Prism or block on inclined plane

2 \includegraphics[max width=\textwidth, alt={}, center]{3e7472a8-df1e-45c4-81fb-e4397bddf5ad-2_316_1065_712_541} Two identical uniform heavy triangular prisms, each of base width 10 cm , are arranged as shown at the ends of a smooth horizontal shelf of length 1 m . Some books, each of width 5 cm , are placed on the shelf between the prisms.

1. Find how far the base of a prism can project beyond an end of the shelf without the prism toppling.
2. Find the greatest number of books that can be stored on the shelf without either of the prisms toppling.

7 \includegraphics[max width=\textwidth, alt={}, center]{d6cb7a28-e8d7-4239-b9d3-120a284d7353-3_519_860_1430_641} \(O A B C\) is the cross-section through the centre of mass of a uniform prism of weight 20 N . The crosssection is in the shape of a sector of a circle with centre \(O\), radius \(O A = r \mathrm {~m}\) and angle \(A O C = \frac { 2 } { 3 } \pi\) radians. The prism lies on a plane inclined at an angle \(\theta\) radians to the horizontal, where \(\theta < \frac { 1 } { 3 } \pi\). OC lies along a line of greatest slope with \(O\) higher than \(C\). The prism is freely hinged to the plane at \(O\). A force of magnitude 15 N acts at \(A\), in a direction towards to the plane and at right angles to it (see diagram). Given that the prism remains in equilibrium, find the set of possible values of \(\theta\).

7 \includegraphics[max width=\textwidth, alt={}, center]{5998f4b1-21da-4c25-8b09-91a1cb1eee42-4_357_776_260_680} A small bead \(B\) of mass \(m \mathrm {~kg}\) moves with constant speed in a horizontal circle on a fixed smooth wire. The wire is in the form of a circle with centre \(O\) and radius 0.4 m . One end of a light elastic string of natural length 0.4 m and modulus of elasticity \(42 m \mathrm {~N}\) is attached to \(B\). The other end of the string is attached to a fixed point \(A\) which is 0.3 m vertically above \(O\) (see diagram).

1. Show that the vertical component of the contact force exerted by the wire on the bead is 3.7 mN upwards.
2. Given that the contact force has zero horizontal component, find the angular speed of \(B\).
3. Given instead that the horizontal component of the contact force has magnitude \(2 m \mathrm {~N}\), find the two possible speeds of \(B\). The string is now removed. \(B\) again moves on the wire in a horizontal circle with constant speed. It is given that the vertical and horizontal components of the contact force exerted by the wire on the bead have equal magnitudes.
4. Find the speed of \(B\). \end{document}

5 \includegraphics[max width=\textwidth, alt={}, center]{a03ad6c1-b4a3-4007-8d3b-ce289a998a55-3_499_721_715_712} A uniform solid cube with edges of length 0.4 m rests in equilibrium on a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. \(A B C D\) is a cross-section through the centre of mass of the cube, with \(A B\) along a line of greatest slope. \(B\) lies below the level of \(A\). One end of a light elastic string with modulus of elasticity 12 N and natural length 0.4 m is attached to \(C\). The other end of the string is attached to a point below the level of \(B\) on the same line of greatest slope, such that the string makes an angle of \(30 ^ { \circ }\) with the plane (see diagram). The cube is on the point of toppling. Find

1. the tension in the string,
2. the weight of the cube.

5 \includegraphics[max width=\textwidth, alt={}, center]{8f8492a7-8a83-4eb2-81ee-99b4a385b704-3_876_483_260_840} A uniform triangular prism of weight 20 N rests on a horizontal table. \(A B C\) is the cross-section through the centre of mass of the prism, where \(B C = 0.5 \mathrm {~m} , A B = 0.4 \mathrm {~m} , A C = 0.3 \mathrm {~m}\) and angle \(B A C = 90 ^ { \circ }\). The vertical plane \(A B C\) is perpendicular to the edge of the table. The point \(D\) on \(A C\) is at the edge of the table, and \(A D = 0.25 \mathrm {~m}\). One end of a light elastic string of natural length 0.6 m and modulus of elasticity 48 N is attached to \(C\) and a particle of mass 2.5 kg is attached to the other end of the string. The particle is released from rest at \(C\) and falls vertically (see diagram).

1. Show that the tension in the string is 60 N at the instant when the prism topples.
2. Calculate the speed of the particle at the instant when the prism topples.

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.

6 \includegraphics[max width=\textwidth, alt={}, center]{d9970ad1-a7f4-429a-bad1-43e8d114b968-3_656_757_781_694} The diagram shows the cross-section \(A B C D E F\) through the centre of mass of a uniform prism which rests with \(A B\) on rough horizontal ground. \(A B C D\) is a rectangle with \(A B = C D = 0.4 \mathrm {~m}\) and \(B C = A D = 1.8 \mathrm {~m}\). The other part of the cross-section is a semicircle with diameter \(D F\) and radius \(r \mathrm {~m}\).

1. Given that the prism is on the point of toppling, show that \(r = 0.6\). A force of magnitude \(P \mathrm {~N}\) is applied to the prism, acting at \(60 ^ { \circ }\) to the upwards vertical along a tangent to the semicircle at a point between \(D\) and \(E\). The prism has weight 15 N and is in equilibrium on the point of toppling about \(B\).
2. Show that \(P = 3.26\), correct to 3 significant figures.
3. Find the smallest possible value of the coefficient of friction between the prism and the ground.

