Resultant force on lamina

5 \includegraphics[max width=\textwidth, alt={}, center]{f8961f84-c080-4407-a178-45b76f200111-3_533_698_1343_721} A uniform rectangular lamina \(A B C D\), in which \(A B = 8 a\) and \(B C = 6 a\), has mass \(M\). A uniform circular lamina of radius \(\frac { 5 } { 2 } a\) has mass \(\frac { 1 } { 3 } M\). The two laminas are fixed together in the same plane with their centres coinciding at the point \(O\) (see diagram). A particle \(P\) of mass \(\frac { 1 } { 2 } M\) is attached at \(C\). The system is free to rotate about a fixed smooth horizontal axis through \(A\) and perpendicular to the plane \(A B C D\). Show that the moment of inertia of the system about this axis is \(\frac { 2225 } { 24 } M a ^ { 2 }\). The system is released from rest with \(A C\) horizontal and \(D\) below \(A C\). Find, in the form \(k \sqrt { } \left( \frac { g } { a } \right)\), the greatest angular speed in the subsequent motion, giving the value of \(k\) correct to 3 decimal places.
[0pt] [4]

2 \includegraphics[max width=\textwidth, alt={}, center]{9c82b387-8e5e-48b9-973d-5337b4e56a66-2_536_905_520_621} A uniform lamina \(A B C\) in the shape of an isosceles triangle has weight 24 N . The perpendicular distance from \(A\) to \(B C\) is 12 cm . The lamina rests in a vertical plane in equilibrium, with the vertex \(A\) in contact with a horizontal surface. Angle \(B A C = 100 ^ { \circ }\) and \(A B\) makes an angle of \(10 ^ { \circ }\) with the horizontal. Equilibrium is maintained by a force of magnitude \(F \mathrm {~N}\) acting along \(B C\) (see diagram). Show that \(F = 8\).

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 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.

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.

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.

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

1. Fig. 1.1 and Fig. 1.2 show rigid rods with forces acting as marked. The diagrams are to scale, and in each figure the side length of a grid square is 1 metre. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-2_428_552_443_319} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-2_431_553_440_1005} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure}
   • On the copy of Fig. 1.1 in the Printed Answer Booklet, add, to scale, a force so that the overall system represents an anti-clockwise couple of magnitude 24 Nm .
   • On the copy of Fig. 1.2 in the Printed Answer Booklet, add, to scale, a force so that the overall system represents a clockwise couple of magnitude 1 Nm .
   • Fig. 1.3 shows a rectangular lamina with two coplanar forces acting as marked. Each grid square has side length 1 m .
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-2_561_761_1452_315} \captionsetup{labelformat=empty} \caption{Fig. 1.3}
   \end{figure} A third coplanar force, of magnitude \(T \mathrm {~N}\), acts at A so that the resultant force on the lamina is zero.
   1. Calculate the value of \(T\).
   2. Determine the magnitude and direction of the couple represented by this system of three forces.

3 Fig. 3.1 shows a thin rectangular frame ABCD , with part of it filled by a triangular lamina ABD . \(\mathrm { AD } = 30 \mathrm {~cm}\) and \(\mathrm { AB } = x \mathrm {~cm}\). Together they form the composite structure S . The centre of mass of \(S\) lies at a point \(M , 16.5 \mathrm {~cm}\) from \(A D\) and 11.7 cm from \(A B\). \begin{figure}[h] \end{figure} The frame and the triangular lamina are both uniform but made of different materials. The mass of the frame is 1.7 kg .

1. Show that the triangular lamina has a mass of 3.3 kg .
2. Determine the value of \(x\), correct to \(\mathbf { 3 }\) significant figures. One end of a light inextensible string is attached to S at D . The other end is attached to a fixed point on a vertical wall. For S to hang in equilibrium with AD vertical, a force of magnitude \(Q N\) is applied to S as shown in Fig. 3.2. The line of action of this force lies in the same plane as S . The string is taut and lies in the same plane as S at an angle \(\phi\) to the downward vertical. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-4_611_994_1756_242} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
3. By taking moments about D , show that \(Q = 50.5\), correct to 3 significant figures.
4. Determine, in degrees, the value of \(\phi\).

3 Fig. 3 shows a light square lamina ABCD , of side length 0.75 m , suspended vertically by wires attached to A and B so that AB is horizontal. A particle P of mass \(m \mathrm {~kg}\) is attached to the edge DC . The lamina hangs in equilibrium. \begin{figure}[h] \end{figure} The tension in the wire attached to A is 14 N and the tension in the wire attached to B is \(T \mathrm {~N}\). The wire at A makes an angle of \(25 ^ { \circ }\) with the horizontal and the wire at B makes an angle of \(60 ^ { \circ }\) with the horizontal.

