3.04a Calculate moments: about a point

4 \includegraphics[max width=\textwidth, alt={}, center]{f8961f84-c080-4407-a178-45b76f200111-3_561_606_260_767} A uniform rod \(A B\) has mass \(m\) and length \(2 d\). The rod rests in equilibrium on a smooth peg \(C\), with the end \(A\) resting on a rough horizontal plane. The distance \(A C\) is \(2 a\) and the angle between \(A B\) and the horizontal is \(\alpha\), where \(\cos \alpha = \frac { 3 } { 5 }\). A particle of mass \(\frac { 1 } { 2 } m\) is attached to the rod at \(B\) (see diagram). Find the normal reaction at \(A\) and deduce that \(d < \frac { 25 } { 6 } a\). The coefficient of friction between the rod and the plane is \(\mu\). Show that \(\mu \geqslant \frac { 8 d } { 25 a - 6 d }\).

2 An object is made from two identical uniform rods \(A B\) and \(B C\) each of length 0.6 m and weight 7 N . The rods are rigidly joined to each other at \(B\) and angle \(A B C = 90 ^ { \circ }\).

1. Calculate the distance of the centre of mass of the object from \(B\). The object is freely suspended at \(A\) and a force of magnitude \(F \mathrm {~N}\) is applied to the rod \(B C\) at \(C\). The object is in equilibrium with \(A B\) inclined at \(45 ^ { \circ }\) to the horizontal.
2. (a) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a093cbad-3ba0-45ce-a617-d4ecc8cb1ec9-2_401_314_799_995} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Calculate \(F\) given that the force acts horizontally as shown in Fig. 1.
   (b) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a093cbad-3ba0-45ce-a617-d4ecc8cb1ec9-2_503_273_1446_1014} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Calculate \(F\) given instead that the force acts perpendicular to the rod as shown in Fig. 2.

2 \includegraphics[max width=\textwidth, alt={}, center]{2c6b2e42-09cb-4653-9378-6c6add7771cc-2_463_567_479_790} A uniform rod \(A B\) has weight 6 N and length 0.8 m . The rod rests in limiting equilibrium with \(B\) in contact with a rough horizontal surface and \(A B\) inclined at \(60 ^ { \circ }\) to the horizontal. Equilibrium is maintained by a force, in the vertical plane containing \(A B\), acting at \(A\) at an angle of \(45 ^ { \circ }\) to \(A B\) (see diagram). Calculate

1. the magnitude of the force applied at \(A\),
2. the least possible value of the coefficient of friction at \(B\).

1 \includegraphics[max width=\textwidth, alt={}, center]{e30ba526-db21-4904-96dc-c12a1f67c81a-2_426_531_258_808} A circular object is formed from a uniform semicircular lamina of weight 12 N and a uniform semicircular arc of weight 8 N . The lamina and the arc both have centre \(O\) and radius 0.6 m and are joined at the ends of their common diameter \(A B\). The object is freely pivoted to a fixed point at \(A\) with \(A B\) inclined at \(30 ^ { \circ }\) to the vertical. The object is in equilibrium acted on by a horizontal force of magnitude \(F\) N applied at the lowest point of the object, and acting in the plane of the object (see diagram).

1. Show that the centre of mass of the object is at \(O\).
2. Calculate \(F\).

5. \begin{figure}[h] \end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying on calculator technology are not acceptable.} The straight line \(l _ { 1 }\), shown in Figure 1, passes through the points \(P ( - 2,9 )\) and \(Q ( 10,6 )\).

1. Find the equation of \(l _ { 1 }\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found. The straight line \(l _ { 2 }\) passes through the origin \(O\) and is perpendicular to \(l _ { 1 }\) The lines \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(R\) as shown in Figure 1.
2. Find the coordinates of \(R\)
3. Find the exact area of triangle \(O P Q\).

4. \begin{figure}[h] \end{figure} A non-uniform \(\operatorname { rod } A B\), of mass \(m\) and length \(5 d\), rests horizontally in equilibrium on two supports at \(C\) and \(D\), where \(A C = D B = d\), as shown in Figure 1. The centre of mass of the rod is at the point \(G\). A particle of mass \(\frac { 5 } { 2 } m\) is placed on the rod at \(B\) and the rod is on the point of tipping about \(D\).

