20 kilograms and is placed on the rod at a distance of 0.3 metres from \(B\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{4fdb2637-6368-422c-99da-85b80efe31c5-04_163_1323_767_402} 3

1. Find the distance of the centre of mass of the body from \(A\). 3
2. The body rests in equilibrium in a horizontal position on two supports, \(C\) and \(D\).
   The support \(C\) is 0.5 metres from \(A\) and the support \(D\) is 1 metre from \(B\). Find the magnitudes of the forces exerted on the body by the supports.
   [0pt] [4 marks]

1. Figure 1 A thin uniform rod, of total length \(30 a\) and mass \(M\), is bent to form a frame. The frame is in the shape of a triangle \(A B C\), where \(A B = 12 a , B C = 5 a\) and \(C A = 13 a\), as shown in Figure 1.

1. Show that the centre of mass of the frame is \(\frac { 3 } { 2 } a\) from \(A B\). The frame is freely suspended from \(A\). A horizontal force of magnitude \(k M g\), where \(k\) is a constant, is applied to the frame at \(B\). The line of action of the force lies in the vertical plane containing the frame. The frame hangs in equilibrium with \(A B\) vertical.
2. Find the value of \(k\).

3. \begin{figure}[h] \end{figure} The lamina \(L\), shown in Figure 2, consists of a uniform square lamina \(A B D F\) and two uniform triangular laminas \(B D C\) and \(F D E\). The square has sides of length \(2 a\). The two triangles are identical. The straight lines \(B D E\) and \(F D C\) are perpendicular with \(B D = D F = 2 a\) and \(D C = D E = a\).
The mass per unit of area of the square is \(M\).
The mass per unit area of each triangle is \(3 M\).
The centre of mass of \(L\) is at the point \(G\).

1. Without doing any calculations, explain why \(G\) lies on \(A D\).
2. Show that the distance of \(G\) from \(D\) is \(\frac { \sqrt { 2 } } { 2 } a\) The lamina \(L\) is freely suspended from \(B\) and hangs in equilibrium.
3. Find the size of the angle between \(B E\) and the downward vertical.

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1. \begin{figure}[h] \end{figure} Five identical uniform rods are joined together to form the rigid framework \(A B C D\) shown in Figure 1. Each rod has weight \(W\) and length 4a. The points \(A , B , C\) and \(D\) all lie in the same plane. The centre of mass of the framework is at the point \(G\).

1. Explain why \(G\) is the midpoint of \(A C\). The framework is suspended from the ceiling by two vertical light inextensible strings. One string is attached to the framework at \(A\) and the other string is attached to the framework at \(B\). The framework hangs freely in equilibrium with \(A B\) horizontal.
2. Find
   1. the tension in the string attached at \(A\),
   2. the tension in the string attached at \(B\). A particle of weight \(k W\) is now attached to the framework at \(D\) and a particle of weight \(2 k W\) is now attached to the framework at \(C\). The framework remains in equilibrium with \(A B\) horizontal and the strings vertical. Either string will break if the tension in it exceeds \(6 W\).
3. Find the greatest possible value of \(k\).

4. \begin{figure}[h] \end{figure} The uniform triangular lamina \(A B C D E\) is such that angle \(C E A = 90 ^ { \circ } , C E = 9 a\) and \(E A = 6 a\). The point \(D\) lies on \(C E\), with \(D E = 3 a\). The point \(B\) on \(C A\) is such that \(D B\) is parallel to \(E A\) and \(D B = 4 a\). The triangular lamina is folded along the line \(D B\) to form the folded lamina \(A B D E C F\), as shown in Figure 2. The distance of the centre of mass of the triangular lamina from \(D C\) is \(d _ { 1 }\) The distance of the centre of mass of the folded lamina from \(D C\) is \(d _ { 2 }\)

1. Explain why \(d _ { 1 } = d _ { 2 }\) The folded lamina is freely suspended from \(B\) and hangs in equilibrium with \(B A\) inclined at an angle \(\alpha\) to the downward vertical through \(B\).
2. Find, to the nearest degree, the size of angle \(\alpha\).