1. Determine the value of \(T\).
2. Determine
   1. the value of \(m\),
   2. the distance of P from D . P is moved to the midpoint of CD . A couple is applied to the lamina so that it remains in equilibrium with AB horizontal and the tension in both wires unchanged.
3. Determine

2 A triangular lamina, ABC , is cut from a piece of thin uniform plane sheet metal. The dimensions of ABC are shown in Fig. 2. \begin{figure}[h] \end{figure} This piece of metal is freely suspended from a string attached to C and hangs in equilibrium. Calculate the angle of BC with the downward vertical, giving your answer in degrees.

2. \begin{figure}[h] \end{figure} A uniform rod of length \(28 a\) is cut into seven identical rods each of length \(4 a\). These rods are joined together to form the rigid framework \(A B C D E A\) shown in Figure 1. All seven rods lie in the same plane.
The distance of the centre of mass of the framework from \(E D\) is \(d\).

1. Show that \(d = \frac { 8 \sqrt { 3 } } { 7 } a\) The weight of the framework is \(W\).
   The framework is freely pivoted about a horizontal axis through \(C\).
   The framework is held in equilibrium in a vertical plane, with \(A C\) vertical and \(A\) below \(C\), by a horizontal force that is applied to the framework at \(A\). The force acts in the same vertical plane as the framework and has magnitude \(F\).
2. Find \(F\) in terms of \(W\).

4 Particles \(A , B\) and \(C\) of masses \(2 \mathrm {~kg} , 3 \mathrm {~kg}\) and 5 kg respectively are joined by light rigid rods to form a triangular frame. The frame is placed at rest on a horizontal plane with \(A\) at the point \(( 0,0 )\), \(B\) at the point ( \(0.6,0\) ) and \(C\) at the point ( \(0.4,0.2\) ), where distances in the coordinate system are measured in metres (see Fig. 1). \begin{figure}[h] \end{figure} \(G\), which is the centre of mass of the frame, is at the point \(( \bar { x } , \bar { y } )\).

1. - Show that \(\bar { x } = 0.38\).
   A rough plane, \(\Pi\), is inclined at an angle \(\theta\) to the horizontal where \(\sin \theta = \frac { 3 } { 5 }\). The frame is placed on \(\Pi\) with \(A B\) vertical and \(B\) in contact with \(\Pi . C\) is in the same vertical plane as \(A B\) and a line of greatest slope of \(\Pi . C\) is on the down-slope side of \(A B\). The frame is kept in equilibrium by a horizontal light elastic string whose natural length is \(l \mathrm {~m}\) and whose modulus of elasticity is \(g \mathrm {~N}\). The string is attached to \(A\) at one end and to a fixed point on \(\Pi\) at the other end (see Fig. 2). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{709f3a7a-d857-4813-98ab-de6b41a3a8dc-03_605_828_1525_248} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The coefficient of friction between \(B\) and \(\Pi\) is \(\mu\).
2. Show that \(l = 0.3\).
3. Show that \(\mu \geqslant \frac { 14 } { 27 }\).

Answer only one of the following two alternatives. **EITHER** \includegraphics{figure_11a} A uniform plane object consists of three identical circular rings, \(X\), \(Y\) and \(Z\), enclosed in a larger circular ring \(W\). Each of the inner rings has mass \(m\) and radius \(r\). The outer ring has mass \(3m\) and radius \(R\). The centres of the inner rings lie at the vertices of an equilateral triangle of side \(2r\). The outer ring touches each of the inner rings and the rings are rigidly joined together. The fixed axis \(AB\) is the diameter of \(W\) that passes through the centre of \(X\) and the point of contact of \(Y\) and \(Z\) (see diagram). It is given that \(R = \left(1 + \frac{2}{3}\sqrt{3}\right)r\).

1. Show that the moment of inertia of the object about \(AB\) is \(\left(7 + 2\sqrt{3}\right)mr^2\). [8]

The line \(CD\) is the diameter of \(W\) that is perpendicular to \(AB\). A particle of mass \(9m\) is attached to \(D\). The object is now held with its plane horizontal. It is released from rest and rotates freely about the fixed horizontal axis \(AB\).

1. Find, in terms of \(g\) and \(r\), the angular speed of the object when it has rotated through \(60°\). [6]

**OR** Fish of a certain species live in two separate lakes, \(A\) and \(B\). A zoologist claims that the mean length of fish in \(A\) is greater than the mean length of fish in \(B\). To test his claim, he catches a random sample of 8 fish from \(A\) and a random sample of 6 fish from \(B\). The lengths of the 8 fish from \(A\), in appropriate units, are as follows. $$15.3 \quad 12.0 \quad 15.1 \quad 11.2 \quad 14.4 \quad 13.8 \quad 12.4 \quad 11.8$$ Assuming a normal distribution, find a 95% confidence interval for the mean length of fish in \(A\). [5] The lengths of the 6 fish from \(B\), in the same units, are as follows. $$15.0 \quad 10.7 \quad 13.6 \quad 12.4 \quad 11.6 \quad 12.6$$ Stating any assumptions that you make, test at the 5% significance level whether the mean length of fish in \(A\) is greater than the mean length of fish in \(B\). [7] Calculate a 95% confidence interval for the difference in the mean lengths of fish from \(A\) and from \(B\). [2]

