Framework or multiple rod structures

6 \includegraphics[max width=\textwidth, alt={}, center]{6fe2c5e0-0496-4fb4-95d2-354b90607b5b-4_620_899_644_623} A rigid rod consists of two parts. The part \(B C\) is in the form of an arc of a circle of radius 2 m and centre \(O\), with angle \(B O C = \frac { 1 } { 4 } \pi\) radians. \(B C\) is uniform and has weight 3 N . The part \(A B\) is straight and of length 2 m ; it is uniform and has weight 4 N . The part \(A B\) of the rod is a tangent to the arc \(B C\) at \(B\). The end \(A\) of the rod is freely hinged to a fixed point of a vertical wall. The rod is held in equilibrium, with the straight part \(A B\) making an angle of \(\frac { 1 } { 4 } \pi\) radians with the wall, by means of a horizontal string attached to \(C\). The string is in the same vertical plane as the rod, and the tension in the string is \(T \mathrm {~N}\) (see diagram).

1. Show that the centre of mass \(G\) of the part \(B C\) of the rod is at a distance of 2.083 m from the wall, correct to 4 significant figures.
2. Find the value of \(T\).
3. State the magnitude of the horizontal component and the magnitude of the vertical component of the force exerted on the rod by the hinge. \includegraphics[max width=\textwidth, alt={}, center]{6fe2c5e0-0496-4fb4-95d2-354b90607b5b-5_579_1118_264_516} A particle \(A\) is released from rest at time \(t = 0\), at a point \(P\) which is 7 m above horizontal ground. At the same instant as \(A\) is released, a particle \(B\) is projected from a point \(O\) on the ground. The horizontal distance of \(O\) from \(P\) is 24 m . Particle \(B\) moves in the vertical plane containing \(O\) and \(P\), with initial speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and initial direction making an angle of \(\theta\) above the horizontal (see diagram). Write down
4. an expression for the height of \(A\) above the ground at time \(t \mathrm {~s}\),
5. an expression in terms of \(V , \theta\) and \(t\) for
   1. the horizontal distance of \(B\) from \(O\),
   2. the height of \(B\) above the ground. At time \(t = T\) the particles \(A\) and \(B\) collide at a point above the ground.
   3. Show that \(\tan \theta = \frac { 7 } { 24 }\) and that \(V T = 25\).
   4. Deduce that \(7 V ^ { 2 } > 3125\).

4 \includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-3_375_627_1448_758} Uniform rods \(A B , A C\) and \(B C\) have lengths \(3 \mathrm {~m} , 4 \mathrm {~m}\) and 5 m respectively, and weights \(15 \mathrm {~N} , 20 \mathrm {~N}\) and 25 N respectively. The rods are rigidly joined to form a right-angled triangular frame \(A B C\). The frame is hinged at \(B\) to a fixed point and is held in equilibrium, with \(A C\) horizontal, by means of an inextensible string attached at \(C\). The string is at right angles to \(B C\) and the tension in the string is \(T \mathrm {~N}\) (see diagram).

1. Find the value of \(T\). A uniform triangular lamina \(P Q R\), of weight 60 N , has the same size and shape as the frame \(A B C\). The lamina is now attached to the frame with \(P , Q\) and \(R\) at \(A , B\) and \(C\) respectively. The composite body is held in equilibrium with \(A , B\) and \(C\) in the same positions as before. Find
2. the new value of \(T\),
3. the magnitude of the vertical component of the force acting on the composite body at \(B\).

3 \includegraphics[max width=\textwidth, alt={}, center]{d6cb7a28-e8d7-4239-b9d3-120a284d7353-2_373_759_1119_694} A uniform object \(A B C\) is formed from two rods \(A B\) and \(B C\) joined rigidly at right angles at \(B\). The rod \(A B\) has length 0.3 m and the rod \(B C\) has length 0.2 m . The object rests with the end \(A\) on a rough horizontal surface and the \(\operatorname { rod } A B\) vertical. The object is held in equilibrium by a horizontal force of magnitude 4 N applied at \(B\) and acting in the direction \(C B\) (see diagram).

1. Find the distance of the centre of mass of the object from \(A B\).
2. Calculate the weight of the object.
3. Find the least possible value of the coefficient of friction between the surface and the object.

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.

