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OCR MEI M2 2008 January Q3
18 marks Standard +0.3
A lamina is made from uniform material in the shape shown in Fig. 3.1. BCJA, DZOJ, ZEIO and FGHI are all rectangles. The lengths of the sides are shown in centimetres. \includegraphics{figure_3}
  1. Find the coordinates of the centre of mass of the lamina, referred to the axes shown in Fig. 3.1. [5]
The rectangles BCJA and FGHI are folded through 90° about the lines CJ and FI respectively to give the fire-screen shown in Fig. 3.2.
  1. Show that the coordinates of the centre of mass of the fire-screen, referred to the axes shown in Fig. 3.2, are (2.5, 0, 57.5). [4]
The \(x\)- and \(y\)-axes are in a horizontal floor. The fire-screen has a weight of 72 N. A horizontal force \(P\) N is applied to the fire-screen at the point Z. This force is perpendicular to the line DE in the positive \(x\) direction. The fire-screen is on the point of tipping about the line AH.
  1. Calculate the value of \(P\). [5]
The coefficient of friction between the fire-screen and the floor is \(\mu\).
  1. For what values of \(\mu\) does the fire-screen slide before it tips? [4]
OCR MEI M2 2008 January Q4
18 marks Standard +0.3
Fig. 4.1 shows a uniform beam, CE, of weight 2200 N and length 4.5 m. The beam is freely pivoted on a fixed support at D and is supported at C. The distance CD is 2.75 m. \includegraphics{figure_4} The beam is horizontal and in equilibrium.
  1. Show that the anticlockwise moment of the weight of the beam about D is 1100 N m. Find the value of the normal reaction on the beam of the support at C. [6]
The support at C is removed and spheres at P and Q are suspended from the beam by light strings attached to the points C and R. The sphere at P has weight 440 N and the sphere at Q has weight \(W\) N. The point R of the beam is 1.5 m from D. This situation is shown in Fig. 4.2.
  1. The beam is horizontal and in equilibrium. Show that \(W = 1540\). [3]
The sphere at P is changed for a lighter one with weight 400 N. The sphere at Q is unchanged. The beam is now held in equilibrium at an angle of 20° to the horizontal by means of a light rope attached to the beam at E. This situation (but without the rope at E) is shown in Fig. 4.3. \includegraphics{figure_5}
  1. Calculate the tension in the rope when it is
    1. at 90° to the beam, [6]
    2. horizontal. [3]
OCR MEI M2 2011 January Q1
19 marks Standard +0.3
Fig. 1.1 shows block A of mass 2.5 kg which has been placed on a long, uniformly rough slope inclined at an angle \(\alpha\) to the horizontal, where \(\cos \alpha = 0.8\). The coefficient of friction between A and the slope is 0.85. \includegraphics{figure_1}
  1. Calculate the maximum possible frictional force between A and the slope. Show that A will remain at rest. [6]
With A still at rest, block B of mass 1.5 kg is projected down the slope, as shown in Fig. 1.2. B has a speed of 16 m s\(^{-1}\) when it collides with A. In this collision the coefficient of restitution is 0.4, the impulses are parallel to the slope and linear momentum parallel to the slope is conserved.
  1. Show that the velocity of A immediately after the collision is 8.4 m s\(^{-1}\) down the slope. Find the velocity of B immediately after the collision. [6]
  2. Calculate the impulse on B in the collision. [3]
The blocks do not collide again.
  1. For what length of time after the collision does A slide before it comes to rest? [4]
OCR MEI M2 2011 January Q2
17 marks Standard +0.3
  1. A firework is instantaneously at rest in the air when it explodes into two parts. One part is the body B of mass 0.06 kg and the other a cap C of mass 0.004 kg. The total kinetic energy given to B and C is 0.8 J. B moves off horizontally in the \(\mathbf{i}\) direction. By considering both kinetic energy and linear momentum, calculate the velocities of B and C immediately after the explosion. [8]
  2. A car of mass 800 kg is travelling up some hills. In one situation the car climbs a vertical height of 20 m while its speed decreases from 30 m s\(^{-1}\) to 12 m s\(^{-1}\). The car is subject to a resistance to its motion but there is no driving force and the brakes are not being applied.
    1. Using an energy method, calculate the work done by the car against the resistance to its motion. [4]
    In another situation the car is travelling at a constant speed of 18 m s\(^{-1}\) and climbs a vertical height of 20 m in 25 s up a uniform slope. The resistance to its motion is now 750 N.
    1. Calculate the power of the driving force required. [5]
OCR MEI M2 2011 January Q3
19 marks Standard +0.8
\includegraphics{figure_3} Fig. 3 shows a framework in equilibrium in a vertical plane. The framework is made from the equal, light, rigid rods AB, AD, BC, BD and CD so that ABD and BCD are equilateral triangles of side 2 m. AD and BC are horizontal. The rods are freely pin-jointed to each other at A, B, C and D. The pin-joint at A is fixed to a wall and the pin-joint at B rests on a smooth horizontal support. Fig. 3 also shows the external forces acting on the framework: there is a vertical load of 45 N at C and a horizontal force of 50 N applied at D; the normal reaction of the support on the framework at B is \(R\) N; horizontal and vertical forces \(X\) N and \(Y\) N act at A.