1. Show that \(G D = \frac { 5 } { 2 } d\). The particle is moved from \(B\) to the mid-point of the rod and the rod remains in equilibrium.
2. Find the magnitude of the normal reaction between the support at \(D\) and the rod.

5. \begin{figure}[h] \end{figure} A large \(\log A B\) is 6 m long. It rests in a horizontal position on two smooth supports \(C\) and \(D\), where \(A C = 1 \mathrm {~m}\) and \(B D = 1 \mathrm {~m}\), as shown in Figure 4. David needs an estimate of the weight of the log, but the log is too heavy to lift off both supports. When David applies a force of magnitude 1500 N vertically upwards to the \(\log\) at \(A\), the \(\log\) is about to tilt about \(D\).

1. State the value of the reaction on the \(\log\) at \(C\) for this case. David initially models the log as uniform rod. Using this model,
2. estimate the weight of the log The shape of the log convinces David that his initial modelling assumption is too simple. He removes the force at \(A\) and applies a force acting vertically upwards at \(B\). He finds that the log is about to tilt about \(C\) when this force has magnitude 1000 N. David now models the log as a non-uniform rod, with the distance of the centre of mass of the \(\log\) from \(C\) as \(x\) metres. Using this model, find
3. a new estimate for the weight of the log,
4. the value of \(x\).
5. State how you have used the modeling assumption that the log is a rod.

1. \begin{figure}[h] \end{figure} A uniform plank \(A B\) has mass 40 kg and length 4 m . It is supported in a horizontal position by two smooth pivots, one at the end \(A\), the other at the point \(C\) of the plank where \(A C = 3 \mathrm {~m}\), as shown in Fig. 1. A man of mass 80 kg stands on the plank which remains in equilibrium. The magnitudes of the reactions at the two pivots are each equal to \(R\) newtons. By modelling the plank as a rod and the man as a particle, find

1. the value of \(R\),
2. the distance of the man from \(A\).
   (4)

5. A steel girder \(A B\) has weight 210 N . It is held in equilibrium in a horizontal position by two vertical cables. One cable is attached to the end \(A\). The other cable is attached to the point \(C\) on the girder, where \(A C = 90 \mathrm {~cm}\), as shown in Figure 3. The girder is modelled as a uniform rod, and the cables as light inextensible strings. Given that the tension in the cable at \(C\) is twice the tension in the cable at \(A\), find

1. the tension in the cable at \(A\),
2. show that \(A B = 120 \mathrm {~cm}\). A small load of weight \(W\) newtons is attached to the girder at \(B\). The load is modelled as a particle. The girder remains in equilibrium in a horizontal position. The tension in the cable at \(C\) is now three times the tension in the cable at \(A\).
3. Find the value of \(W\).

3. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has length 1.5 m and mass 8 kg . A particle of mass \(m \mathrm {~kg}\) is attached to the rod at \(B\). The rod is supported at the point \(C\), where \(A C = 0.9 \mathrm {~m}\), and the system is in equilibrium with \(A B\) horizontal, as shown in Figure 2.

1. Show that \(m = 2\). A particle of mass 5 kg is now attached to the rod at \(A\) and the support is moved from \(C\) to a point \(D\) of the rod. The system, including both particles, is again in equilibrium with \(A B\) horizontal.
2. Find the distance \(A D\).

6. \begin{figure}[h] \end{figure} A plank \(A B\) has mass 12 kg and length 2.4 m . A load of mass 8 kg is attached to the plank at the point \(C\), where \(A C = 0.8 \mathrm {~m}\). The loaded plank is held in equilibrium, with \(A B\) horizontal, by two vertical ropes, one attached at \(A\) and the other attached at \(B\), as shown in Figure 2. The plank is modelled as a uniform rod, the load as a particle and the ropes as light inextensible strings.