4. \begin{figure}[h] \end{figure} The number "4", shown in Figure 2, is a rigid framework made from three uniform rods, \(A B , B C\) and \(C D\), where $$A B = 6 a , B C = 5 a \text { and } C D = 4 a$$ The point \(E\) is on \(A B\) and \(C D\), where \(B E = 4 a , C E = 3 a\) and angle \(C E B = 90 ^ { \circ }\) The three rods are all made from the same material and they all lie in the same plane. The framework is suspended from \(B\) and hangs in equilibrium with \(B A\) at an angle \(\theta\) to the downward vertical. Find \(\theta\) to the nearest degree.
VILU SIHI NI JAIUM ION OC
VIUV SIHI NI JAHM ION OC
VIIV SIHI NI EIIIM ION OC

The points \(C\) and \(D\) are at a distance \(( 2 \sqrt { } 3 ) a\) apart on a horizontal surface. A rough peg \(A\) is fixed at a vertical distance \(6 a\) above \(C\) and a smooth peg \(B\) is fixed at a vertical distance \(4 a\) above \(D\). A uniform rectangular frame \(P Q R S\), with \(P Q = 3 a\) and \(Q R = 6 a\), is made of rigid thin wire and has weight \(W\). It rests in equilibrium in a vertical plane with \(P S\) on \(A\) and \(S R\) on \(B\), and with angle \(S A C = 30 ^ { \circ }\) (see diagram).

1. Show that \(A B = 4 a\) and that angle \(S A B = 30 ^ { \circ }\).
2. Show that the normal reaction at \(A\) is \(\frac { 1 } { 2 } W\).
3. Find the frictional force at \(A\).

The points \(C\) and \(D\) are at a distance \(( 2 \sqrt { } 3 ) a\) apart on a horizontal surface. A rough peg \(A\) is fixed at a vertical distance \(6 a\) above \(C\) and a smooth peg \(B\) is fixed at a vertical distance \(4 a\) above \(D\). A uniform rectangular frame \(P Q R S\), with \(P Q = 3 a\) and \(Q R = 6 a\), is made of rigid thin wire and has weight \(W\). It rests in equilibrium in a vertical plane with \(P S\) on \(A\) and \(S R\) on \(B\), and with angle \(S A C = 30 ^ { \circ }\) (see diagram).

1. Show that \(A B = 4 a\) and that angle \(S A B = 30 ^ { \circ }\).
2. Show that the normal reaction at \(A\) is \(\frac { 1 } { 2 } W\).
3. Find the frictional force at \(A\).

3 \includegraphics[max width=\textwidth, alt={}, center]{58728f93-bfdb-4f76-a9b9-3a1d1592bfc9-3_898_1116_258_518} The end \(P\) of a uniform rod \(P Q\), of weight \(k W\) and length \(8 a\), is rigidly attached to a point on the surface of a uniform sphere with centre \(C\), weight \(W\) and radius \(a\). The end \(Q\) is rigidly attached to a point on the surface of an identical sphere with centre \(D\). The points \(C , P , Q\) and \(D\) are in a straight line. The object consisting of the rod and two spheres rests with one sphere in contact with a rough horizontal surface, at the point \(A\), and the other sphere in contact with a smooth vertical wall, at the point \(B\). The angle between \(C D\) and the horizontal is \(\theta\). The point \(B\) is at a height of \(7 a\) above the base of the wall (see diagram). The points \(A , B , C , D , P\) and \(Q\) are all in the same vertical plane.

1. Show that \(\sin \theta = \frac { 3 } { 5 }\). The object is in limiting equilibrium and the coefficient of friction at \(A\) is \(\mu\).
2. Find the numerical value of \(\mu\).
3. Given that the resultant force on the object at \(A\) is \(W \sqrt { } ( 65 )\), show that \(k = 5\).

3 \includegraphics[max width=\textwidth, alt={}, center]{62d0d8cb-8f8c-4298-9705-71a735a9a4e7-3_898_1116_258_518} The end \(P\) of a uniform rod \(P Q\), of weight \(k W\) and length \(8 a\), is rigidly attached to a point on the surface of a uniform sphere with centre \(C\), weight \(W\) and radius \(a\). The end \(Q\) is rigidly attached to a point on the surface of an identical sphere with centre \(D\). The points \(C , P , Q\) and \(D\) are in a straight line. The object consisting of the rod and two spheres rests with one sphere in contact with a rough horizontal surface, at the point \(A\), and the other sphere in contact with a smooth vertical wall, at the point \(B\). The angle between \(C D\) and the horizontal is \(\theta\). The point \(B\) is at a height of \(7 a\) above the base of the wall (see diagram). The points \(A , B , C , D , P\) and \(Q\) are all in the same vertical plane.

