6.04e Rigid body equilibrium: coplanar forces

3. \begin{figure}[h] \end{figure} A uniform \(\operatorname { rod } A B\) of weight \(W\) is freely hinged at end \(A\) to a vertical wall. The rod is supported in equilibrium at an angle of \(60 ^ { \circ }\) to the wall by a light rigid strut \(C D\). The strut is freely hinged to the rod at the point \(D\) and to the wall at the point \(C\), which is vertically below \(A\), as shown in Figure 1. The rod and the strut lie in the same vertical plane, which is perpendicular to the wall. The length of the rod is \(4 a\) and \(A C = A D = 2.5 a\).

1. Show that the magnitude of the thrust in the strut is \(\frac { 4 \sqrt { 3 } } { 5 } W\).
2. Find the magnitude of the force acting on the \(\operatorname { rod }\) at \(A\).

4. \begin{figure}[h] \end{figure} The uniform square lamina \(A B C D\) shown in Figure 2 has sides of length 4a. The points \(E\) and \(F\), on \(D A\) and \(D C\) respectively, are both at a distance \(3 a\) from \(D\). The portion \(D E F\) of the lamina is folded through \(180 ^ { \circ }\) about \(E F\) to form the folded lamina \(A B C F E\) shown in Figure 3 below. \begin{figure}[h] \end{figure}

1. Show that the distance from \(A B\) of the centre of mass of the folded lamina is \(\frac { 55 } { 32 } a\).
   (6) The folded lamina is freely suspended from \(E\) and hangs in equilibrium.
2. Find the size of the angle between \(E D\) and the downward vertical.

4. \begin{figure}[h] \end{figure} The uniform lamina \(A B C D E\) is made by joining a rectangular lamina \(A B D E\) to a triangular lamina \(B C D\) along the edge \(B D\). The rectangle has length \(6 a\) and width \(3 a\). The triangle is isosceles, with \(B C = C D\), and the distance from \(C\) to \(B D\) is \(3 a\), as shown in Figure 2.

1. Find the distance of the centre of mass of the lamina, \(A B C D E\), from \(A E\). The lamina \(A B C D E\) is freely suspended from \(A\). A horizontal force of magnitude \(F\) newtons is applied to the lamina at \(D\). The line of action of the force lies in the vertical plane containing the lamina. The lamina is in equilibrium with \(A E\) vertical. The mass of the lamina is 4 kg .
2. Find the magnitude of the force exerted on the lamina at \(A\).
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5. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has mass 6 kg and length 2 m . The end \(A\) of the rod rests against a rough vertical wall. One end of a light string is attached to the rod at \(B\). The other end of the string is attached to the wall at \(C\), which is vertically above \(A\). The angle between the rod and the string is \(30 ^ { \circ }\) and the angle between the rod and the wall is \(70 ^ { \circ }\), as shown in Figure 3. The rod is in a vertical plane perpendicular to the wall and rests in limiting equilibrium. Find

1. the tension in the string,
2. the coefficient of friction between the rod and the wall,
3. the direction of the force exerted on the rod by the wall at \(A\).
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4. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has mass 5 kg and length 4 m . The rod is held in a horizontal position by a light inextensible string. The end \(A\) of the rod rests against a rough vertical wall. One end of the string is attached to the rod at \(B\) and the other end is attached to the wall at a point \(D\). The point \(D\) is vertically above \(A\), with \(A D = 3 \mathrm {~m}\). A particle of mass 2 kg is attached to the rod at \(C\), where \(A C = 0.5 \mathrm {~m}\), as shown in Figure 1. The rod is in equilibrium in a vertical plane perpendicular to the wall. The coefficient of friction between the rod and the wall is \(\mu\). Find

1. the tension in the string,
2. the magnitude of the force exerted by the wall on the rod at \(A\),
3. the range of possible values of \(\mu\).

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 \(k\).

2. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of mass 6 kg and length 1.6 m , rests with its end \(A\) on rough horizontal ground. The rod is held in equilibrium at \(30 ^ { \circ }\) to the horizontal by a light string attached to the rod at \(B\). The string is at \(40 ^ { \circ }\) to the horizontal and lies in the same vertical plane as the rod, as shown in Figure 1. The tension in the string is \(T\) newtons. The coefficient of friction between the ground and the rod is \(\mu\).

1. Show that, to 3 significant figures, \(T = 27.1\)
2. Find the set of values of \(\mu\) for which equilibrium is possible. \includegraphics[max width=\textwidth, alt={}, center]{a3f28425-4acf-4878-b0e3-15b5bc8a92d7-07_27_40_2802_1893}

3. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of mass 25 kg and length 3 m , has end \(A\) resting on rough horizontal ground. The end \(B\) rests against a rough vertical wall. The rod is in a vertical plane perpendicular to the wall.
The coefficient of friction between the rod and the ground is \(\frac { 4 } { 5 }\) The coefficient of friction between the rod and the wall is \(\frac { 3 } { 5 }\) The rod rests in limiting equilibrium.
The rod is at an angle of \(\theta\) to the ground, as shown in Figure 1. Find the exact value of \(\tan \theta\).
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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 \(k\).