1. \begin{figure}[h] \end{figure} Figure 1 shows a uniform rectangular lamina \(A B C D\) with \(A B = 2 a\) and \(A D = a\) The mass of the lamina is \(6 m\). A particle of mass \(2 m\) is attached to the lamina at \(A\), a particle of mass \(m\) is attached to the lamina at \(B\) and a particle of mass \(3 m\) is attached to the lamina at \(D\), to form a loaded lamina \(L\) of total mass \(12 m\).

1. Write down the distance of the centre of mass of \(L\) from \(A B\). You must give a reason for your answer.
2. Show that the distance of the centre of mass of \(L\) from \(A D\) is \(\frac { 2 a } { 3 }\) A particle of mass \(k m\) is now also attached to \(L\) at \(D\) to form a new loaded lamina \(N\).
3. Show that the distance of the centre of mass of \(N\) from \(A B\) is \(\frac { ( k + 6 ) a } { ( k + 12 ) }\) When \(N\) is freely suspended from \(A\) and is hanging in equilibrium, the side \(A B\) makes an angle \(\alpha\) with the vertical, where \(\tan \alpha = \frac { 3 } { 2 }\)
4. Find the value of \(k\).

1. \begin{figure}[h] \end{figure} A uniform rod of length \(72 a\) is cut into pieces. The pieces are used to make two rigid squares, \(A B C D\) and \(P Q R S\), with sides of length \(10 a\) and \(8 a\) respectively. The two squares are joined to form the rigid framework shown in Figure 1. The squares both lie in the same plane with the rod \(A B\) parallel to the rod \(P Q\).
Given that

• \(A D\) cuts \(P Q\) in the ratio \(3 : 5\)
• \(D C\) cuts \(Q R\) in the ratio 5:3
  1. explain why the centre of mass of square \(A B C D\) is at \(Q\).
  2. Find the distance of the centre of mass of the framework from \(B\).

3. \begin{figure}[h] \end{figure} The uniform lamina \(A B C D E F G H I J\) is shown in Figure 3.
The lamina has \(A J = 8 a , A B = 5 a\) and \(B C = D E = E F = F G = G H = H I = I J = 2 a\).
All the corners are right angles.

1. Show that the distance of the centre of mass of the lamina from \(A J\) is \(\frac { 49 } { 26 } a\) A light inextensible rope is attached to the lamina at \(A\) and another light inextensible rope is attached to the lamina at \(B\). The lamina hangs in equilibrium with both ropes vertical and \(A B\) horizontal. The weight of the lamina is \(W\).
2. Find, in terms of \(W\), the tension in the rope attached to the lamina at \(B\). The rope attached to \(B\) breaks and subsequently the lamina hangs freely in equilibrium, suspended from \(A\).
3. Find the size of the angle between \(A J\) and the downward vertical.

1. \begin{figure}[h] \end{figure} A uniform plane lamina is in the shape of an isosceles trapezium \(A B C D E F\), as shown shaded in Figure 1.

• \(B C E F\) is a square
• \(A B = C D = a\)
• \(B C = 3 a\)
  1. Show that the distance of the centre of mass of the lamina from \(A D\) is \(\frac { 11 a } { 8 }\)

The mass of the lamina is \(M\) The lamina is suspended by two light vertical strings, one attached to the lamina at \(A\) and the other attached to the lamina at \(F\) The lamina hangs freely in equilibrium, with \(B F\) horizontal.

Find, in terms of \(M\) and \(g\), the tension in the string attached at \(A\)

2. \begin{figure}[h] \end{figure} Uniform wire is used to form the framework shown in Figure 2.
In the framework

• \(A B C D\) is a rectangle with \(A D = 2 a\) and \(D C = a\)
• \(B E C\) is a semicircular arc of radius \(a\) and centre \(O\), where \(O\) lies on \(B C\)

The diameter of the semicircle is \(B C\) and the point \(E\) is such that \(O E\) is perpendicular to \(B C\). The points \(A , B , C , D\) and \(E\) all lie in the same plane.

1. Show that the distance of the centre of mass of the framework from \(B C\) is $$\frac { a } { 6 + \pi }$$ The framework is freely suspended from \(A\) and hangs in equilibrium with \(A E\) at an angle \(\theta ^ { \circ }\) to the downward vertical.
2. Find the value of \(\theta\). The mass of the framework is \(M\).
   A particle of mass \(k M\) is attached to the framework at \(B\).
   The centre of mass of the loaded framework lies on \(O A\).
3. Find the value of \(k\).