1. Show that \(\sin \theta = \frac { 3 } { 5 }\). The object is in limiting equilibrium and the coefficient of friction at \(A\) is \(\mu\).
2. Find the numerical value of \(\mu\).
3. Given that the resultant force on the object at \(A\) is \(W \sqrt { } ( 65 )\), show that \(k = 5\).

3 \includegraphics[max width=\textwidth, alt={}, center]{184020e1-7ff2-4172-8d33-baff963afa76-3_898_1116_258_518} The end \(P\) of a uniform rod \(P Q\), of weight \(k W\) and length \(8 a\), is rigidly attached to a point on the surface of a uniform sphere with centre \(C\), weight \(W\) and radius \(a\). The end \(Q\) is rigidly attached to a point on the surface of an identical sphere with centre \(D\). The points \(C , P , Q\) and \(D\) are in a straight line. The object consisting of the rod and two spheres rests with one sphere in contact with a rough horizontal surface, at the point \(A\), and the other sphere in contact with a smooth vertical wall, at the point \(B\). The angle between \(C D\) and the horizontal is \(\theta\). The point \(B\) is at a height of \(7 a\) above the base of the wall (see diagram). The points \(A , B , C , D , P\) and \(Q\) are all in the same vertical plane.

1. Show that \(\sin \theta = \frac { 3 } { 5 }\). The object is in limiting equilibrium and the coefficient of friction at \(A\) is \(\mu\).
2. Find the numerical value of \(\mu\).
3. Given that the resultant force on the object at \(A\) is \(W \sqrt { } ( 65 )\), show that \(k = 5\).

8 In Fig. 8, OAB is a thin bent rod, with \(\mathrm { OA } = 1 \mathrm {~m} , \mathrm { AB } = 2 \mathrm {~m}\) and angle \(\mathrm { OAB } = 120 ^ { \circ }\). Angles \(\theta , \phi\) and \(h\) are as shown in Fig. 8. \begin{figure}[h] \end{figure}

1. Show that \(h = \sin \theta + 2 \sin \left( \theta + 60 ^ { \circ } \right)\). The rod is free to rotate about the origin so that \(\theta\) and \(\phi\) vary. You may assume that the result for \(h\) in part (a) holds for all values of \(\theta\).
2. Find an angle \(\theta\) for which \(h = 0\).

4 \includegraphics[max width=\textwidth, alt={}, center]{b96a99a6-3df4-4000-9bf1-aab7ab954b4a-3_563_707_274_721} A rigid body \(A B C\) consists of two uniform rods \(A B\) and \(B C\), rigidly joined at \(B\). The lengths of \(A B\) and \(B C\) are 13 cm and 20 cm respectively, and their weights are 13 N and 20 N respectively. The distance of \(B\) from \(A C\) is 12 cm . The body hangs in equilibrium, with \(A C\) horizontal, from two vertical strings attached at \(A\) and \(C\). Find the tension in each string.

4

1. A uniform, rigid beam, AB , has a weight of 600 N . It is horizontal and in equilibrium resting on two small smooth pegs at P and Q . Fig. 4.1 shows the positions of the pegs; lengths are in metres. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3865b4b3-97c7-412b-aabd-2705a954a847-5_229_647_404_790} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure}
   1. Calculate the forces exerted by the pegs on the beam. A force of \(L \mathrm {~N}\) is applied vertically downwards at B . The beam is in equilibrium but is now on the point of tipping.
   2. Calculate the value of \(L\).
2. Fig. 4.2 shows a framework in a vertical plane constructed of light, rigid rods \(\mathrm { AB } , \mathrm { BC }\) and CA . The rods are freely pin-jointed to each other at \(\mathrm { A } , \mathrm { B }\) and C and to a fixed point at A . The pin-joint at C rests on a smooth, horizontal support. The dimensions of the framework are shown in metres. There is a force of 340 N acting at B in the plane of the framework. This force and the \(\operatorname { rod } \mathrm { BC }\) are both inclined to the vertical at an angle \(\alpha\), which is defined in triangle BCX . The force on the framework exerted by the support at C is \(R \mathrm {~N}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3865b4b3-97c7-412b-aabd-2705a954a847-5_675_869_1434_678} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure}
   1. Show that \(R = 600\).
   2. Draw a diagram showing all the forces acting on the framework and also the internal forces in the rods.
      [0pt]
   3. Calculate the internal forces in the three rods, indicating whether each rod is in tension or in compression (thrust). [Your working in this part should correspond to your diagram in part (ii).]

