3.04b Equilibrium: zero resultant moment and force

4. \begin{figure}[h] \end{figure} A ramp, \(A B\), of length 8 m and mass 20 kg , rests in equilibrium with the end \(A\) on rough horizontal ground. The ramp rests on a smooth solid cylindrical drum which is partly under the ground. The drum is fixed with its axis at the same horizontal level as \(A\). The point of contact between the ramp and the drum is \(C\), where \(A C = 5 \mathrm {~m}\), as shown in Figure 2. The ramp is resting in a vertical plane which is perpendicular to the axis of the drum, at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 7 } { 24 }\) The ramp is modelled as a uniform rod.

1. Explain why the reaction from the drum on the ramp at point \(C\) acts in a direction which is perpendicular to the ramp.
2. Find the magnitude of the resultant force acting on the ramp at \(A\). The ramp is still in equilibrium in the position shown in Figure 2 but the ramp is not now modelled as being uniform. Given that the centre of mass of the ramp is assumed to be closer to \(A\) than to \(B\),
3. state how this would affect the magnitude of the normal reaction between the ramp and the drum at \(C\).

4. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has mass \(M\) and length \(2 a\) A particle of mass \(2 M\) is attached to the rod at the point \(C\), where \(A C = 1.5 a\) The rod rests with its end \(A\) on rough horizontal ground.
The rod is held in equilibrium at an angle \(\theta\) to the ground by a light string that is attached to the end \(B\) of the rod. The string is perpendicular to the rod, as shown in Figure 2.

1. Explain why the frictional force acting on the rod at \(A\) acts horizontally to the right on the diagram. The tension in the string is \(T\)
2. Show that \(T = 2 M g \cos \theta\) Given that \(\cos \theta = \frac { 3 } { 5 }\)
3. show that the magnitude of the vertical force exerted by the ground on the rod at \(A\) is \(\frac { 57 M g } { 25 }\) The coefficient of friction between the rod and the ground is \(\mu\) Given that the rod is in limiting equilibrium,
4. show that \(\mu = \frac { 8 } { 19 }\)

6. \begin{figure}[h] \end{figure} A \(\operatorname { rod } A B\) has mass \(M\) and length \(2 a\).
The rod has its end \(A\) on rough horizontal ground and its end \(B\) against a smooth vertical wall. The rod makes an angle \(\theta\) with the ground, as shown in Figure 3.
The rod is at rest in limiting equilibrium.

1. State the direction (left or right on Figure 3 above) of the frictional force acting on the \(\operatorname { rod }\) at \(A\). Give a reason for your answer. The magnitude of the normal reaction of the wall on the rod at \(B\) is \(S\).
   In an initial model, the rod is modelled as being uniform.
   Use this initial model to answer parts (b), (c) and (d).
2. By taking moments about \(A\), show that $$S = \frac { 1 } { 2 } M g \cot \theta$$ The coefficient of friction between the rod and the ground is \(\mu\) Given that \(\tan \theta = \frac { 3 } { 4 }\)
3. find the value of \(\mu\)
4. find, in terms of \(M\) and \(g\), the magnitude of the resultant force acting on the rod at \(A\). In a new model, the rod is modelled as being non-uniform, with its centre of mass closer to \(B\) than it is to \(A\). A new value for \(S\) is calculated using this new model, with \(\tan \theta = \frac { 3 } { 4 }\)
5. State whether this new value for \(S\) is larger, smaller or equal to the value that \(S\) would take using the initial model. Give a reason for your answer.

6. \begin{figure}[h] \end{figure} Figure 5 shows a uniform rod \(A B\) of mass \(M\) and length \(2 a\).

• the rod has its end \(A\) on rough horizontal ground
• the rod rests in equilibrium against a small smooth fixed horizontal peg \(P\)
• the point \(C\) on the rod, where \(A C = 1.5 a\), is the point of contact between the rod and the peg
• the rod is at an angle \(\theta\) to the ground, where \(\tan \theta = \frac { 4 } { 3 }\)

The rod lies in a vertical plane perpendicular to the peg.
The magnitude of the normal reaction of the peg on the rod at \(C\) is \(S\).

1. Show that \(S = \frac { 2 } { 5 } M g\) The coefficient of friction between the rod and the ground is \(\mu\).
   Given that the rod is in limiting equilibrium,
2. find the value of \(\mu\).

