6.04b Find centre of mass: using symmetry

1. \hspace{0pt} [The centre of mass of a semicircular arc of radius \(r\) is \(\frac { 2 r } { \pi }\) from the centre.]

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

• \(A B C\) is straight and has length \(25 a\)
• \(A D E\) is straight and has length \(24 a\)
• \(A B D\) is a semicircular arc of radius \(7 a\)
• \(E C = 7 a\)
• angle \(A E C = 90 ^ { \circ }\)
• the points \(A , B , C , D\) and \(E\) all lie in the same plane

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

1. Show that \(d = \frac { 53 } { 2 ( 7 + \pi ) } a\) The framework is freely suspended from \(A\) and hangs in equilibrium with \(A C\) at angle \(\alpha ^ { \circ }\) to the downward vertical.
2. Find the value of \(\alpha\).

3. \begin{figure}[h] \end{figure} The uniform lamina \(A B D E\) is in the shape of a rectangle with \(A B = 8 a\) and \(B D = 6 a\). The triangle \(B C D\) is isosceles and has base \(6 a\) and perpendicular height \(6 a\). The template \(A B C D E\), shown shaded in Figure 1, is formed by removing the triangular lamina \(B C D\) from the lamina \(A B D E\).

1. Show that the centre of mass of the template is \(\frac { 14 } { 5 } a\) from \(A E\). The template is freely suspended from \(A\) and hangs in equilibrium with \(A B\) at an angle of \(\theta ^ { \circ }\) to the downward vertical.
2. Find the value of \(\theta\), giving your answer to the nearest whole number.

5. \begin{figure}[h] \end{figure} A beam \(A B\) has mass 12 kg and length 5 m . It is held in equilibrium in a horizontal position by two vertical ropes attached to the beam. One rope is attached to \(A\), the other to the point \(C\) on the beam, where \(B C = 1 \mathrm {~m}\), as shown in Figure 2. The beam is modelled as a uniform rod, and the ropes as light strings.

1. Find
   1. the tension in the rope at \(C\),
   2. the tension in the rope at \(A\). A small load of mass 16 kg is attached to the beam at a point which is \(y\) metres from \(A\). The load is modelled as a particle. Given that the beam remains in equilibrium in a horizontal position,
2. find, in terms of \(y\), an expression for the tension in the rope at \(C\). The rope at \(C\) will break if its tension exceeds 98 N. The rope at \(A\) cannot break.
3. Find the range of possible positions on the beam where the load can be attached without the rope at \(C\) breaking.

4. \begin{figure}[h] \end{figure} A bench consists of a plank which is resting in a horizontal position on two thin vertical legs. The plank is modelled as a uniform rod \(P S\) of length 2.4 m and mass 20 kg . The legs at \(Q\) and \(R\) are 0.4 m from each end of the plank, as shown in Figure 1. Two pupils, Arthur and Beatrice, sit on the plank. Arthur has mass 60 kg and sits at the middle of the plank and Beatrice has mass 40 kg and sits at the end \(P\). The plank remains horizontal and in equilibrium. By modelling the pupils as particles, find

1. the magnitude of the normal reaction between the plank and the leg at \(Q\) and the magnitude of the normal reaction between the plank and the leg at \(R\). Beatrice stays sitting at \(P\) but Arthur now moves and sits on the plank at the point \(X\). Given that the plank remains horizontal and in equilibrium, and that the magnitude of the normal reaction between the plank and the leg at \(Q\) is now twice the magnitude of the normal reaction between the plank and the leg at \(R\),
2. find the distance \(Q X\).

6. \begin{figure}[h] \end{figure} The uniform lamina \(A B C D\) is a square with sides of length \(6 a\). The point \(E\) is the midpoint of side \(A D\). The triangle \(C D E\) is removed from the square to form the uniform lamina \(L\), shown in Figure 3. The centre of mass of \(L\) is at the point \(G\).