1. Three particles of masses \(4 m , 2 m\) and \(k m\) are placed at the points with coordinates \(( - 3 , - 1 ) , ( 6,1 )\) and \(( - 1,5 )\) respectively.

Given that the centre of mass of the three particles is at the point with coordinates \(( \bar { x } , \bar { y } )\)

1. show that \(\bar { x } = \frac { - k } { k + 6 }\)
2. find \(\bar { y }\) in terms of \(k\). Given that the centre of mass of the three particles lies on the line with equation \(y = 2 x + 3\)
3. find the value of \(k\). A fourth particle is placed at the point with coordinates \(( \lambda , 4 )\).
   Given that the centre of mass of the four particles also lies on the line with equation \(y = 2 x + 3\)
4. find the value of \(\lambda\).

4. \begin{figure}[h] \end{figure} A uniform triangular lamina \(A B C\) is isosceles, with \(A C = B C\). The midpoint of \(A B\) is \(M\). The length of \(A B\) is \(18 a\) and the length of \(C M\) is \(18 a\). The triangular lamina \(C D E\), with \(D E = 6 a\) and \(C D = 12 a\), has \(E D\) parallel to \(A B\) and \(M D C\) is a straight line. Triangle \(C D E\) is removed from triangle \(A B C\) to form the lamina \(L\), shown shaded in Figure 1. The distance of the centre of mass of \(L\) from \(M C\) is \(d\).

1. Show that \(d = \frac { 4 } { 7 } a\) The lamina \(L\) is suspended by two light inextensible strings. One string is attached to \(L\) at \(A\) and the other string is attached to \(L\) at \(B\).
   The lamina hangs in equilibrium in a vertical plane with the strings vertical and \(A B\) horizontal.
   The weight of \(L\) is \(W\)
2. Find, in terms of \(W\), the tension in the string attached to \(L\) at \(B\) The string attached to \(L\) at \(B\) breaks, so that \(L\) is now suspended from \(A\). When \(L\) is hanging in equilibrium in a vertical plane, the angle between \(A B\) and the downward vertical through \(A\) is \(\theta ^ { \circ }\)
3. Find the value of \(\theta\)

1. \begin{figure}[h] \end{figure} A uniform rod of length \(24 a\) is cut into seven pieces which are used to form the framework \(A B C D E F\) shown in Figure 1. It is given that

• \(A F = B E = C D = A B = F E = 4 a\)
• \(B C = E D = 2 a\)
• the rods \(A F , B E\) and \(C D\) are parallel
• the rods \(A B , B C , F E\) and \(E D\) are parallel
• \(A F\) is perpendicular to \(A B\)
• the rods all lie in the same plane

The distance of the centre of mass of the framework from \(A F\) is \(d\).

1. Show that \(d = \frac { 19 } { 6 } a\)
2. Find the distance of the centre of mass of the framework from \(A\).

4. \begin{figure}[h] \end{figure} The uniform triangular lamina \(A B C\) has \(A B\) perpendicular to \(A C\), \(A B = 9 a\) and \(A C = 6 a\). The point \(D\) on \(A B\) is such that \(A D = a\). The rectangle \(D E F G\), with \(D E = 2 a\) and \(E F = 3 a\), is removed from the lamina to form the template shown shaded in Figure 3. The distance of the centre of mass of the template from \(A C\) is \(d\).

1. Show that \(d = \frac { 23 } { 7 } a\) The template is freely suspended from \(A\) and hangs in equilibrium with \(A B\) at an angle \(\theta ^ { \circ }\) to the downward vertical through \(A\).
2. Find the value of \(\theta\) A new piece, of exactly the same size and shape as the template, is cut from a lamina of a different uniform material. The template and the new piece are joined together to form the model shown in Figure 4. Both parts of the model lie in the same plane. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fd8bc7b5-adee-4d67-b15d-571255b00b83-12_369_1185_1667_440} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} The weight of \(C P Q R S T A\) is \(W\) The weight of \(A D G F E B C\) is \(4 W\) The model is freely suspended from \(A\).
   A horizontal force of magnitude \(X\), acting in the same vertical plane as the model, is now applied to the model at \(T\) so that \(A C\) is vertical, as shown in Figure 4.
3. Find \(X\) in terms of \(W\).

