3.04b Equilibrium: zero resultant moment and force

6 A light elastic string has natural length 4 m and modulus of elasticity 2 N . One end of the string is attached to a fixed point \(O\) of a smooth plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle \(P\) of mass \(0.1 \mathrm {~kg} . P\) is held at rest at \(O\) and then released. The speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the extension of the string is \(x \mathrm {~m}\).

1. Show that \(v ^ { 2 } = 45 - 5 ( x - 1 ) ^ { 2 }\). Hence find
2. the distance of \(P\) from \(O\) when \(P\) is at its lowest point,
3. the maximum speed of \(P\).

6 \includegraphics[max width=\textwidth, alt={}, center]{fe5c198d-5d05-4241-98f5-894ba92f7afe-4_447_736_269_701} \(P\) is the vertex of a uniform solid cone of mass 5 kg , and \(O\) is the centre of its base. Strings are attached to the cone at \(P\) and at \(O\). The cone hangs in equilibrium with \(P O\) horizontal and the strings taut. The strings attached at \(P\) and \(O\) make angles of \(\theta ^ { \circ }\) and \(20 ^ { \circ }\), respectively, with the vertical (see diagram, which shows a cross-section).

1. By taking moments about \(P\) for the cone, find the tension in the string attached at \(O\).
2. Find the value of \(\theta\) and the tension in the string attached at \(P\).

4 \includegraphics[max width=\textwidth, alt={}, center]{af7d1fc8-5552-48b8-a359-895b2b5d3d6c-3_560_894_258_628} A uniform beam \(A B\) has length 2 m and weight 70 N . The beam is hinged at \(A\) to a fixed point on a vertical wall, and is held in equilibrium by a light inextensible rope. One end of the rope is attached to the wall at a point 1.7 m vertically above the hinge. The other end of the rope is attached to the beam at a point 0.8 m from \(A\). The rope is at right angles to \(A B\). The beam carries a load of weight 220 N at \(B\) (see diagram).

1. Find the tension in the rope.
2. Find the direction of the force exerted on the beam at \(A\).

1 \includegraphics[max width=\textwidth, alt={}, center]{d1f1f036-1676-443e-b733-ca1fe79972d4-2_334_679_258_731} A non-uniform \(\operatorname { rod } A B\), of length 0.6 m and weight 9 N , has its centre of mass 0.4 m from \(A\). The end \(A\) of the rod is in contact with a rough vertical wall. The rod is held in equilibrium, perpendicular to the wall, by means of a light string attached to \(B\). The string is inclined at \(30 ^ { \circ }\) to the horizontal. The tension in the string is \(T \mathrm {~N}\) (see diagram).

1. Calculate \(T\).
2. Find the least possible value of the coefficient of friction at \(A\).

4 A uniform solid cylinder has radius 0.7 m and height \(h \mathrm {~m}\). A uniform solid cone has base radius 0.7 m and height 2.4 m . The cylinder and the cone both rest in equilibrium each with a circular face in contact with a horizontal plane. The plane is now tilted so that its inclination to the horizontal, \(\theta ^ { \circ }\), is increased gradually until the cone is about to topple.

1. Find the value of \(\theta\) at which the cone is about to topple.
2. Given that the cylinder does not topple, find the greatest possible value of \(h\). The plane is returned to a horizontal position, and the cone is fixed to one end of the cylinder so that the plane faces coincide. It is given that the weight of the cylinder is three times the weight of the cone. The curved surface of the cone is placed on the horizontal plane (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{d1f1f036-1676-443e-b733-ca1fe79972d4-3_476_1211_836_466}
3. Given that the solid immediately topples, find the least possible value of \(h\).

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 \(F\).

6 \includegraphics[max width=\textwidth, alt={}, center]{6503ebb1-5649-4ca5-9500-da4fb28009dd-3_454_1029_1379_557} \(A B C D\) is the cross-section through the centre of mass of a uniform rectangular block of weight 260 N . The lengths \(A B\) and \(B C\) are 1.5 m and 0.8 m respectively. The block rests in equilibrium with the point \(D\) on a rough horizontal floor. Equilibrium is maintained by a light rope attached to the point \(A\) on the block and the point \(E\) on the floor. The points \(E , A\) and \(B\) lie in a straight line inclined at \(30 ^ { \circ }\) to the horizontal (see diagram).

1. By taking moments about \(D\), show that the tension in the rope is 146 N , correct to 3 significant figures.
2. Given that the block is in limiting equilibrium, calculate the coefficient of friction between the block and the floor.

