6.04e Rigid body equilibrium: coplanar forces

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.

6. \begin{figure}[h] \end{figure} Figure 2 shows a uniform rod \(A B\) of mass \(m\) and length 4a. The end \(A\) of the rod is freely hinged to a point on a vertical wall. A particle of mass \(m\) is attached to the rod at \(B\). One end of a light inextensible string is attached to the rod at C , where \(\mathrm { AC } = 3 \mathrm { a }\). The other end of the string is attached to the wall at D , where \(\mathrm { AD } = 2 \mathrm { a }\) and D is vertically above A . The rod rests horizontally in equilibrium in a vertical plane perpendicular to the wall and the tension in the string is T .

1. Show that \(\mathrm { T } = \mathrm { mg } \sqrt { } 13\).
   (5) The particle of mass \(m\) at \(B\) is removed from the rod and replaced by a particle of mass \(M\) which is attached to the rod at B . The string breaks if the tension exceeds \(2 \mathrm { mg } \sqrt { } 13\). Given that the string does not break,
2. show that \(M \leqslant \frac { 5 } { 2 } m\).

4. \begin{figure}[h] \end{figure} A uniform ladder, of weight \(W\) and length \(2 a\), rests in equilibrium with one end \(A\) on a smooth horizontal floor and the other end \(B\) on a rough vertical wall. The ladder is in a vertical plane perpendicular to the wall. The coefficient of friction between the wall and the ladder is \(\mu\). The ladder makes an angle \(\theta\) with the floor, where \(\tan \theta = 2\). A horizontal light inextensible string \(C D\) is attached to the ladder at the point \(C\), where \(A C = \frac { 1 } { 2 } a\). The string is attached to the wall at the point \(D\), with \(B D\) vertical, as shown in Fig. 2. The tension in the string is \(\frac { 1 } { 4 } W\). By modelling the ladder as a rod,

1. find the magnitude of the force of the floor on the ladder,
2. show that \(\mu \geqslant \frac { 1 } { 2 }\).
3. State how you have used the modelling assumption that the ladder is a rod.

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 \(O\).

2. \begin{figure}[h] \end{figure} Figure 1 shows a ladder \(A B\), of mass 25 kg and length 4 m , resting in equilibrium with one end \(A\) on rough horizontal ground and the other end \(B\) against a smooth vertical wall. The ladder is in a vertical plane perpendicular to the wall. The coefficient of friction between the ladder and the ground is \(\frac { 11 } { 25 }\). The ladder makes an angle \(\beta\) with the ground. When Reece, who has mass 75 kg , stands at the point \(C\) on the ladder, where \(A C = 2.8 \mathrm {~m}\), the ladder is on the point of slipping. The ladder is modelled as a uniform rod and Reece is modelled as a particle.

1. Find the magnitude of the frictional force of the ground on the ladder.
2. Find, to the nearest degree, the value of \(\beta\).
3. State how you have used the modelling assumption that Reece is a particle.

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.

5. \begin{figure}[h] \end{figure} The uniform L-shaped lamina \(A B C D E F\), shown in Figure 2, has sides \(A B\) and \(F E\) parallel, and sides \(B C\) and \(E D\) parallel. The pairs of parallel sides are 9 cm apart. The points \(A , F\), \(D\) and \(C\) lie on a straight line. \(A B = B C = 36 \mathrm {~cm} , F E = E D = 18 \mathrm {~cm} . \angle A B C = \angle F E D = 90 ^ { \circ }\), and \(\angle B C D = \angle E D F = \angle E F D = \angle B A C = 45 ^ { \circ }\).

1. Find the distance of the centre of mass of the lamina from
   1. side \(A B\),
   2. side \(B C\). 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 vertical.

4. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of length 1.5 m and mass 3 kg , is smoothly hinged to a vertical wall at \(A\). The rod is held in equilibrium in a horizontal position by a light strut \(C D\) as shown in Figure 1. The rod and the strut lie in the same vertical plane, which is perpendicular to the wall. The end \(C\) of the strut is freely jointed to the wall at a point 0.5 m vertically below \(A\). The end \(D\) is freely joined to the rod so that \(A D\) is 0.5 m .