1. Find the tension in the rope attached at \(B\). The plank is now modelled as a non-uniform rod. With the new model, the tension in the rope attached at \(A\) is 10 N greater than the tension in the rope attached at \(B\).
2. Find the distance of the centre of mass of the plank from \(A\).

2. \begin{figure}[h] \end{figure} A non-uniform rod \(A B\) has length 3 m and mass 4.5 kg . The rod rests in equilibrium, in a horizontal position, on two smooth supports at \(P\) and at \(Q\), where \(A P = 0.8 \mathrm {~m}\) and \(Q B = 0.6 \mathrm {~m}\), as shown in Figure 1. The centre of mass of the rod is at \(G\). Given that the magnitude of the reaction of the support at \(P\) on the rod is twice the magnitude of the reaction of the support at \(Q\) on the rod, find

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

6. \begin{figure}[h] \end{figure} A non-uniform beam \(A D\) has weight \(W\) newtons and length 4 m . It is held in equilibrium in a horizontal position by two vertical ropes attached to the beam. The ropes are attached to two points \(B\) and \(C\) on the beam, where \(A B = 1 \mathrm {~m}\) and \(C D = 1 \mathrm {~m}\), as shown in Figure 3. The tension in the rope attached to \(C\) is double the tension in the rope attached to \(B\). The beam is modelled as a rod and the ropes are modelled as light inextensible strings.

1. Find the distance of the centre of mass of the beam from \(A\). A small load of weight \(k W\) newtons is attached to the beam at \(D\). The beam remains in equilibrium in a horizontal position. The load is modelled as a particle. Find
2. an expression for the tension in the rope attached to \(B\), giving your answer in terms of \(k\) and \(W\),
3. the set of possible values of \(k\) for which both ropes remain taut.

4. \begin{figure}[h] \end{figure} A beam \(A B\) has weight \(W\) newtons and length 4 m . The beam is held in equilibrium in a horizontal position by two vertical ropes attached to the beam. One rope is attached to \(A\) and the other rope is attached to the point \(C\) on the beam, where \(A C = d\) metres, as shown in Figure 3. The beam is modelled as a uniform rod and the ropes as light inextensible strings. The tension in the rope attached at \(C\) is double the tension in the rope attached at \(A\).

1. Find the value of \(d\). A small load of weight \(k W\) newtons is attached to the beam at \(B\). The beam remains in equilibrium in a horizontal position. The load is modelled as a particle. The tension in the rope attached at \(C\) is now four times the tension in the rope attached at \(A\).
2. Find the value of \(k\).

5. \begin{figure}[h] \end{figure} A beam \(A B\) has length 5 m and mass 25 kg . The beam is suspended in equilibrium in a horizontal position by two vertical ropes. One rope is attached to the beam at \(A\) and the other rope is attached to the point \(C\) on the beam where \(C B = 0.5 \mathrm {~m}\), as shown in Figure 3. A particle \(P\) of mass 60 kg is attached to the beam at \(B\) and the beam remains in equilibrium in a horizontal position. The beam is modelled as a uniform rod and the ropes are modelled as light strings.

1. Find
   1. the tension in the rope attached to the beam at \(A\),
   2. the tension in the rope attached to the beam at \(C\). Particle \(P\) is removed and replaced by a particle \(Q\) of mass \(M \mathrm {~kg}\) at \(B\). Given that the beam remains in equilibrium in a horizontal position,
2. find
   1. the greatest possible value of \(M\),
   2. the greatest possible tension in the rope attached to the beam at \(C\).

6. A non-uniform plank \(A B\) has length 6 m and mass 30 kg . The plank rests in equilibrium in a horizontal position on supports at the points \(S\) and \(T\) of the plank where \(A S = 0.5 \mathrm {~m}\) and \(T B = 2 \mathrm {~m}\). When a block of mass \(M \mathrm {~kg}\) is placed on the plank at \(A\), the plank remains horizontal and in equilibrium and the plank is on the point of tilting about \(S\). When the block is moved to \(B\), the plank remains horizontal and in equilibrium and the plank is on the point of tilting about \(T\). The distance of the centre of mass of the plank from \(A\) is \(d\) metres. The block is modelled as a particle and the plank is modelled as a non-uniform rod. Find

1. the value of \(d\),
2. the value of \(M\).
   