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3. \begin{figure}[h] \end{figure} The uniform lamina \(A B C D\) is a square of side \(6 a\). The template \(T\), shown shaded in Figure 1, is formed by removing the right-angled triangle \(E F G\) and the circle, centre \(H\) and radius \(a\), from the square lamina. Triangle \(E F G\) has \(E F = E G = 4 a\), with \(E F\) parallel to \(A B\) and \(E G\) parallel to \(A D\). The distance between \(A B\) and \(E F\) is \(a\) and the distance between \(A D\) and \(E G\) is \(a\). The point \(H\) lies on \(A C\) and the distance of \(H\) from \(B C\) is \(2 a\).

1. Show that the centre of mass of \(T\) is a distance \(\frac { 4 ( 67 - 3 \pi ) } { 3 ( 28 - \pi ) } a\) from \(A D\). The template \(T\) is suspended from the ceiling by two light inextensible vertical strings. One string is attached to \(T\) at \(A\) and the other string is attached to \(T\) at \(B\) so that \(T\) hangs in equilibrium with \(A B\) horizontal. The weight of \(T\) is \(W\). The tension in the string attached to \(T\) at \(B\) is \(k W\), where \(k\) is a constant.
2. Find the value of \(k\), giving your answer to 2 decimal places.

7. \begin{figure}[h] \end{figure} The template shown in Figure 3 is formed by joining together three separate laminas. All three laminas lie in the same plane.

• PQUV is a uniform square lamina with sides of length \(3 a\)
• URST is a uniform square lamina with sides of length \(6 a\)
• \(Q R U\) is a uniform triangular lamina with \(U Q = 3 a , U R = 6 a\) and angle \(Q U R = 90 ^ { \circ }\)

The mass per unit area of \(P Q U V\) is \(k\), where \(k\) is a constant.
The mass per unit area of URST is \(k\).
The mass per unit area of \(Q R U\) is \(2 k\).
The distance of the centre of mass of the template from \(Q T\) is \(d\).

1. Show that \(d = \frac { 29 } { 14 } a\) The template is freely suspended from the point \(Q\) and hangs in equilibrium with \(Q R\) at \(\theta ^ { \circ }\) to the downward vertical.
2. Find the value of \(\theta\)

3. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} The uniform triangular lamina \(A B C\), shown in Figure 1, has height \(9 y\), base \(B C = 6 x\), and \(A B = A C\) The points \(P\) and \(Q\) are such that \(A P : P C = A Q : Q B = 2 : 1\) The lamina is folded along \(P Q\) to form the folded lamina \(F\) The distance of the centre of mass of \(F\) from \(P Q\) is \(d\)

1. Show that \(d = \frac { 16 } { 9 } y\) The folded lamina is suspended from \(P\) and hangs freely in equilibrium with \(P Q\) at an angle \(\alpha\) to the downward vertical.
   Given that \(\tan \alpha = \frac { 64 } { 81 }\)
2. find \(x\) in terms of \(y\)

3. \begin{figure}[h] \end{figure} A uniform circular disc \(C\) has centre \(X\) and radius \(R\).
A disc with centre \(Y\) and radius \(r\), where \(0 < r < R\) and \(X Y = R - r\), is removed from \(C\) to form the template shown shaded in Figure 1. The centre of mass of the template is a distance \(k r\) from \(X\).

1. Show that \(r = \frac { k } { 1 - k } R\)
2. Hence find the range of possible values of \(k\). The point \(P\) is on the outer edge of the template and \(P X\) is perpendicular to \(X Y\).
   The template is freely suspended from \(P\) and hangs in equilibrium.
   Given that \(k = \frac { 4 } { 9 }\)
3. find the angle that \(X Y\) makes with the vertical. The mass of the template is \(M\).
4. Find, in terms of \(M\), the mass of the lightest particle that could be attached to the template so that it would hang in equilibrium from \(P\) with \(X Y\) horizontal.

6. \begin{figure}[h] \end{figure} A uniform rod, \(A B\), of mass \(m\) and length \(2 a\), rests in limiting equilibrium with its end \(A\) on rough horizontal ground and its end \(B\) against a smooth vertical wall.
The vertical plane containing the rod is at right angles to the wall.
The rod is inclined to the wall at an angle \(\alpha\), as shown in Figure 2.
The coefficient of friction between the rod and the ground is \(\frac { 1 } { 3 }\)

1. Show that \(\tan \alpha = \frac { 2 } { 3 }\) With the rod in the same position, a horizontal force of magnitude \(k m g\) is applied to the \(\operatorname { rod }\) at \(A\), towards the wall. The line of action of this force is at right angles to the wall. The rod remains in equilibrium.
2. Find the largest possible value of \(k\).

5. \begin{figure}[h] \end{figure} A uniform rod \(A B\) of length 8 m and weight \(W\) newtons rests in equilibrium against a rough horizontal peg \(P\). The end \(A\) is on rough horizontal ground. The friction is limiting at both \(A\) and \(P\). The distance \(A P\) is 5 m , as shown in Figure 1. The rod rests at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 4 } { 3 }\). The rod is in a vertical plane which is perpendicular to \(P\). The coefficient of friction between the rod and \(P\) is \(\frac { 1 } { 4 }\) and the coefficient of friction between the rod and the ground is \(\mu\).