1. Find the thrust in \(C D\).
2. Find the magnitude and direction of the force exerted on the \(\operatorname { rod } A B\) at \(A\).

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.

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 \(D C\).

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.

7. \begin{figure}[h] \end{figure} A uniform rod \(A B\) of weight \(W\) has its end \(A\) freely hinged to a point on a fixed vertical wall. The rod is held in equilibrium, at angle \(\theta\) to the horizontal, by a force of magnitude \(P\). The force acts perpendicular to the rod at \(B\) and in the same vertical plane as the rod, as shown in Figure 3. The rod is in a vertical plane perpendicular to the wall. The magnitude of the vertical component of the force exerted on the rod by the wall at \(A\) is \(Y\).

1. Show that \(Y = \frac { W } { 2 } \left( 2 - \cos ^ { 2 } \theta \right)\). Given that \(\theta = 45 ^ { \circ }\)
2. find the magnitude of the force exerted on the rod by the wall at \(A\), giving your answer in terms of \(W\).

2. \begin{figure}[h] \end{figure} The uniform lamina \(O A B C D\), shown in Figure 1, is formed by removing the triangle \(O A D\) from the square \(A B C D\) with centre \(O\). The square has sides of length \(2 a\).

1. Show that the centre of mass of \(O A B C D\) is \(\frac { 2 } { 9 } a\) from \(O\). The mass of the lamina is \(M\). A particle of mass \(k M\) is attached to the lamina at \(D\) to form the system \(S\). The system \(S\) is freely suspended from \(A\) and hangs in equilibrium with \(A O\) vertical.
2. Find the value of \(k\).

3. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} The uniform rectangular lamina \(A B D E\), shown in Figure 1, has side \(A B\) of length \(2 a\) and side \(B D\) of length \(6 a\). The point \(C\) divides \(B D\) in the ratio 1:2 and the point \(F\) divides \(E A\) in the ratio \(1 : 2\). The rectangular lamina is folded along \(F C\) to produce the folded lamina \(L\), shown in Figure 2.

1. Show that the centre of mass of \(L\) is \(\frac { 16 } { 9 } a\) from \(E F\). The folded lamina, \(L\), is freely suspended from \(C\) and hangs in equilibrium.
2. Find the size of the angle between \(C F\) and the downward vertical.

3. [The centre of mass of a semicircular lamina of radius \(r\) is \(\frac { 4 r } { 3 \pi }\) from the centre.] \begin{figure}[h] \end{figure} Figure 2 shows the uniform lamina \(A B C D E\), such that \(A B D E\) is a square with sides of length \(2 a\) and \(B C D\) is a semicircle with diameter \(B D\).

1. Show that the distance of the centre of mass of the lamina from \(B D\) is \(\frac { 20 a } { 3 ( 8 + \pi ) }\). The lamina is freely suspended from \(D\) and hangs in equilibrium.
2. Find, to the nearest degree, the angle that \(D E\) makes with the downward vertical.

3. \begin{figure}[h] \end{figure} A uniform plane lamina is in the shape of an isosceles triangle \(A B C\), where \(A B = A C\). The mid-point of \(B C\) is \(M , A M = 30 \mathrm {~cm}\) and \(B M = 40 \mathrm {~cm}\). The mid-points of \(A C\) and \(A B\) are \(D\) and \(E\) respectively. The triangular portion \(A D E\) is removed leaving a uniform plane lamina \(B C D E\) as shown in Fig. 2.

1. Show that the centre of mass of the lamina \(B C D E\) is \(6 \frac { 2 } { 3 } \mathrm {~cm}\) from \(B C\).
   (6 marks)
   The lamina \(B C D E\) is freely suspended from \(D\) and hangs in equilibrium.
2. Find, in degrees to one decimal place, the angle which \(D E\) makes with the vertical.
   (3 marks)

7. \includegraphics[max width=\textwidth, alt={}, center]{0d3d35b1-e3c5-47ac-b05e-78cdf1eb3083-4_360_472_1105_815} A uniform plane lamina \(A B C D E\) is formed by joining a uniform square \(A B D E\) with a uniform triangular lamina \(B C D\), of the same material, along the side \(B D\), as shown in Fig. 2. The lengths \(A B , B C\) and \(C D\) are \(18 \mathrm {~cm} , 15 \mathrm {~cm}\) and 15 cm respectively.

