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OCR M1 2007 June Q1
6 marks Easy -1.2
1 \includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-2_415_823_264_660} Two horizontal forces \(\mathbf { P }\) and \(\mathbf { Q }\) act at the origin O of rectangular coordinates Oxy (see diagram). The components of \(\mathbf { P }\) in the \(x\) - and \(y\)-directions are 14 N and 5 N respectively. The components of \(\mathbf { Q }\) in the \(x\) - and \(y\)-directions are - 9 N and 7 N respectively.
  1. Write down the components, in the \(x\) - and \(y\)-directions, of the resultant of \(\mathbf { P }\) and \(\mathbf { Q }\).
  2. Hence find the magnitude of this resultant, and the angle the resultant makes with the positive \(x\)-axis.
OCR M1 2007 June Q2
7 marks Moderate -0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-2_714_1048_1231_552} A particle starts from the point A and travels in a straight line. The diagram shows the ( \(\mathrm { t } , \mathrm { v }\) ) graph, consisting of three straight line segments, for the motion of the particle during the interval \(0 \leqslant t \leqslant 290\).
  1. Find the value of ther which the distance of the particle from A is greatest.
  2. Find the displacement of the particle from A when \(\mathrm { t } = 290\).
  3. Find the total distance travelled by the particle during the interval \(0 \leqslant \mathrm { t } \leqslant 290\).
OCR M1 2007 June Q3
8 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-3_437_846_274_651} A block of mass 50 kg is in equilibrium on smooth horizontal ground with one end of a light wire attached to its upper surface. The other end of the wire is attached to an object of mass mkg . The wire passes over a small smooth pulley, and the object hangs vertically below the pulley. The part of the wire between the block and the pulley makes an angle of \(72 ^ { \circ }\) with the horizontal. A horizontal force of magnitude X N acts on the block in the vertical plane containing the wire (see diagram). The tension in the wire is T N and the contact force exerted by the ground on the block is R N.
  1. By resolving forces on the block vertically, find a relationship between T and R . It is given that the block is on the point of lifting off the ground.
  2. Show that \(\mathrm { T } = 515\), correct to 3 significant figures, and hence find the value of m .
  3. By resolving forces on the block horizontally, write down a relationship between T and X , and hence find the value of \(X\).
OCR M1 2007 June Q4
10 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-3_149_606_1626_772} Two particles of masses 0.18 kg and m kg move on a smooth horizontal plane. They are moving towards each other in the same straight line when they collide. Immediately before the impact the speeds of the particles are \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram).
  1. Given that the particles are brought to rest by the impact, find m .
  2. Given instead that the particles move with equal speeds of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after the impact, find
    1. the value of m , assuming that the particles move in opposite directions after the impact,
    2. the two possible values of m , assuming that the particles coalesce.
OCR M1 2007 June Q5
11 marks Moderate -0.3
5 A particle \(P\) is projected vertically upwards, from horizontal ground, with speed \(8.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that the greatest height above the ground reached by P is 3.6 m . A particle Q is projected vertically upwards, from a point 2 m above the ground, with speed \(\mathrm { um } \mathrm { s } ^ { - 1 }\). The greatest height abovetheground reached by Q is also 3.6 m .
  2. Find the value of \(u\). It is given that P and Q are projected simultaneously.
  3. Show that, at the instant when P and Q are at the same height, the particles have the same speed and are moving in opposite directions.
OCR M1 2007 June Q6
14 marks Standard +0.3
6 A particle starts from rest at the point A and travels in a straight line. The displacement sm of the particle from A at time ts after leaving A is given by $$s = 0.001 t ^ { 4 } - 0.04 t ^ { 3 } + 0.6 t ^ { 2 } , \quad \text { for } 0 \leqslant t \leqslant 10 .$$
  1. Show that the velocity of the particle is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(\mathrm { t } = 10\). The acceleration of the particle for \(t \geqslant 10\) is \(( 0.8 - 0.08 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  2. Show that the velocity of the particle is zero when \(\mathrm { t } = 20\).