4 \includegraphics[max width=\textwidth, alt={}, center]{833c338f-53c1-436e-a772-0cdaf17fa72d-3_559_1303_255_422} The diagram shows a central cross-section CDEF of a uniform solid cube of weight \(W\) and with edges of length \(2 a\). The cube rests on a rough horizontal plane. A thin uniform \(\operatorname { rod } A B\), of weight \(W\) and length \(6 a\), is hinged to the plane at \(A\). The rod rests in smooth contact with the cube at \(C\), with angle \(C A D\) equal to \(30 ^ { \circ }\). The rod is in the same vertical plane as \(C D E F\). The coefficient of friction between the plane and the cube is \(\mu\). Given that the system is in equilibrium, show that \(\mu \geqslant \frac { 3 } { 25 } \sqrt { } 3\). [6] Find the magnitude of the force acting on the \(\operatorname { rod }\) at \(A\).

6 A uniform solid cylinder, L, has base radius 5 cm , height 24 cm and mass 5 kg . L is placed on a rough plane inclined at an angle \(\alpha\) to the horizontal, as shown in Fig. 6. \begin{figure}[h] \end{figure}

1. On the copy of Fig. 6 in the Printed Answer Booklet mark the forces acting on L . The coefficient of friction between L and the plane is 0.3 . Initially \(\alpha\) is \(15 ^ { \circ }\).
2. Show that L rests in equilibrium on the plane. A couple is applied to L . It is given that L will topple if the couple is applied in an anticlockwise sense, but L will not topple if the couple is applied in a clockwise sense.
3. Find the range of possible values of the magnitude of the couple. The couple is now removed and the plane is slowly tilted so that \(\alpha\) increases.
4. Determine whether L topples first without sliding or slides first without toppling.

2 The diagram below shows the cross-section through the centre of mass of a uniform block of weight \(W \mathrm {~N}\), resting on a slope inclined at an angle \(\alpha\) to the horizontal. The cross-section is a rectangle ABCD . The slope exerts a frictional force of magnitude \(F \mathrm {~N}\) and a normal contact force of magnitude \(R \mathrm {~N}\). \includegraphics[max width=\textwidth, alt={}, center]{9b624694-edb6-4000-838f-3557e078952d-3_546_940_450_242}

1. Explain why a triangle of forces may be used to model the scenario.
2. In the space provided in the Printed Answer Booklet, draw such a triangle, fully annotated, including the angle \(\alpha\) in the correct position. The coefficient of friction between the block and the slope is \(\mu\).
3. Given that the block is in limiting equilibrium, use your diagram in part (b) to show that \(\mu = \tan \alpha\). It is given that \(\mathrm { AB } = 8.9 \mathrm {~cm}\) and \(\mathrm { AD } = 11.6 \mathrm {~cm}\). The coefficient of friction between the slope and the block is 1.35 . The slope is slowly tilted so that \(\alpha\) increases.
4. Determine whether the block topples first without sliding or slides first without toppling.

\includegraphics{figure_7} The diagram shows the cross-section \(OABCDE\) through the centre of mass of a uniform prism on a rough inclined plane. The portion \(ADEO\) is a rectangle in which \(AD = OE = 0.6\) m and \(DE = AO = 0.8\) m; the portion \(BCD\) is an isosceles triangle in which angle \(BCD\) is a right angle, and \(A\) is the mid-point of \(BD\). The plane is inclined at \(45°\) to the horizontal, \(BC\) lies along a line of greatest slope of the plane and \(DE\) is horizontal.

1. Calculate the distance of the centre of mass of the prism from \(BD\). [3]

The weight of the prism is \(21\) N, and it is held in equilibrium by a horizontal force of magnitude \(P\) N acting along \(ED\).

1. 1. Find the smallest value of \(P\) for which the prism does not topple. [2]
   2. It is given that the prism is about to slip for this smallest value of \(P\). Calculate the coefficient of friction between the prism and the plane. [3]

The value of \(P\) is gradually increased until the prism ceases to be in equilibrium.

1. Show that the prism topples before it begins to slide, stating the value of \(P\) at which equilibrium is broken. [5]

\includegraphics{figure_7} A small object is projected with speed \(24\text{ m s}^{-1}\) from a point \(O\) at the foot of a plane inclined at \(45°\) to the horizontal. The angle of projection of the object is \(15°\) above a line of greatest slope of the plane (see diagram). At time \(t\) s after projection, the horizontal and vertically upwards displacements of the object from \(O\) are \(x\) m and \(y\) m respectively.

1. Express \(x\) and \(y\) in terms of \(t\), and hence find the value of \(t\) for the instant when the object strikes the plane. [4]
2. Express the vertical height of the object above the plane in terms of \(t\) and hence find the greatest vertical height of the object above the plane. [5]

\includegraphics{figure_11} The diagram shows the cross-section through the centre of mass of a uniform solid prism. The cross-section is a right-angled triangle ABC, with AB perpendicular to AC, which lies in a vertical plane. The length of AB is 3 cm, and the length of AC is 12 cm. The prism is resting in equilibrium on a horizontal surface and against a vertical wall. The side AC of the prism makes an angle \(\theta\) with the horizontal. A horizontal force of magnitude \(P\) N is now applied to the prism at B. This force acts towards the wall in the vertical plane which passes through the centre of mass G of the prism and is perpendicular to the wall. The weight of the prism is 15 N and the coefficients of friction between the prism and the surface, and between the prism and the wall, are each \(\frac{1}{2}\).

1. Show that the least value of \(P\) needed to move the prism is given by $$P = \frac{40 \cos \theta + 95 \sin \theta}{16 \sin \theta - 13 \cos \theta}.$$ [8]
2. Determine the range in which the value of \(\theta\) must lie. [4]