2 The rigid object shown in Fig. 2.1 is made of thin non-uniform rods. ABC is a straight line; \(\mathrm { BC } , \mathrm { BE }\) and ED form three sides of a rectangle. The centre of mass of the object is at G. The lengths are in centimetres. The weight of the object is 15 N . \begin{figure}[h] \end{figure} Initially, the object is suspended by light vertical strings attached to B and to C and hangs in equilibrium with AC horizontal.

1. Calculate the tensions in each of the strings. In a new situation the strings are removed. The object can rotate freely in a vertical plane about a fixed horizontal axis through A and perpendicular to ABCDE. The object is held in equilibrium with AC horizontal by a force of magnitude \(T \mathrm {~N}\) in the plane ABCDE acting at C at an angle of \(30 ^ { \circ }\) to CA . This situation is shown in Fig. 2.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ea3c0177-bf3b-4475-9ab1-ae628aeb0bf0-3_356_451_1292_808} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
2. Calculate \(T\). Calculate also the magnitude of the force exerted on the object by the axis at A . The object is now placed on a rough horizontal table and is in equilibrium with ABCDE in a vertical plane and DE in contact with the table. The coefficient of friction between the edge DE and the table is 0.65 . A force of slowly increasing magnitude (starting at 0 N ) is applied at A in the direction AB . Assume that the object remains in a vertical plane.
3. Determine whether the object slips before it tips.

4

1. Fig. 4.1 shows a framework constructed from 4 uniform heavy rigid rods \(\mathrm { OP } , \mathrm { OQ } , \mathrm { PR }\) and RS , rigidly joined at \(\mathrm { O } , \mathrm { P } , \mathrm { Q } , \mathrm { R }\) and S and with OQ perpendicular to PR . Fig. 4.1 also shows the dimensions of the rods and axes \(\mathrm { O } x\) and \(\mathrm { O } y\) : the units are metres. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-5_454_994_408_548} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure} Each rod has a mass of 0.8 kg per metre.
   1. Show that, referred to the axes in Fig. 4.1, the \(x\)-coordinate of the centre of mass of the framework is 1.5 and calculate the \(y\)-coordinate. The framework is freely suspended from S and a small object of mass \(m \mathrm {~kg}\) is attached to it at O . The framework is in equilibrium with OQ horizontal.
   2. Calculate \(m\).
2. Fig. 4.2 shows a framework in equilibrium in a vertical plane. The framework is made from 5 light, rigid rods \(\mathrm { OP } , \mathrm { OQ } , \mathrm { OR } , \mathrm { PQ }\) and QR . Its dimensions are indicated. PQ is horizontal and OR vertical. The rods are freely pin-jointed to each other at \(\mathrm { O } , \mathrm { P } , \mathrm { Q }\) and R . The pin-joint at O is fixed to a wall.
   Fig. 4.2 also shows the external forces acting on the framework: there are vertical loads of 120 N and 60 N at Q and P respectively; a horizontal string attached to Q has tension \(T \mathrm {~N}\); horizontal and vertical forces \(X \mathrm {~N}\) and \(Y \mathrm {~N}\) act on the framework from the pin-joint at O . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-6_566_453_625_788} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure}
   1. By considering only the pin-joint at R , explain why the rods OR and RQ must have zero internal force.
   2. Find the values of \(T , X\) and \(Y\).
   3. Using the diagram in your printed answer book, show all the forces acting on the pin-joints, including those internal to the rods.
      [0pt]
   4. Calculate the forces internal to the rods OP and PQ , stating whether each rod is in tension or compression (thrust). [You may leave answers in surd form. Your working in this part should correspond to your diagram in part (iii).]

1. \begin{figure}[h] \end{figure} Five identical uniform rods are joined together to form the rigid framework \(A B C D\) shown in Figure 1. Each rod has weight \(W\) and length 4a. The points \(A , B , C\) and \(D\) all lie in the same plane. The centre of mass of the framework is at the point \(G\).

1. Explain why \(G\) is the midpoint of \(A C\). The framework is suspended from the ceiling by two vertical light inextensible strings. One string is attached to the framework at \(A\) and the other string is attached to the framework at \(B\). The framework hangs freely in equilibrium with \(A B\) horizontal.
2. Find
   1. the tension in the string attached at \(A\),
   2. the tension in the string attached at \(B\). A particle of weight \(k W\) is now attached to the framework at \(D\) and a particle of weight \(2 k W\) is now attached to the framework at \(C\). The framework remains in equilibrium with \(A B\) horizontal and the strings vertical. Either string will break if the tension in it exceeds \(6 W\).
3. Find the greatest possible value of \(k\).