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

5. A solid \(S\) consists of a uniform solid hemisphere of radius \(r\) and a uniform solid circular cylinder of radius \(r\) and height \(3 r\). The circular face of the hemisphere is joined to one of the circular faces of the cylinder, so that the centres of the two faces coincide. The other circular face of the cylinder has centre \(O\). The mass per unit volume of the hemisphere is \(3 k\) and the mass per unit volume of the cylinder is \(k\).

1. Show that the distance of the centre of mass of \(S\) from \(O\) is \(\frac { 9 r } { 4 }\) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2c0bb9ea-31a6-42f1-9e2e-d792eee8fd10-07_501_1082_653_422} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} The solid \(S\) is held in equilibrium by a horizontal force of magnitude \(P\). The circular face of \(S\) has one point in contact with a fixed rough horizontal plane and is inclined at an angle \(\alpha\) to the horizontal. The force acts through the highest point of the circular face of \(S\) and in the vertical plane through the axis of the cylinder, as shown in Figure 2. The coefficient of friction between \(S\) and the plane is \(\mu\). Given that \(S\) is on the point of slipping along the plane in the same direction as \(P\),
2. show that \(\mu = \frac { 1 } { 8 } ( 9 - 4 \cot \alpha )\).

5. \begin{figure}[h] \end{figure} Figure 3 shows a uniform solid \(S\) formed by joining the plane faces of two solid right circular cones, of base radius \(r\), so that the centres of their bases coincide at \(O\). One cone, with vertex \(V\), has height \(4 r\) and the other cone has height \(k r\), where \(k > 4\)

1. Find the distance of the centre of mass of \(S\) from \(O\).
   (4) The point \(A\) lies on the circumference of the common base of the cones. The solid is placed on a horizontal surface with VA in contact with the surface. Given that \(S\) rests in equilibrium,
2. find the greatest possible value of \(k\). When \(S\) is suspended from \(A\) and hangs freely in equilibrium, \(O A\) makes an angle of \(12 ^ { \circ }\) with the downward vertical.
3. Find the value of \(k\).

6. (a) Use algebraic integration to show that the centre of mass of a uniform solid hemisphere of radius \(r\) is at a distance \(\frac { 3 } { 8 } r\) from the centre of its plane face.
[0pt] [You may assume that the volume of a sphere of radius \(r\) is \(\frac { 4 } { 3 } \pi r ^ { 3 }\) ]
(5) \begin{figure}[h] \end{figure} A uniform solid hemisphere of mass \(m\) and radius \(r\) is joined to a uniform solid right circular cone to form a solid \(S\). The cone has mass \(M\), base radius \(r\) and height \(4 r\). The vertex of the cone is \(O\). The plane face of the cone coincides with the plane face of the hemisphere, as shown in Figure 3.
(b) Find the distance of the centre of mass of \(S\) from \(O\). The point \(A\) lies on the circumference of the base of the cone. The solid is placed on a horizontal table with \(O A\) in contact with the table. The solid remains in equilibrium in this position.
(c) Show that \(M \geqslant \frac { 1 } { 10 } m\)

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 \(\alpha\)

1. (a) Use algebraic integration to show that the centre of mass of a uniform semicircular disc of radius \(r\) and centre \(O\) is at a distance \(\frac { 4 r } { 3 \pi }\) from the diameter through \(O\) [You may assume, without proof, that the area of a circle of radius \(r\) is \(\pi r ^ { 2 }\) ]

A uniform lamina L is in the shape of a semicircle with centre \(B\) and diameter \(A C = 8 a\). The semicircle with diameter \(A B\) is removed from \(L\) and attached to the straight edge \(B C\) to form the template \(T\), shown shaded in Figure 4. \begin{figure}[h] \end{figure} The distance of the centre of mass of \(T\) from \(A C\) is \(d\).
(b) Show that \(d = \frac { 4 a } { \pi }\) The template \(T\) is freely suspended from \(A\) and hangs in equilibrium with \(A C\) at an angle \(\theta\) to the downward vertical.
(c) Find the exact value of \(\tan \theta\)