1. Show that the distance of \(G\) from the side \(A B\) is \(\frac { 7 } { 3 } a\).
2. Find the distance of \(G\) from the side \(A E\). The mass of \(L\) is \(M\). A particle of mass \(k M\) is attached to \(L\) at the point \(E\). The lamina, with the particle attached, is freely suspended from \(A\) and hangs in equilibrium with the diagonal \(A C\) vertical.
3. Find the value of \(k\).

4. \begin{figure}[h] \end{figure} The uniform lamina \(A B C\) is an isosceles triangle with \(A B = B C , A C = 6 a\) and the distance from \(B\) to \(A C\) is \(3 a\). The uniform lamina \(M N C\) is an isosceles triangle with \(M N = N C\) and \(M C = 3 a\). Triangles \(A B C\) and \(M N C\) are similar and are made of the same material. The lamina \(L\) is formed by fixing triangle \(M N C\) on top of triangle \(A B C\), as shown in Figure 2.

1. Show that the distance of the centre of mass of \(L\) from \(A C\) is \(\frac { 9 } { 10 } a\) The lamina \(L\) is freely suspended from \(B\) and hangs in equilibrium.
2. Find, to the nearest degree, the size of the angle between \(A B\) and the downward vertical.

4. \begin{figure}[h] \end{figure} The uniform lamina \(L\), shown shaded in Figure 2, is formed by removing the square \(P Q R V\), of side \(2 a\), and the square \(R S T U\), of side \(4 a\), from a uniform square lamina \(A B C D\), of side \(8 a\). The lines \(Q R U\) and \(V R S\) are straight. The side \(A D\) is parallel to \(P V\) and the side \(A B\) is parallel to \(P Q\). The distance between \(A D\) and \(P V\) is \(a\) and the distance between \(A B\) and \(P Q\) is \(a\). The centre of mass of \(L\) is at the point \(G\).

1. Show that the distance of \(G\) from the side \(A D\) is \(\frac { 42 } { 11 } a\) The mass of \(L\) is \(M\). A particle of mass \(k M\) is attached to \(L\) at \(C\). The lamina, with the attached particle, is freely suspended from \(B\) and hangs in equilibrium with \(B C\) making an angle of \(45 ^ { \circ }\) with the horizontal.
2. Find the value of \(k\).

   VIXV SIHIANI III IM IONOO

   VIAV SIHI NI JYHAM ION OO

   VI4V SIHI NI JLIYM ION OO

1. Three particles of masses \(m , 4 m\) and \(k m\) are placed at the points whose coordinates are \(( - 3,2 ) , ( 4,3 )\) and \(( 6 , - 4 )\) respectively. The centre of mass of the three particles is at the point with coordinates \(( c , 0 )\).

Find

1. the value of \(k\),
2. the value of \(c\).

6. \begin{figure}[h] \end{figure} The uniform lamina \(L\) shown shaded in Figure 2 is formed by removing two circular discs, \(C _ { 1 }\) and \(C _ { 2 }\), from a circular disc with centre \(O\) and radius \(8 a\). Disc \(C _ { 1 }\) has centre \(A\) and radius \(a\). Disc \(C _ { 2 }\) has centre \(B\) and radius \(2 a\). The diameters \(P R\) and \(Q S\) are perpendicular. The midpoint of \(P O\) is \(A\) and the midpoint of \(O R\) is \(B\).

1. Show that the centre of mass of \(L\) is \(\frac { 484 } { 59 } a\) from \(R\). The mass of \(L\) is \(M\). A particle of mass \(k M\) is attached to \(L\) at \(S\). The lamina with the attached particle is suspended from \(R\) and hangs freely in equilibrium with the diameter \(P R\) at an angle of arctan \(\left( \frac { 1 } { 4 } \right)\) to the downward vertical through \(R\).
2. Find the value of \(k\).