\includegraphics{figure_4} \(ABCDEF\) is the cross-section through the centre of mass of a uniform solid prism. \(ABCF\) is a rectangle in which \(AB = CF = 1.6\) m, and \(BC = AF = 0.4\) m. \(CDE\) is a triangle in which \(CD = 1.8\) m, \(CE = 0.4\) m, and angle \(DCE = 90°\). The prism stands on a rough horizontal surface. A horizontal force of magnitude \(T\) N acts at \(B\) in the direction \(CB\) (see diagram). The prism is in equilibrium.

1. Show that the distance of the centre of mass of the prism from \(AB\) is \(0.488\) m. [4]
2. Given that the weight of the prism is \(100\) N, find the greatest and least possible values of \(T\). [3]

\includegraphics{figure_4} The diagram shows the cross-section \(ABCD\) through the centre of mass of a uniform solid prism. \(AB = 0.9\) m, \(BC = 2a\) m, \(AD = a\) m and angle \(ABC =\) angle \(BAD = 90°\).

1. Calculate the distance of the centre of mass of the prism from \(AD\). [2]
2. Express the distance of the centre of mass of the prism from \(AB\) in terms of \(a\). [2]

The prism has weight 18 N and rests in equilibrium on a rough horizontal surface, with \(AD\) in contact with the surface. A horizontal force of magnitude 6 N is applied to the prism. This force acts through the centre of mass in the direction \(BC\).

1. Given that the prism is on the point of toppling, calculate \(a\). [3]

\includegraphics{figure_6} Fig. 1 shows the cross-section \(ABCDE\) through the centre of mass \(G\) of a uniform prism. The cross-section consists of a rectangle \(ABCF\) from which a triangle \(DEF\) has been removed; \(AB = 0.6\text{ m}\), \(BC = 0.7\text{ m}\) and \(DF = EF = 0.3\text{ m}\).

1. Show that the distance of \(G\) from \(BC\) is \(0.276\text{ m}\), and find the distance of \(G\) from \(AB\). [5]
2. The prism is placed with \(CD\) on a rough horizontal surface. A force of magnitude \(2\text{ N}\) acting in the plane of the cross-section is applied to the prism. The line of action of the force passes through \(G\) and is perpendicular to \(DE\) (see Fig. 2). The prism is on the point of toppling about the edge through \(D\). Calculate the weight of the prism. [3]

\includegraphics{figure_1} Figure 1 shows an aerial view of a revolving door consisting of 4 panels, each of width 1.2 m and set at 90° intervals, which are free to rotate about a fixed central column, \(O\). The revolving door is situated outside a lecture theatre and four students are trying to push the door. Two of the students are pushing panels \(OA\) and \(OD\) clockwise (as viewed from above) with horizontal forces of 70 N and 90 N respectively, whilst the other two are pushing panels \(OB\) and \(OC\) anti-clockwise with horizontal forces of 80 N and 60 N respectively.

1. Calculate the total moment about \(O\) when the four students are pushing the panels at their outer edge, 1.2 m from \(O\). [3 marks]

The student at \(C\) moves her hand 0.2 m closer to \(O\) and the student at \(D\) moves his hand \(x\) m closer to \(O\). Given that the students all push in the same directions and with the same forces as in part (a), and that the door is in equilibrium,

1. Find the value of \(x\). [5 marks]

\includegraphics{figure_4} A rigid lamina of negligible mass is in the form of a rhombus ABCD, where AC = 6 m and BD = 8 m. Forces of magnitude 2 N, 4 N, 3 N and 5 N act along its sides AB, BC, CD and DA, respectively, as shown in the diagram. A further force F N, acting at A, and a couple of magnitude G N m are also applied to the lamina so that it is in equilibrium.

1. Determine the magnitude and direction of F. [4]
2. Determine the value of G. [2]

The vertices of a triangular lamina, which is in the \(x\)–\(y\) plane, are at the origin O and the points A\((2, 3)\) and B\((-2, 1)\). Forces \(2\mathbf{i} + \mathbf{j}\) and \(-3\mathbf{i} + 2\mathbf{j}\) are applied to the lamina at A and B, respectively, and a force \(\mathbf{F}\), whose line of action is in the \(x\)–\(y\) plane, is applied at O. The three forces form a couple.

1. Determine the magnitude and the direction of \(\mathbf{F}\). [4]
2. Determine the magnitude and direction of the additional couple that must be applied to the lamina in order to keep it in equilibrium. [3]