1. Show that the magnitude of the normal reaction between the rod and \(P\) is \(0.48 W\) newtons.
2. Find the value of \(\mu\).

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 \(k\).

4. \begin{figure}[h] \end{figure} A uniform \(\operatorname { rod } A B\) has mass \(m\) and length \(6 a\). The end \(A\) rests against a rough vertical wall. One end of a light inextensible string is attached to the rod at the point \(C\), where \(A C = 2 a\). The other end of the string is attached to the wall at the point \(D\), where \(D\) is vertically above \(A\), with the string perpendicular to the rod. A particle of mass \(m\) is attached to the rod at the end \(B\). The rod is in equilibrium in a vertical plane which is perpendicular to the wall. The rod is inclined at \(60 ^ { \circ }\) to the wall, as shown in Figure 1. Find, in terms of \(m\) and \(g\),

1. the tension in the string,
2. the magnitude of the horizontal component of the force exerted by the wall on the rod. The coefficient of friction between the wall and the rod is \(\mu\). Given that the rod is in limiting equilibrium,
3. find the value of \(\mu\).

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 \(k\).

6. \begin{figure}[h] \end{figure} A uniform rod, \(A B\), of mass \(8 m\) and length \(2 a\), has its end \(A\) resting against a rough vertical wall. One end of a light inextensible string is attached to the rod at \(B\) and the other end of the string is attached to the wall at the point \(D\), which is vertically above \(A\). The angle between the rod and the string is \(30 ^ { \circ }\). A particle of mass \(k m\) is fixed to the rod at \(C\), where \(A C = 0.5 a\). The rod is in equilibrium in a vertical plane perpendicular to the wall, and is at an angle of \(60 ^ { \circ }\) to the wall, as shown in Figure 5. The tension in the string is \(T\).

1. Show that \(T = \frac { \sqrt { 3 } } { 4 } ( 16 + k ) m g\) The coefficient of friction between the wall and the rod is \(\frac { 2 } { 3 } \sqrt { 3 }\).
   Given that the rod is in limiting equilibrium,
2. find the value of \(k\). \includegraphics[max width=\textwidth, alt={}, center]{99d06f7b-f5cc-4c19-ae26-8f715eda8ee8-23_67_65_2656_1886}

7. In this question you may use, without proof, the formula for the centre of mass of a uniform sector of a circle, as given in the formulae book. \begin{figure}[h] \end{figure} The uniform lamina \(A B C D E\), shown shaded in Figure 3, is formed by joining a rectangle to a sector of a circle.

• The rectangle \(A B C E\) has \(A B = E C = a\) and \(A E = B C = d\)
• The sector \(C D E\) has centre \(C\) and radius \(a\)
• Angle \(E C D = \frac { \pi } { 3 }\) radians

The centre of mass of the lamina lies on EC.

1. Show that \(a = \sqrt { 3 } d\) The lamina is freely suspended from \(B\) and hangs in equilibrium with \(B C\) at an angle \(\beta\) radians to the downward vertical.
2. Find the value of \(\beta\)

6. \begin{figure}[h] \end{figure} The uniform lamina \(P Q R S T U V\) shown in Figure 2 is formed from two identical rectangles, \(P Q U V\) and \(Q R S T U\).
The rectangles have sides \(P Q = R S = 2 a\) and \(P V = Q R = k a\).

1. Show that the centre of mass of the lamina is \(\left( \frac { 6 + k } { 4 } \right) a\) from \(P V\) The lamina is freely suspended from \(P\) and hangs in equilibrium with \(P R\) at an angle of \(\alpha\) to the downward vertical. Given that \(\tan \alpha = \frac { 7 } { 15 }\)
2. find the value of \(k\).

2. \begin{figure}[h] \end{figure} Figure 1 shows a template where

• PQUY is a uniform square lamina with sides of length \(4 a\)
• RSTU is a uniform square lamina with sides of length \(2 a\)
• VWXY is a uniform square lamina with sides of length \(2 a\)
• the three squares all lie in the same plane
• the mass per unit area of \(V W X Y\) is double the mass per unit area of \(P Q U Y\)
• the mass per unit area of \(R S T U\) is double the mass per unit area of \(P Q U Y\)
• the distance of the centre of mass of the template from \(P X\) is \(d\)
  1. Show that \(d = \frac { 5 } { 2 } a\)

The template is freely pivoted about \(Q\) and hangs in equilibrium with \(P Q\) at an angle of \(\theta\) to the downward vertical.

Find the value of \(\tan \theta\) The mass of the template is \(M\) The template is still freely pivoted about \(Q\), but it is now held in equilibrium, with \(P Q\) vertical, by a horizontal force of magnitude \(F\) which acts on the template at \(X\). The line of action of the force lies in the same plane as the template.

Find \(F\) in terms of \(M\) and \(g\)

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 \(a\).