5. \begin{figure}[h] \end{figure} The uniform lamina \(A B C\) is in the shape of an equilateral triangle with sides of length \(4 a\). The midpoint of \(B C\) is \(D\). The point \(E\) lies on \(A D\) with \(D E = \frac { 3 a } { 2 }\). A circular hole, with centre \(E\) and radius \(a\), is made in the lamina \(A B C\) to form the lamina \(L\), shown shaded in Figure 2.

1. Find the distance of the centre of mass of \(L\) from \(D\). The lamina \(L\) is freely suspended from the point \(B\) and hangs in equilibrium.
2. Find, to the nearest degree, the size of the acute angle between \(A D\) and the downward vertical.

5. \begin{figure}[h] \end{figure} Figure 3 shows a uniform rectangular lamina \(A B C D\) with sides of length \(3 a\) and \(k a\), where \(k > 3\). The point \(E\) on side \(A D\) is such that \(D E = 3 a\). Rectangle \(A B C D\) is folded along the line \(C E\) to produce the folded lamina \(L\) shown in Figure 4. \begin{figure}[h] \end{figure} Find, in terms of \(a\) and \(k\),

1. the distance of the centre of mass of \(L\) from \(A B\),
2. the distance of the centre of mass of \(L\) from \(A E\). The folded lamina \(L\) is freely suspended from \(A\) and hangs in equilibrium with \(A B\) at \(45 ^ { \circ }\) to the downward vertical.
3. Find, to 3 significant figures, the value of \(k\).

5. Figure 2 The uniform lamina \(A B C D E F\), shown in Figure 2, consists of two identical rectangles with sides of length \(a\) and \(3 a\). The mass of the lamina is \(M\). A particle of mass \(k M\) is attached to the lamina at \(E\). The lamina, with the attached particle, is freely suspended from \(A\) and hangs in equilibrium with \(A F\) at an angle \(\theta\) to the downward vertical. Given that \(\tan \theta = \frac { 4 } { 7 }\), find the value of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{f30ed5b8-880e-42de-860e-d1538fa68f11-16_677_677_244_580}

6. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of mass \(3 m\) and length \(2 a\), is freely hinged at \(A\) to a fixed point on horizontal ground. A particle of mass \(m\) is attached to the rod at the end \(B\). The system is held in equilibrium by a force \(\mathbf { F }\) acting at the point \(C\), where \(A C = b\). The rod makes an acute angle \(\theta\) with the ground, as shown in Figure 3. The line of action of \(\mathbf { F }\) is perpendicular to the rod and in the same vertical plane as the rod.

1. Show that the magnitude of \(\mathbf { F }\) is \(\frac { 5 m g a } { b } \cos \theta\) The force exerted on the rod by the hinge at \(A\) is \(\mathbf { \mathbf { R } }\), which acts upwards at an angle \(\phi\) above the horizontal, where \(\phi > \theta\).
2. Find
   1. the component of \(\mathbf { R }\) parallel to the rod, in terms of \(m , g\) and \(\theta\),
   2. the component of \(\mathbf { R }\) perpendicular to the rod, in terms of \(a , b , m , g\) and \(\theta\).
3. Hence, or otherwise, find the range of possible values of \(b\), giving your answer in terms of \(a\).

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

1. Find the distance of the centre of mass of the frame from \(B C\). 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.
2. Find the size of the angle between BC and the vertical.

7. \begin{figure}[h] \end{figure} A loaded plate \(L\) is modelled as a uniform rectangular lamina \(A B C D\) and three particles. The sides \(C D\) and \(A D\) of the lamina have lengths \(5 a\) and \(2 a\) respectively and the mass of the lamina is \(3 m\). The three particles have mass \(4 m , m\) and \(2 m\) and are attached at the points \(A , B\) and \(C\) respectively, as shown in Fig. 3.