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.

1. Three particles of masses \(3 m\), \(4 m\) and \(2 m\) are placed at the points \(( - 2,2 ) , ( 3,1 )\) and ( \(p , p\) ) respectively.

The value of \(p\) is such that the distance of the centre of mass of the three particles from the point ( 0,0 ) is as small as possible. Find the value of \(p\).

1. \begin{figure}[h] \end{figure} A letter P from a shop sign is modelled as a uniform plane lamina which consists of a rectangular lamina, \(O A B D E\), joined to a semicircular lamina, \(B C D\), along its diameter \(B D\). $$O A = E D = a , A B = 2 a , O E = 4 a \text {, and the diameter } B D = 2 a \text {, as shown in Figure } 1 .$$ Using the model,

1. find, in terms of \(\pi\) and \(a\), the distance of the centre of mass of the letter P ,
   from (i) \(O E\) (ii) \(O A\) The letter P is freely suspended from \(O\) and hangs in equilibrium. The angle between \(O E\) and the downward vertical is \(\alpha\). Using the model,
2. find the exact value of \(\tan \alpha\)

1. Three particles of masses \(2 m , 3 m\) and \(k m\) are placed at the points with coordinates (3a, 2a), (a, -4a) and (-3a, 4a) respectively.

The centre of mass of the three particles lies at the point with coordinates \(( \bar { x } , \bar { y } )\).

1. 1. Find \(\bar { x }\) in terms of \(a\) and \(k\)
   2. Find \(\bar { y }\) in terms of \(a\) and \(k\) Given that the distance of the centre of mass of the three particles from the point ( 0,0 ) is \(\frac { 1 } { 3 } a\)
2. find the possible values of \(k\)

2 The region bounded by the \(x\)-axis and the curve \(y = a x ( 2 - x )\), where \(a\) is a constant, is occupied by a uniform lamina \(L _ { 1 }\) (see Fig. 1). Units on the axes are metres. \begin{figure}[h] \end{figure}

1. Write down the value of the \(x\)-coordinate of the centre of mass of \(L _ { 1 }\).
2. Show that the \(y\)-coordinate of the centre of mass of \(L _ { 1 }\) is \(\frac { 2 } { 5 } a\). The mass of \(L _ { 1 }\) is \(M \mathrm {~kg}\). A uniform rectangular lamina of width 2 m and height \(a \mathrm {~m}\) is made from a different material from that of \(L _ { 1 }\) and has a mass of \(2 M \mathrm {~kg}\). A new lamina, \(L _ { 2 }\), is formed by joining the straight edge of \(L _ { 1 }\) to an edge of the rectangular lamina of length 2 m (see Fig. 2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a8c9d007-e67f-4637-9e74-630ba9a91442-2_547_273_1772_890} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} \(L _ { 2 }\) is freely suspended from one of its right-angled corners and hangs in equilibrium with its edge of length 2 m making an angle of \(20 ^ { \circ }\) with the horizontal.
3. Find the value of \(a\), giving your answer correct to 3 significant figures.

7 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]{493f11f4-e25c-4eeb-a0ab-20ec6d7a7a7d-5_611_842_1649_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 \(\mu \geqslant \frac { 14 } { 27 }\). \section*{OCR} Oxford Cambridge and RSA

4 A uniform T-shaped lamina is formed by rigidly joining two rectangles \(A B C H\) and \(D E F G\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-4_748_652_456_644}

1. Show that the centre of mass of the lamina is 26 cm from the edge \(A B\).
2. Explain why the centre of mass of the lamina is 5 cm from the edge \(G F\).
3. The point \(X\) is on the edge \(A B\) and is 7 cm from \(A\), as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-4_697_534_1576_753} The lamina is freely suspended from \(X\) and hangs in equilibrium.
   Find the angle between the edge \(A B\) and the vertical, giving your answer to the nearest degree.
   (4 marks)