4. \begin{figure}[h] \end{figure} A ladder \(A B\) has mass \(M\) and length \(6 a\).
The end \(A\) of the ladder is on rough horizontal ground.
The ladder rests against a fixed smooth horizontal rail at the point \(C\).
The point \(C\) is at a vertical height \(4 a\) above the ground.
The vertical plane containing \(A B\) is perpendicular to the rail.
The ladder is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 4 } { 5 }\), as shown in Figure 1.
The coefficient of friction between the ladder and the ground is \(\mu\).
The ladder rests in limiting equilibrium.
The ladder is modelled as a uniform rod.
Using the model,

1. show that the magnitude of the force exerted on the ladder by the rail at \(C\) is \(\frac { 9 M g } { 25 }\)
2. Hence, or otherwise, find the value of \(\mu\).

3. \begin{figure}[h] \end{figure} A beam \(A B\) has mass \(m\) and length \(2 a\).
The beam rests in equilibrium with \(A\) on rough horizontal ground and with \(B\) against a smooth vertical wall. The beam is inclined to the horizontal at an angle \(\theta\), as shown in Figure 2.
The coefficient of friction between the beam and the ground is \(\mu\) The beam is modelled as a uniform rod resting in a vertical plane that is perpendicular to the wall. Using the model,

1. show that \(\mu \geqslant \frac { 1 } { 2 } \cot \theta\) A horizontal force of magnitude \(k m g\), where \(k\) is a constant, is now applied to the beam at \(A\). This force acts in a direction that is perpendicular to the wall and towards the wall.
   Given that \(\tan \theta = \frac { 5 } { 4 } , \mu = \frac { 1 } { 2 }\) and the beam is now in limiting equilibrium,
2. use the model to find the value of \(k\).

7 A rod of length 2 m hangs vertically in equilibrium. Parallel horizontal forces of 30 N and 50 N are applied to the top and bottom and the rod is held in place by a horizontal force \(F \mathrm {~N}\) applied \(x \mathrm {~m}\) below the top of the rod as shown in Fig. 7. \begin{figure}[h] \end{figure}

1. Find the value of \(F\).
2. Find the value of \(x\).

4 Fig. 4 shows a uniform beam of mass 4 kg and length 2.4 m resting on two supports P and Q . P is at one end of the beam and Q is 0.3 m from the other end.
Determine whether a person of mass 50 kg can tip the beam by standing on it. \begin{figure}[h] \end{figure}

6 A shelf consists of a horizontal uniform plank AB of length 0.8 m and mass 5 kg with light inextensible vertical strings attached at each end. A stack of bricks each of mass 2.3 kg is placed on the plank as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{9dd6fc6d-b51e-4a73-ace5-d26a7558032c-06_397_734_641_242}

1. Explain the meaning of each of the following modelling assumptions.
   Either of the strings will break if the tension exceeds 75 N.
2. Find the greatest number of bricks that can be placed at the centre of the plank without breaking the strings.
3. Find an expression for the moment about A of the weight of a stack of \(n\) bricks when the stack is at a distance of \(x \mathrm {~m}\) from A . State the units for your answer.
4. Calculate the greatest distance from A that the largest stack of bricks can be placed without a string breaking.

4 A ruler PQRS is a uniform rectangular lamina with mass 20 grams. The length of PQ is 30 cm and the length of PS is 4 cm . The ruler is attached at P to a smooth hinge and held with S vertically below P by a horizontal force of magnitude \(F \mathrm {~N}\) as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{8eeff88d-8b05-43c6-86a5-bd82221c0bea-04_303_1495_1363_239}

1. Calculate the value of \(F\).
2. Explain what would happen to the lamina if the force at S were removed.

7 A rectangular book ABCD rests on a smooth horizontal table. The length of AB is 28 cm and the length of AD is 18 cm . The following five forces act on the book, as shown in the diagram.

• 4 N at A in the direction AD
• 5 N at B in the direction BC
• 3 N at B in the direction BA
• 9 N at D in the direction DA
• 3 N at D in the direction DC \includegraphics[max width=\textwidth, alt={}, center]{1d0ca3d5-6529-435f-a0b8-50ea4859adde-06_663_830_774_242}
  1. Show that the resultant of the forces acting on the book has zero magnitude.
  2. Find the total moment of the forces about the centre of the book. Give your answer in Nm .
  3. Describe how the book will move under the action of these forces.

6 A uniform ruler AB has mass 28 g and length 30 cm . As shown in Fig. 6, the ruler is placed on a horizontal table so that it overhangs a point C at the edge of the table by 25 cm . A downward force of \(F \mathrm {~N}\) is applied at A . This force just holds the ruler in equilibrium so that the contact force between the table and the ruler acts through C . \begin{figure}[h] \end{figure}

1. Complete the force diagram in the Printed Answer Booklet, labelling the forces and all relevant distances.
2. Calculate the value of \(F\). Answer all the questions.
   Section B (78 marks)

5 ABCD is a rectangular lamina in which AB is 30 cm and AD is 10 cm .
Three forces of 20 N and one force of 30 N act along the sides of the lamina as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{4fac72cb-85cb-48d9-8817-899ef3f80a0f-05_558_981_1263_233} An additional force \(F \mathrm {~N}\) is also applied at right angles to AB to a point on the edge \(\mathrm { AB } x \mathrm {~cm}\) from A .