1. Show that the distance of the centre of mass of \(L\) from \(A D\) is \(2.25 a\).
2. Find the distance of the centre of mass of \(L\) from \(A B\). The point \(O\) is the mid-point of \(A B\). The loaded plate \(L\) is freely suspended from \(O\) and hangs at rest under gravity.
3. Find, to the nearest degree, the size of the angle that \(A B\) makes with the horizontal. A horizontal force of magnitude \(P\) is applied at \(C\) in the direction \(C D\). The loaded plate \(L\) remains suspended from \(O\) and rests in equilibrium with \(A B\) horizontal and \(C\) vertically below \(B\).
4. Show that \(P = \frac { 5 } { 4 } \mathrm { mg }\).
5. Find the magnitude of the force on \(L\) at \(O\).

5. \begin{figure}[h] \end{figure} A uniform lamina \(A B C D\) is made by joining a uniform triangular lamina \(A B D\) to a uniform semi-circular lamina \(D B C\), of the same material, along the edge \(B D\), as shown in Figure 2. Triangle \(A B D\) is right-angled at \(D\) and \(A D = 18 \mathrm {~cm}\). The semi-circle has diameter \(B D\) and \(B D = 12 \mathrm {~cm}\).

1. Show that, to 3 significant figures, the distance of the centre of mass of the lamina \(A B C D\) from \(A D\) is 4.69 cm . Given that the centre of mass of a uniform semicircular lamina, radius \(r\), is at a distance \(\frac { 4 r } { 3 \pi }\) from the centre of the bounding diameter,
2. find, in cm to 3 significant figures, the distance of the centre of mass of the lamina \(A B C D\) from \(B D\). The lamina is freely suspended from \(B\) and hangs in equilibrium.
3. Find, to the nearest degree, the angle which \(B D\) makes with the vertical.

3. \begin{figure}[h] \end{figure} A uniform lamina \(A B C D E F\) is formed by taking a uniform sheet of card in the form of a square \(A X E F\), of side \(2 a\), and removing the square \(B X D C\) of side \(a\), where \(B\) and \(D\) are the mid-points of \(A X\) and \(X E\) respectively, as shown in Figure 1.

1. Find the distance of the centre of mass of the lamina from \(A F\). The lamina is freely suspended from \(A\) and hangs in equilibrium.
2. Find, in degrees to one decimal place, the angle which \(A F\) makes with the vertical.

6. \begin{figure}[h] \end{figure} Figure 3 shows a rectangular lamina \(O A B C\). The coordinates of \(O , A , B\) and \(C\) are ( 0,0 ), \(( 8,0 ) , ( 8,5 )\) and \(( 0,5 )\) respectively. Particles of mass \(k m , 5 m\) and \(3 m\) are attached to the lamina at \(A , B\) and \(C\) respectively. The \(x\)-coordinate of the centre of mass of the three particles without the lamina is 6.4.

1. Show that \(k = 7\). The lamina \(O A B C\) is uniform and has mass \(12 m\).
2. Find the coordinates of the centre of mass of the combined system consisting of the three particles and the lamina. The combined system is freely suspended from \(O\) and hangs at rest.
3. Find the angle between \(O C\) and the horizontal.

5. \begin{figure}[h] \end{figure} A shop sign \(A B C D E F G\) is modelled as a uniform lamina, as illustrated in Figure 2. \(A B C D\) is a rectangle with \(B C = 120 \mathrm {~cm}\) and \(D C = 90 \mathrm {~cm}\). The shape \(E F G\) is an isosceles triangle with \(E G = 60 \mathrm {~cm}\) and height 60 cm . The mid-point of \(A D\) and the mid-point of \(E G\) coincide.

1. Find the distance of the centre of mass of the sign from the side \(A D\). The sign is freely suspended from \(A\) and hangs at rest.
2. Find the size of the angle between \(A B\) and the vertical.