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8. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\) have masses 1.5 kg and 3 kg respectively. The particles are attached to the ends of a light inextensible string. Particle \(P\) is held at rest on a fixed rough horizontal table. The coefficient of friction between \(P\) and the table is \(\frac { 1 } { 5 }\). The string is parallel to the table and passes over a small smooth light pulley which is fixed at the edge of the table. Particle \(Q\) hangs freely at rest vertically below the pulley, as shown in Figure 3. Particle \(P\) is released from rest with the string taut and slides along the table. Assuming that \(P\) has not reached the pulley, find

1. the tension in the string during the motion,
2. the magnitude and direction of the resultant force exerted on the pulley by the string.

4. A uniform plank \(A B\) has weight 80 N and length \(x\) metres. The plank rests in equilibrium horizontally on two smooth supports at \(A\) and \(C\), where \(A C = 2 \mathrm {~m}\), as shown in Fig. 2. A rock of weight 20 N is placed at \(B\) and the plank remains in equilibrium. The reaction on the plank at \(C\) has magnitude 90 N . The plank is modelled as a rod and the rock as a particle.

1. Find the value of \(x\).
2. State how you have used the model of the rock as a particle. The support at \(A\) is now moved to a point \(D\) on the plank and the plank remains in equilibrium with the rock at \(B\). The reaction on the plank at \(C\) is now three times the reaction at \(D\).
3. Find the distance \(A D\).

6. \begin{figure}[h] \end{figure} A plank \(A B\) rests in equilibrium against a fixed horizontal pole. The plank has length 4 m and weight 20 N and rests on the pole at \(C\), where \(A C = 2.5 \mathrm {~m}\). The end \(A\) of the plank rests on rough horizontal ground and \(A B\) makes an angle \(\theta\) with the ground, as shown in
Figure 2. The coefficient of friction between the plank and the ground is \(\frac { 1 } { 4 }\).
The plank is modelled as a uniform rod and the pole as a rough horizontal peg that is perpendicular to the vertical plane containing \(A B\). Given that \(\cos \theta = \frac { 4 } { 5 }\) and that the friction is limiting at both \(A\) and \(C\),

1. find the magnitude of the normal reaction on the plank at \(C\),
2. find the coefficient of friction between the plank and the pole.

6. \begin{figure}[h] \end{figure} A uniform rod, \(A B\), of weight \(W\) and length \(8 a\), rests in equilibrium with the end \(A\) on rough horizontal ground. The rod rests on a smooth cylinder. The cylinder is fixed to the ground with its axis horizontal. The point of contact between the rod and the cylinder is \(C\), where \(A C = 7 a\), as shown in Figure 4. The rod is resting in a vertical plane that is perpendicular to the axis of the cylinder. The rod makes an angle \(\alpha\) with the horizontal .

1. Show that the normal reaction of the ground on the rod at \(A\) has $$\text { magnitude } W \left( 1 - \frac { 4 } { 7 } \cos ^ { 2 } \alpha \right)$$ Given that the coefficient of friction between the rod and the ground is \(\mu\) and that \(\cos \alpha = \frac { 3 } { \sqrt { 10 } }\)
2. find the range of possible values of \(\mu\).
   \section*{\textbackslash section*\{Question 6 continued\}} \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-19_147_142_2606_1816}

2. \begin{figure}[h] \end{figure} The uniform lamina \(A B C\) has sides \(A B = A C = 13 a\) and \(B C = 10 a\). The lamina is freely suspended from \(A\). A horizontal force of magnitude \(F\) is applied to the lamina at \(B\), as shown in Figure 1. The line of action of the force lies in the vertical plane containing the lamina. The lamina is in equilibrium with \(A B\) vertical. The weight of the lamina is \(W\). Find \(F\) in terms of \(W\).
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6. \begin{figure}[h] \end{figure} A ladder \(A B\) has length 6 m and mass 30 kg . The ladder rests in equilibrium at \(60 ^ { \circ }\) to the horizontal with the end \(A\) on rough horizontal ground and the end \(B\) against a smooth vertical wall, as shown in Figure 3. A man of mass 70 kg stands on the ladder at the point \(C\), where \(A C = 2 \mathrm {~m}\), and the ladder remains in equilibrium. The ladder is modelled as a uniform rod in a vertical plane perpendicular to the wall. The man is modelled as a particle.

