3.04b Equilibrium: zero resultant moment and force

4. \begin{figure}[h] \end{figure} A non-uniform beam \(A B\) has length 8 m and mass \(M \mathrm {~kg}\). The centre of mass of the beam is \(d\) metres from \(A\). The beam is supported in equilibrium in a horizontal position by two vertical light ropes. One rope is attached to the beam at \(C\), where \(A C = 2.5 \mathrm {~m}\) and the other rope is attached to the beam at \(D\), where \(D B = 2 \mathrm {~m}\), as shown in Figure 2. A gymnast, of mass 64 kg , stands on the beam at the point \(X\), where \(A X = 1.875 \mathrm {~m}\), and the beam remains in equilibrium in a horizontal position but is now on the point of tilting about \(C\). The gymnast then dismounts from the beam. A second gymnast, of mass 48 kg , now stands on the beam at the point \(Y\), where \(Y B = 0.5 \mathrm {~m}\), and the beam remains in equilibrium in a horizontal position but is now on the point of tilting about \(D\). The beam is modelled as a non-uniform rod and the gymnasts are modelled as particles. Find the value of \(M\).

7. \begin{figure}[h] \end{figure} A non-uniform beam \(A B\), of mass 60 kg and length \(8 a\) metres, rests in equilibrium in a horizontal position on two vertical supports. One support is at \(C\), where \(A C = a\) metres and the other support is at \(D\), where \(D B = 2 a\) metres, as shown in Figure 2. The magnitude of the normal reaction between the beam and the support at \(D\) is three times the magnitude of the normal reaction between the beam and the support at \(C\). By modelling the beam as a non-uniform rod whose centre of mass is at a distance \(x\) metres from \(A\),

1. find an expression for \(x\) in terms of \(a\). A box of mass \(M \mathrm {~kg}\) is placed on the beam at \(E\), where \(A E = 2 a\) metres.
   The beam remains in equilibrium in a horizontal position.
   The magnitude of the normal reaction between the beam and the support at \(C\) is now equal to the magnitude of the normal reaction between the beam and the support at \(D\). By modelling the box as a particle,
2. find the value of \(M\).

5. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has length 5 m and mass 5 kg . The rod rests in equilibrium in a horizontal position on two supports \(C\) and \(D\), where \(A C = 1 \mathrm {~m}\) and \(D B = 2 \mathrm {~m}\), as shown in Figure 2 . A particle of mass 10 kg is placed on the rod at \(A\) and a particle of mass \(M \mathrm {~kg}\) is placed on the rod at \(B\). The rod remains horizontal and in equilibrium.

1. Find, in terms of \(M\), the magnitude of the reaction on the rod at \(C\).
2. Find, in terms of \(M\), the magnitude of the reaction on the rod at \(D\).
3. Hence, or otherwise, find the range of possible values of \(M\). \includegraphics[max width=\textwidth, alt={}, center]{61cb5bce-2fad-48f0-b6a4-e9899aa0acec-14_2256_51_310_1983}

4. \begin{figure}[h] \end{figure} Figure 1 shows a beam \(A B\), of mass \(m \mathrm {~kg}\) and length 2 m , suspended by two light vertical ropes.
The ropes are attached to the points \(C\) and \(D\) on the beam, where \(A C = 0.6 \mathrm {~m}\) and \(D B = 0.2 \mathrm {~m}\) The beam is in equilibrium in a horizontal position.
A particle of mass pmkg is attached to the beam at \(A\) and the beam remains in equilibrium in a horizontal position. The beam is modelled as a uniform rod.

1. Given that the tension in the rope attached at \(C\) is four times the tension in the rope attached at \(D\), use the model to find the exact value of \(p\). The particle of mass \(p m \mathrm {~kg}\) at \(A\) is removed and replaced by a particle of mass \(q m \mathrm {~kg}\) at \(A\).
   The beam remains in equilibrium in a horizontal position but is now on the point of tilting.
2. Using the model, find the exact value of \(q\)
3. State how you have used the modelling assumption that the beam is uniform.

4. \begin{figure}[h] \end{figure} A non-uniform rod \(A B\) has length 6.5 m and mass 1.2 kg . The centre of mass of the rod is 3 m from \(A\). The rod rests on a horizontal step and overhangs the end of the step \(C\) by 1.5 m , as shown in Figure 2. The rod is perpendicular to the edge of the step.
A particle of mass 4 kg is placed on the rod at \(B\) and another particle, whose mass is \(M \mathrm {~kg}\), is placed on the rod at \(D\), where \(A D = 0.5 \mathrm {~m}\). The rod remains in equilibrium in a horizontal position.