4. \begin{figure}[h] \end{figure} A uniform circular disc has centre \(O\) and radius 4a. The lines \(P Q\) and \(S T\) are perpendicular diameters of the disc. A circular hole of radius \(2 a\) is made in the disc, with the centre of the hole at the point \(R\) on \(O P\) where \(O R = 2 a\), to form the lamina \(L\), shown shaded in Figure 2.

1. Show that the distance of the centre of mass of \(L\) from \(P\) is \(\frac { 14 a } { 3 }\). The mass of \(L\) is \(m\) and a particle of mass \(k m\) is now fixed to \(L\) at the point \(P\). The system is now suspended from the point \(S\) and hangs freely in equilibrium. The diameter \(S T\) makes an angle \(\alpha\) with the downward vertical through \(S\), where \(\tan \alpha = \frac { 5 } { 6 }\).
2. Find the value of \(k\).

4. \begin{figure}[h] \end{figure} The uniform lamina \(A B C D E F\) is a regular hexagon with centre \(O\) and sides of length 2 m , as shown in Figure 1. \begin{figure}[h] \end{figure} The triangles \(O A F\) and \(O E F\) are removed to form the uniform lamina \(O A B C D E\), shown in Figure 2.

1. Find the distance of the centre of mass of \(O A B C D E\) from \(O\). The lamina \(O A B C D E\) is freely suspended from \(E\) and hangs in equilibrium.
2. Find the size of the angle between \(E O\) and the downward vertical.

3. \begin{figure}[h] \end{figure} A non-uniform rod, \(A B\), of mass \(m\) and length 2l, rests in equilibrium with one end \(A\) on a rough horizontal floor and the other end \(B\) against a rough vertical wall. The rod is in a vertical plane perpendicular to the wall and makes an angle of \(60 ^ { \circ }\) with the floor as shown in Figure 1. The coefficient of friction between the rod and the floor is \(\frac { 1 } { 4 }\) and the coefficient of friction between the rod and the wall is \(\frac { 2 } { 3 }\). The rod is on the point of slipping at both ends.

1. Find the magnitude of the vertical component of the force exerted on the rod by the floor. The centre of mass of the rod is at \(G\).
2. Find the distance \(A G\).

4. \begin{figure}[h] \end{figure} Figure 2 shows a lamina \(L\). It is formed by removing a square \(P Q R S\) from a uniform triangle \(A B C\). The triangle \(A B C\) is isosceles with \(A C = B C\) and \(A B = 12 \mathrm {~cm}\). The midpoint of \(A B\) is \(D\) and \(D C = 8 \mathrm {~cm}\). The vertices \(P\) and \(Q\) of the square lie on \(A B\) and \(P Q = 4 \mathrm {~cm}\). The centre of the square is \(O\). The centre of mass of \(L\) is at \(G\).

1. Find the distance of \(G\) from \(A B\). When \(L\) is freely suspended from \(A\) and hangs in equilibrium, the line \(A B\) is inclined at \(25 ^ { \circ }\) to the vertical.
2. Find the distance of \(O\) from \(D C\).

1. Three particles of mass \(3 m , 2 m\) and \(k m\) are placed at the points whose coordinates are \(( 1,5 ) , ( 6,4 )\) and \(( a , 1 )\) respectively. The centre of mass of the three particles is at the point with coordinates \(( 3,3 )\).

Find

1. the value of \(k\),
2. the value of \(a\).

3. \begin{figure}[h] \end{figure} The uniform lamina \(A B C D E F\), shown shaded in Figure 1, is symmetrical about the line through \(B\) and \(E\). It is formed by removing the isosceles triangle \(F E D\), of height \(6 a\) and base \(8 a\), from the isosceles triangle \(A B C\) of height \(9 a\) and base \(12 a\).

1. Find, in terms of \(a\), the distance of the centre of mass of the lamina from \(A C\). The lamina is freely suspended from \(A\) and hangs in equilibrium.
2. Find, to the nearest degree, the size of the angle between \(A B\) and the downward vertical.