1. Given that the lamina is in equilibrium, calculate the values of \(F\) and \(x\). The point of application of the force \(F \mathrm {~N}\) is now moved to B , but the magnitude and direction of the force remain the same.
2. Explain the effect of the new system of forces on the lamina.

15 Fig. 15 shows a uniform shelf AB of weight \(W \mathrm {~N}\).
The shelf is 180 cm long and rests on supports at points C and D . Point C is 30 cm from A and point D is 60 cm from B .
side view \begin{figure}[h] \end{figure} Determine the range of positions a point load of \(3 W\) could be placed on the shelf without the shelf tipping. \section*{Copyright Information:} }{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, The Triangle Building, Shaftesbury Road, Cambridge CB2 8EA.
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5 The cover of a children's book is modelled as being a uniform lamina \(L . L\) occupies the region bounded by the \(x\)-axis, the curve \(\mathrm { y } = 6 + \sin \mathrm { x }\) and the lines \(x = 0\) and \(x = 5\) (see Fig. 5.1). The centre of mass of \(L\) is at the point ( \(\mathrm { x } , \mathrm { y }\) ). \begin{figure}[h] \end{figure}

1. Show that \(\bar { X } = 2.36\), correct to 3 significant figures.
2. Find \(\bar { y }\), giving your answer correct to 3 significant figures. The cover of the book weighs 6 N . \(A\) is the point on the cover with coordinates \(( 3 , \bar { y } )\) and \(B\) is the point on the cover with coordinates \(( 5 , \bar { y } )\). A small badge of weight 2 N is attached to the cover at \(A\). The side of \(L\) along the \(y\)-axis is attached to the rest of the book and the book is placed on a rough horizontal plane. The attachment of the cover to the book is modelled as a hinge. The cover is held in equilibrium at an angle of \(\frac { 1 } { 3 } \pi\) radians to the horizontal by a force of magnitude \(P N\) acting at \(B\) perpendicular to the cover (see Fig. 5.2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{831ba5da-df19-43bb-b163-02bbddb4e2b8-4_444_899_1889_246} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure}
3. State two additional modelling assumptions, one about the attachment of the cover and one about the badge, which are necessary to allow the value of \(P\) to be determined.
4. Using the modelling assumptions, determine the value of \(P\) giving your answer correct to 3 significant figures.

1. \includegraphics[max width=\textwidth, alt={}, center]{31efa627-5114-4797-9d46-7f1311c18ff8-1_490_254_354_347} A vertical pole \(X Y\), of length 2.5 m and mass 0.5 kg , has its lower end \(Y\) free to move in a smooth horizontal groove. Forces of magnitude 0.2 N and 0.14 N are applied to the pole horizontally at the points \(V\) and \(W\) respectively, where \(X V = 1.5 \mathrm {~m}\) and \(V W = 0.5 \mathrm {~m}\).
Find, to the nearest cm , the distance from \(X\) at which an opposing horizontal force must be applied to keep the pole at rest in equilibrium, and state the magnitude of this force.

3. A lump of clay, of mass 0.8 kg , is attached to the end \(A\) of a light \(\operatorname { rod } A B\), which is pivoted at its other end \(B\) so that it can rotate smoothly in a vertical plane. A force is applied to \(A\) at an angle of \(60 ^ { \circ }\) to the vertical, as shown, the magnitude \(F \mathrm {~N}\) of this force being just enough to hold the lump of clay in equilibrium with \(A B\) inclined \includegraphics[max width=\textwidth, alt={}, center]{31efa627-5114-4797-9d46-7f1311c18ff8-1_309_335_1453_1590}
at an angle of \(30 ^ { \circ }\) to the upward vertical.

1. Find the value of \(F\),
2. Find the magnitude of the force in the \(\operatorname { rod } A B\).
3. State the modelling assumption that you have made about the lump of clay.
   (6 marks)
   (2 marks)
   (1 mark)

2. A plank of wood \(X Y\) has length \(5 a\) m and mass 5 kg . It rests on a support at \(Q\), where \(X Q = 3 a\) m . When a kitten of mass 8 kg sits on the plank at \(P\), where \(P Y = a \mathrm {~m}\), the plank just remains horizontal. By modelling the plank as a non-uniform rod and the kitten as a particle, find

1. the magnitude of the reaction at the support,
2. the distance from \(X\) to the centre of mass of the plank, in terms of \(a\).