  1. Write down equations for the horizontal and vertical equilibrium of the framework. [2]
  2. Show that \(R = 135\) and \(Y = 90\). [3]
  3. On the diagram in your printed answer book, show the forces internal to the rods acting on the pin-joints. [2]
  4. Calculate the forces internal to the five rods, stating whether each rod is in tension or compression (thrust). [You may leave your answers in surd form. Your working in this part should correspond to your diagram in part (iii).] [10]
  5. Suppose that the force of magnitude 50 N applied at D is no longer horizontal, and the system remains in equilibrium in the same position. By considering the equilibrium at C, show that the forces in rods CD and BC are not changed. [2]
OCR MEI M2 2011 January Q4
17 marks Standard +0.3
You are given that the centre of mass, G, of a uniform lamina in the shape of an isosceles triangle lies on its axis of symmetry in the position shown in Fig. 4.1. \includegraphics{figure_4_1} Fig. 4.2 shows the cross-section OABCD of a prism made from uniform material. OAB is an isosceles triangle, where OA = AB, and OBCD is a rectangle. The distance OD is \(h\) cm, where \(h\) can take various positive values. All coordinates refer to the axes Ox and Oy shown. The units of the axes are centimetres. \includegraphics{figure_4_2}
  1. Write down the coordinates of the centre of mass of the triangle OAB. [1]
  2. Show that the centre of mass of the region OABCD is \(\left(\frac{12-h^2}{2(h+3)}, 2.5\right)\). [6]
The \(x\)-axis is horizontal. The prism is placed on a horizontal plane in the position shown in Fig. 4.2.
  1. Find the values of \(h\) for which the prism would topple. [3]
The following questions refer to the case where \(h = 3\) with the prism held in the position shown in Fig. 4.2. The cross-section OABCD contains the centre of mass of the prism. The weight of the prism is 15 N. You should assume that the prism does not slide.
  1. Suppose that the prism is held in this position by a vertical force applied at A. Given that the prism is on the point of tipping clockwise, calculate the magnitude of this force. [3]
  2. Suppose instead that the prism is held in this position by a force in the plane of the cross-section OABCD, applied at 30° below the horizontal at C, as shown in Fig. 4.3. Given that the prism is on the point of tipping anti-clockwise, calculate the magnitude of this force. [4]
\includegraphics{figure_4_3}
CAIE M2 2014 June Q3
Standard +0.3
3 A light elastic string has natural length 0.8 m and modulus of elasticity 16 N . One end of the string is attached to a fixed point \(O\), and a particle \(P\) of mass 0.4 kg is attached to the other end of the string. The particle \(P\) hangs in equilibrium vertically below \(O\).
  1. Show that the extension of the string is 0.2 m . \(P\) is projected vertically downwards from the equilibrium position. \(P\) first comes to instantaneous rest at the point where \(O P = 1.4 \mathrm {~m}\).
  2. Calculate the speed at which \(P\) is projected.
  3. Find the speed of \(P\) at the first instant when the string subsequently becomes slack.
CAIE M2 2014 June Q4
Standard +0.8
4 A particle \(P\) is projected with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) above the horizontal from a point \(O\) on horizontal ground.
  1. Find the height of \(P\) above the ground when \(P\) has speed \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Calculate the length of time for which the speed of \(P\) is less than \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and find the horizontal distance travelled by \(P\) during this time.
CAIE M2 2013 June Q1
Easy -1.8
1 hour 15 minutes \section*{
\includegraphics[max width=\textwidth, alt={}]{10abedc3-c814-47c0-8ed4-849ef325feca-1_403_143_792_68}
} Additional Materials:
Answer Booklet/Paper
Graph Paper
List of Formulae (MF9) \section*{READ THESE INSTRUCTIONS FIRST} If you have been given an Answer Booklet, follow the instructions on the front cover of the Booklet.
Write your Centre number, candidate number and name on all the work you hand in.
Write in dark blue or black pen.
You may use a soft pencil for any diagrams or graphs.
Do not use staples, paper clips, highlighters, glue or correction fluid.
Answer all the questions.
Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question.
Where a numerical value for the acceleration due to gravity is needed, use \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
The use of an electronic calculator is expected, where appropriate.
You are reminded of the need for clear presentation in your answers.
At the end of the examination, fasten all your work securely together.
[0pt] The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 50.
Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper.
CAIE M2 2013 June Q2
Standard +0.8
2 A particle \(P\) of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 45 N . The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at \(O\) and falls vertically. Find the extension of the string when \(P\) is at its lowest position.
CAIE M2 2013 June Q3
Moderate -0.8
3 A ball is projected horizontally with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the top of a tower which is 30 m high. The tower stands on horizontal ground.
  1. Find the speed and direction of motion of the ball when it reaches the ground.
  2. Calculate the distance from the foot of the tower to the point where the ball reaches the ground.
CAIE M2 2013 June Q7
Challenging +1.2
7 A small ball \(B\) of mass 0.2 kg moves in a narrow fixed smooth cylindrical tube \(O A\) of length 1 m , closed at the end \(A\). When the ball has displacement \(x \mathrm {~m}\) from \(O\), it has velocity \(v \mathrm {~ms} ^ { - 1 }\) in the direction \(O A\) and experiences a resisting force of magnitude \(\frac { k } { 1 - x } \mathrm {~N}\).
  1. \includegraphics[max width=\textwidth, alt={}, center]{10abedc3-c814-47c0-8ed4-849ef325feca-4_186_805_488_715} The tube is fixed in a horizontal position and \(B\) is projected from \(O\) towards \(A\) with velocity \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Given that \(B\) comes to instantaneous rest after travelling 0.55 m , show that \(k = 0.1803\), correct to 4 significant figures.
  2. The tube is now fixed in a vertical position with \(O\) above \(A\). The ball \(B\) is released from rest at \(O\). Calculate the speed of \(B\) after it has descended 0.1 m . \end{document}