1. Find the magnitude of the force exerted on the ladder by the ground. The man climbs further up the ladder. When he is at the point \(D\) on the ladder, the ladder is about to slip. Given that the coefficient of friction between the ladder and the ground is 0.4
2. find the distance \(A D\).
3. State how you have used the modelling assumption that the ladder is a rod.

5. A smooth solid hemisphere is fixed with its flat surface in contact with rough horizontal ground. The hemisphere has centre \(O\) and radius \(5 a\).
A uniform rod \(A B\), of length \(16 a\) and weight \(W\), rests in equilibrium on the hemisphere with end \(A\) on the ground. The rod rests on the hemisphere at the point \(C\), where \(A C = 12 a\) and angle \(C A O = \alpha\), as shown in Figure 1. Points \(A , C , B\) and \(O\) all lie in the same vertical plane.

1. Explain why \(A O = 13 a\) The normal reaction on the rod at \(C\) has magnitude \(k W\)
2. Show that \(k = \frac { 8 } { 13 }\) The resultant force acting on the rod at \(A\) has magnitude \(R\) and acts upwards at \(\theta ^ { \circ }\) to the horizontal.
3. Find
   1. an expression for \(R\) in terms of \(W\)
   2. the value of \(\theta\) (8) 5 \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{0762451f-b951-4d66-9e01-61ecb7b30d95-16_426_1001_125_475}
      \end{figure} . T a and angle \(C A O = \alpha\), as shown in Figure 1.
      Points \(A , C , B\) and \(O\) all lie in the same vertical plane.
      1. Explain why \(A O = 13 a\)

6. Figure 3 A uniform pole \(A B\), of weight 50 N and length 6 m , has a particle of weight \(W\) newtons attached at its end \(B\). The pole has its end \(A\) freely hinged to a vertical wall.
A light rod holds the particle and pole in equilibrium with the pole at \(60 ^ { \circ }\) to the wall. One end of the light rod is attached to the pole at \(C\), where \(A C = 4 \mathrm {~m}\).
The other end of the light rod is attached to the wall at the point \(D\).
The point \(D\) is vertically below \(A\) with \(A D = 4 \mathrm {~m}\), as shown in Figure 3.
The pole and the light rod lie in a vertical plane which is perpendicular to the wall.
The pole is modelled as a rod.
Given that the thrust in the light rod is \(60 \sqrt { 3 } \mathrm {~N}\),

1. show that \(W = 15\)
2. find the magnitude of the resultant force acting on the pole at \(A\).

7. A non-uniform rod \(A B\) has length 6 m and mass 8 kg . The rod rests in equilibrium, in a horizontal position, on two smooth supports at \(C\) and at \(D\), where \(A C = 1 \mathrm {~m}\) and \(D B = 1 \mathrm {~m}\), as shown in Figure 3. The magnitude of the reaction between the rod and the support at \(D\) is twice the magnitude of the reaction between the rod and the support at \(C\). The centre of mass of the rod is at \(G\), where \(A G = x \mathrm {~m}\).

1. Show that \(x = \frac { 11 } { 3 }\). The support at \(C\) is moved to the point \(F\) on the rod, where \(A F = 2 \mathrm {~m}\). A particle of mass 3 kg is placed on the rod at \(A\). The rod remains horizontal and in equilibrium. The magnitude of the reaction between the rod and the support at \(D\) is \(k\) times the magnitude of the reaction between the rod and the support at \(F\).
2. Find the value of \(k\).
   7. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{04b73f81-3316-4f26-ad98-a7be3a4b738f-20_223_1262_127_338}
   \end{figure}