3 \includegraphics[max width=\textwidth, alt={}, center]{d9970ad1-a7f4-429a-bad1-43e8d114b968-2_442_789_941_676} A non-uniform \(\operatorname { rod } A B\) of length 0.5 m is freely hinged to a fixed point at \(A\). The rod is in equilibrium at an angle of \(30 ^ { \circ }\) with the horizontal with \(B\) below the level of \(A\). Equilibrium is maintained by a force of magnitude \(F\) N applied at \(B\) acting at \(45 ^ { \circ }\) above the horizontal in the vertical plane containing \(A B\). The force exerted by the hinge on the rod has magnitude 10 N and acts at an angle of \(60 ^ { \circ }\) above the horizontal (see diagram).

1. By resolving horizontally and vertically, calculate \(F\) and the weight of the rod.
2. Find the distance of the centre of mass of the rod from \(A\).

7 \includegraphics[max width=\textwidth, alt={}, center]{9daebcbe-826e-4eda-afa7-c935c6ea2bfc-10_451_574_258_781} \(A B C D\) is a uniform lamina in the shape of a trapezium which has centre of mass \(G\). The sides \(A D\) and \(B C\) are parallel and 1.8 m apart, with \(A D = 2.4 \mathrm {~m}\) and \(B C = 1.2 \mathrm {~m}\) (see diagram).

1. Show that the distance of \(G\) from \(A D\) is 0.8 m .
   The lamina is freely suspended at \(A\) and hangs in equilibrium with \(A D\) making an angle of \(30 ^ { \circ }\) with the vertical.
2. Calculate the distance \(A G\).
   With the lamina still freely suspended at \(A\) a horizontal force of magnitude 7 N acting in the plane of the lamina is applied at \(D\). The lamina is in equilibrium with \(A G\) making an angle of \(10 ^ { \circ }\) with the downward vertical.
3. Find the two possible values for the weight of the lamina.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 A light elastic string has natural length \(a\) and modulus of elasticity 4 mg . One end of the string is fixed to a point \(O\) on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the other end of the string. The particle \(P\) is projected along the surface in the direction \(O P\). When the length of the string is \(\frac { 5 } { 4 } a\), the speed of \(P\) is \(v\). When the length of the string is \(\frac { 3 } { 2 } a\), the speed of \(P\) is \(\frac { 1 } { 2 } v\).

1. Find an expression for \(v\) in terms of \(a\) and \(g\).
2. Find, in terms of \(g\), the acceleration of \(P\) when the stretched length of the string is \(\frac { 3 } { 2 } a\). \includegraphics[max width=\textwidth, alt={}, center]{7febbd80-4cbb-4b2e-b022-d6a20e7e13aa-04_552_1059_264_502} A smooth cylinder is fixed to a rough horizontal surface with its axis of symmetry horizontal. A uniform rod \(A B\), of length \(4 a\) and weight \(W\), rests against the surface of the cylinder. The end \(A\) of the rod is in contact with the horizontal surface. The vertical plane containing the rod \(A B\) is perpendicular to the axis of the cylinder. The point of contact between the rod and the cylinder is \(C\), where \(A C = 3 a\). The angle between the rod and the horizontal surface is \(\theta\) where \(\tan \theta = \frac { 3 } { 4 }\) (see diagram). The coefficient of friction between the rod and the horizontal surface is \(\frac { 6 } { 7 }\). A particle of weight \(k W\) is attached to the rod at \(B\). The rod is about to slip. The normal reaction between the rod and the cylinder is \(N\).

3. \begin{figure}[h] \end{figure} A uniform beam \(A B\) has mass 20 kg and length 6 m . The beam rests in equilibrium in a horizontal position on two smooth supports. One support is at \(C\), where \(A C = 1 \mathrm {~m}\), and the other is at the end \(B\), as shown in Figure 1. The beam is modelled as a rod.

1. Find the magnitudes of the reactions on the beam at \(B\) and at \(C\). A boy of mass 30 kg stands on the beam at the point \(D\). The beam remains in equilibrium. The magnitudes of the reactions on the beam at \(B\) and at \(C\) are now equal. The boy is modelled as a particle.
2. Find the distance \(A D\).

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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3. A plank \(A B\) has length 6 m and mass 30 kg . The point \(C\) is on the plank with \(C B = 2 \mathrm {~m}\). The plank rests in equilibrium in a horizontal position on supports at \(A\) and \(C\). Two people, each of mass 75 kg , stand on the plank. One person stands at the point \(P\) of the plank, where \(A P = x\) metres, and the other person stands at the point \(Q\) of the plank, where \(A Q = 2 x\) metres. The plank remains horizontal and in equilibrium with the magnitude of the reaction at \(C\) five times the magnitude of the reaction at \(A\). The plank is modelled as a uniform rod and each person is modelled as a particle.

1. Find the value of \(x\).
2. State two ways in which you have used the assumptions made in modelling the plank as a uniform rod.