1. Find the smallest possible value of \(M\). The particle at \(B\) and the particle at \(D\) are now removed.
   A new particle is placed on the rod at the point \(E\), where \(E B = 0.9 \mathrm {~m}\).
   The rod remains in equilibrium in a horizontal position but is on the point of tilting about \(C\).
2. Find the magnitude of the force acting on the rod at \(C\).

3. \begin{figure}[h] \end{figure} A plank \(A B\) has length 8 m and mass 12 kg . The plank rests on two supports. One support is at \(C\), where \(A C = 3 \mathrm {~m}\) and the other support is at \(D\), where \(A D = x\) metres. A block of mass 3 kg is placed on the plank at \(B\), as shown in Figure 1. The plank rests in equilibrium in a horizontal position. The magnitude of the force exerted on the plank by the support at \(D\) is twice the magnitude of the force exerted on the plank by the support at \(C\). The plank is modelled as a uniform rod and the block is modelled as a particle. Find the value of \(x\).

1. \begin{figure}[h] \end{figure} A non-uniform rod \(A B\) has length 9 m and mass \(M \mathrm {~kg}\).
The rod rests in equilibrium in a horizontal position on two supports, one at \(C\) where \(A C = 2.5 \mathrm {~m}\) and the other at \(D\) where \(D B = 2 \mathrm {~m}\), as shown in Figure 1 . The magnitude of the force acting on the rod at \(D\) is twice the magnitude of the force acting on the \(\operatorname { rod }\) at \(C\). The centre of mass of the rod is \(d\) metres from \(A\).
Find the value of \(d\).

1. \begin{figure}[h] \end{figure} Figure 1 shows a beam \(A B\) with weight 24 N and length 6 m .
The beam is suspended by two light vertical ropes. The ropes are attached to the points \(C\) and \(D\) on the beam where \(A C = x\) metres and \(C D = 2 \mathrm {~m}\). The tension in the rope attached to the beam at \(C\) is double the tension in the rope attached to the beam at \(D\). The beam is modelled as a uniform rod, resting horizontally in equilibrium.
Find

1. the tension in the rope attached to the beam at \(D\).
2. the value of \(x\).

4. \begin{figure}[h] \end{figure} A diving board \(A B\) consists of a wooden plank of length 4 m and mass 30 kg . The plank is held at rest in a horizontal position by two supports at the points \(A\) and \(C\), where \(A C = 0.6 \mathrm {~m}\), as shown in Figure 1. The force on the plank at \(A\) acts vertically downwards and the force on the plank at \(C\) acts vertically upwards. A diver of mass 50 kg is standing on the board at the end \(B\). The diver is modelled as a particle and the plank is modelled as a uniform rod. The plank is in equilibrium.

1. Find
   1. the magnitude of the force acting on the plank at \(A\),
   2. the magnitude of the force acting on the plank at \(C\). The support at \(A\) will break if subjected to a force whose magnitude is greater than 5000 N .
2. Find, in kg, the greatest integer mass of a diver who can stand on the board at \(B\) without breaking the support at \(A\).
3. Explain how you have used the fact that the diver is modelled as a particle.
   

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1. \begin{figure}[h] \end{figure} A uniform \(\operatorname { rod } A B\) has weight 70 N and length 3 m . It rests in a horizontal position on two smooth supports placed at \(P\) and \(Q\), where \(A P = 0.5 \mathrm {~m}\), as shown in Fig. 1 . The reaction on the rod at \(P\) has magnitude 20 N . Find

1. the magnitude of the reaction on the rod at \(Q\),
2. the distance \(A Q\).

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 \(Q X\).

5. \begin{figure}[h] \end{figure} A uniform rod, of length \(8 a\) and mass \(M\), has one end freely hinged to a fixed point \(A\) on a vertical wall. One end of a light inextensible string is attached to the rod at the point \(B\), where \(A B = 5 a\). The other end of the string is attached to the wall at the point \(C\), where \(A C = 5 a\) and \(C\) is vertically above \(A\). The rod rests in equilibrium in a vertical plane perpendicular to the wall with angle \(B A C = 70 ^ { \circ }\), as shown in Figure 3.

1. Find, in terms of \(M\) and \(g\), the tension in the string. The magnitude of the force acting on the rod at \(A\) is \(\lambda M g\), where \(\lambda\) is a constant.
2. Find, to 2 significant figures, the value of \(\lambda\).

5. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has length 4 m and weight 50 N .
The rod has its end \(A\) on rough horizontal ground. The rod is held in equilibrium at an angle \(\alpha\) to the ground by a light inextensible cable attached to the rod at \(B\), as shown in Figure 2. The cable and the rod lie in the same vertical plane and the cable is perpendicular to the rod. The tension in the cable is \(T\) newtons. Given that \(\sin \alpha = \frac { 3 } { 5 }\)

1. show that \(T = 20\) Given also that the rod is in limiting equilibrium,
2. find the value of the coefficient of friction between the rod and the ground.

5. \begin{figure}[h] \end{figure} A uniform beam \(A B\), of mass 15 kg and length 6 m , rests with end \(A\) on rough horizontal ground. The end \(B\) of the beam rests against a rough vertical wall. The beam is inclined at \(75 ^ { \circ }\) to the ground, as shown in Figure 2.
The coefficient of friction between the beam and the wall is 0.2
The coefficient of friction between the beam and the ground is \(\mu\) The beam is modelled as a uniform rod which lies in a vertical plane perpendicular to the wall. The beam rests in limiting equilibrium.

1. Find the magnitude of the normal reaction between the beam and the wall at \(B\).
2. Find the value of \(\mu\)

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 \(\mu\).

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 \(\mu\).

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}

5. \begin{figure}[h] \end{figure} A pole \(A B\) has length 2.5 m and weight 70 N .
The pole rests with end \(B\) against a rough vertical wall. One end of a cable of length 4 m is attached to the pole at \(A\). The other end of the cable is attached to the wall at the point \(C\). The point \(C\) is vertically above \(B\) and \(B C = 2.5 \mathrm {~m}\).
The angle between the cable and the wall is \(\alpha\), as shown in Figure 2.
The pole is in a vertical plane perpendicular to the wall.
The cable is modelled as a light inextensible string and the pole is modelled as a uniform rod. Given that \(\tan \alpha = \frac { 3 } { 4 }\)

1. show that the tension in the cable is 56 N . Given also that the pole is in limiting equilibrium,
2. find the coefficient of friction between the pole and the wall. \includegraphics[max width=\textwidth, alt={}, center]{80dceee7-2eea-4082-ad20-7b3fe4e8bb25-15_90_61_2613_1886}

5. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has length \(8 a\) and weight \(W\).
The end \(A\) of the rod is freely hinged to horizontal ground.
The rod rests in equilibrium against a block which is also fixed to the ground.
The block is modelled as a smooth solid hemisphere with radius \(2 a\) and centre \(D\).
The point of contact between the rod and the block is \(C\), where \(A C = 5 a\) The rod is at an angle \(\theta\) to the ground, as shown in Figure 1.
Points \(A , B , C\) and \(D\) all lie in the same vertical plane.

1. Show that \(A D = \sqrt { 29 } a\)
2. Show that the magnitude of the normal reaction at \(C\) between the rod and the block is \(\frac { 4 } { \sqrt { 29 } } W\) The resultant force acting on the rod at \(A\) has magnitude \(k W\) and acts at an angle \(\alpha\) to the ground.
3. Find (i) the exact value of \(k\) (ii) the exact value of \(\tan \alpha\)
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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 \(g\)

6. \begin{figure}[h] \end{figure} A uniform \(\operatorname { rod } A B\) has length \(8 a\) and weight \(W\).
The end \(A\) of the rod is freely hinged to a fixed point on a vertical wall.
A particle of weight \(\frac { 1 } { 4 } W\) is attached to the rod at \(B\).
A light inelastic string of length \(5 a\) has one end attached to the rod at the point \(C\), where \(A C = 5 a\). The other end of the string is attached to the wall at the point \(D\), where \(D\) is above \(A\) and \(A D = 5 a\), as shown in Figure 4. The rod rests in equilibrium.
The tension in the string is \(T\).

1. Show that \(T = \frac { 6 } { 5 } \mathrm {~W}\)
2. Find, in terms of \(W\), the magnitude of the force exerted on the rod by the hinge at \(A\).

7. \begin{figure}[h] \end{figure} A uniform plank \(A B\), of weight 100 N and length 4 m , rests in equilibrium with the end \(A\) on rough horizontal ground. The plank rests on a smooth cylindrical drum. The drum is fixed to the ground and cannot move. The point of contact between the plank and the drum is \(C\), where \(A C = 3 \mathrm {~m}\), as shown in Figure 4. The plank is resting in a vertical plane which is perpendicular to the axis of the drum, at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 3 }\). The coefficient of friction between the plank and the ground is \(\mu\). Modelling the plank as a rod, find the least possible value of \(\mu\).