2. \begin{figure}[h] \end{figure} Figure 2 shows a washing line suspended at either end by vertical rigid poles. A jacket of mass 0.7 kg is suspended in equilibrium part of the way along the line. The sections of the washing line on either side of the jacket make angles of \(35 ^ { \circ }\) and \(40 ^ { \circ }\) with the horizontal.

1. Find the tension in the washing line on each side of the jacket.
2. Explain why, in practice, the angles are likely to be very similar in value.

4. The force \(\mathbf { F } _ { \mathbf { 1 } } = ( 5 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) acts at the point \(A\) on a lamina where the position vector of \(A\), relative to a fixed origin \(O\), is \(( 3 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }\).

1. Calculate the magnitude and the sense of the moment of the force about \(O\). Another force \(\mathbf { F } _ { 2 } = ( p \mathbf { i } + q \mathbf { j } )\), acts at the point \(B\) with position vector ( \({ } ^ { - } \mathbf { i } + 4 \mathbf { j }\) ) m so that the resultant moment of the two forces, \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\), about \(O\) is zero. Given also that the moment of \(\mathbf { F } _ { 2 }\) about \(A\) is 34 Ns in a clockwise sense,
2. find the values of \(p\) and \(q\).

3. \begin{figure}[h] \end{figure} Figure 2 shows 4 points \(A , B , C\) and \(D\) arranged such that they form the corners of a square of side 2 m . Forces of \(5 \mathrm {~N} , 3 \mathrm {~N} , 2 \mathrm {~N}\) and 4 N act in the directions \(\overrightarrow { A B } , \overrightarrow { B C } , \overrightarrow { D C }\) and \(\overrightarrow { D A }\) respectively.

1. Calculate the magnitude and sense of the resultant moment about \(A\). An additional force of magnitude \(X\) Newtons is added in the direction \(\overrightarrow { C A }\). The resultant moment of all the forces about \(D\) is now zero.
2. Find, in the form \(k \sqrt { } 2\), the value of \(X\).

1. \begin{figure}[h] \end{figure} Figure 1 shows a non-uniform beam \(A B\) of mass 10 kg and length 6 m resting in a horizontal position on a single support 2 m from \(A\). The beam is supported at \(B\) by a vertical string. Given that the magnitude of the tension in the string is 1.5 times the magnitude of the reaction at the support, find the distance of the centre of mass of the beam from \(A\).
(6 marks)

6. \begin{figure}[h] \end{figure} Figure 2 shows a picnic bench of mass 20 kg which consists of a horizontal plank of wood of length 2 m resting on two supports, each of which is 0.6 m from the centre of the plank. Luigi sits on the bench at its midpoint and his mother Maria sits at one end. Their masses are 40 kg and 75 kg respectively. By modelling the bench as a uniform rod and Luigi and Maria as particles,

1. find the reaction at each of the two supports. Luigi moves to sit closer to his mother.
2. Find how close Luigi can get to his mother before the reaction at one of the supports becomes zero.
3. Explain the significance of a zero reaction at one of the supports.

6. \begin{figure}[h] \end{figure} Figure 2 shows a bench of length 3 m being used in a gymnasium.
The bench rests horizontally on two identical supports which are 2.2 m apart and equidistant from the middle of the bench.

1. Explain why it is reasonable to model the bench as a uniform rod. When a gymnast of mass 55 kg stands on the bench 0.1 m from one end, the bench is on the point of tilting.
2. Find the mass of the bench. The first gymnast dismounts and a second gymnast of mass 33 kg steps onto the bench at a distance of 0.4 m from its centre.
3. Show that the magnitudes of the reaction forces on the two supports are in the ratio \(5 : 3\).
   (6 marks)

3 A uniform ladder of length 4 metres and mass 20 kg rests in equilibrium with its foot, \(A\), on a rough horizontal floor and its top leaning against a smooth vertical wall. The vertical plane containing the ladder is perpendicular to the wall and the angle between the ladder and the floor is \(60 ^ { \circ }\). A man of mass 80 kg is standing at point \(C\) on the ladder. With the man in this position, the ladder is on the point of slipping. The coefficient of friction between the ladder and the floor is 0.4 . The man may be modelled as a particle at \(C\). \includegraphics[max width=\textwidth, alt={}, center]{1bc18163-b20e-4dc6-bd35-496efec8dc73-3_567_448_708_788}

1. Draw a diagram to show the forces acting on the ladder.
2. Show that the magnitude of the frictional force between the ladder and the ground is 392 N .
3. Find the distance \(A C\).