  3. Find the displacement from A of the particle when \(\mathrm { t } = 20\).
OCR M1 2007 June Q7
16 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-5_488_739_269_703} One end of a light inextensible string is attached to a block of mass 1.5 kg . The other end of the string is attached to an object of mass 1.2 kg . The block is held at rest in contact with a rough plane inclined at \(21 ^ { \circ }\) to the horizontal. The string is taut and passes over a small smooth pulley at the bottom edge of the plane. The part of the string above the pulley is parallel to a line of greatest slope of the plane and the object hangs freely below the pulley (see diagram). The block is released and the object moves vertically downwards with acceleration \(\mathrm { am } \mathrm { s } ^ { - 2 }\). The tension in the string is TN . The coefficient of friction between the block and the plane is 0.8 .
  1. Show that the frictional force acting on the block has magnitude 10.98 N , correct to 2 decimal places.
  2. By applying Newton's second law to the block and to the object, find a pair of simultaneous equations in T and a .
  3. Hence show that \(\mathrm { a } = 2.24\), correct to 2 decimal places.
  4. Given that the object is initially 2 m above a horizontal floor and that the block is 2.8 m from the pulley, find the speed of the block at the instant when
    1. the object reaches the floor,
    2. the block reaches the pulley. {}
      7
OCR M1 2010 June Q1
8 marks Moderate -0.8
1 A block \(B\) of mass 3 kg moves with deceleration \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) in a straight line on a rough horizontal surface. The initial speed of \(B\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate
  1. the time for which \(B\) is in motion,
  2. the distance travelled by \(B\) before it comes to rest,
  3. the coefficient of friction between \(B\) and the surface.
OCR M1 2010 June Q2
9 marks Moderate -0.3
2 Two particles \(P\) and \(Q\) are moving in opposite directions in the same straight line on a smooth horizontal surface when they collide. \(P\) has mass 0.4 kg and speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 } . Q\) has mass 0.6 kg and speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, the speed of \(P\) is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Given that \(P\) and \(Q\) are moving in the same direction after the collision, find the speed of \(Q\).
  2. Given instead that \(P\) and \(Q\) are moving in opposite directions after the collision, find the distance between them 3 s after the collision.
OCR M1 2010 June Q3
9 marks Moderate -0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{4b703cf9-b3d3-4210-b57b-89136595f8a5-02_570_495_1114_826} Three horizontal forces of magnitudes \(12 \mathrm {~N} , 5 \mathrm {~N}\), and 9 N act along bearings \(000 ^ { \circ } , 150 ^ { \circ }\) and \(270 ^ { \circ }\) respectively (see diagram).
  1. Show that the component of the resultant of the three forces along bearing \(270 ^ { \circ }\) has magnitude 6.5 N .
  2. Find the component of the resultant of the three forces along bearing \(000 ^ { \circ }\).
  3. Hence find the magnitude and bearing of the resultant of the three forces.
OCR M1 2010 June Q4
10 marks Moderate -0.3
4 A particle \(P\) moving in a straight line has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after passing through a fixed point \(O\). It is given that \(v = 3.2 - 0.2 t ^ { 2 }\) for \(0 \leqslant t \leqslant 5\). Calculate
  1. the value of \(t\) when \(P\) is at instantaneous rest,
  2. the acceleration of \(P\) when it is at instantaneous rest,
  3. the greatest distance of \(P\) from \(O\).
OCR M1 2010 June Q5
9 marks Moderate -0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{4b703cf9-b3d3-4210-b57b-89136595f8a5-03_508_1397_255_374} The diagram shows the ( \(t , v\) ) graph for a lorry delivering waste to a recycling centre. The graph consists of six straight line segments. The lorry reverses in a straight line from a stationary position on a weighbridge before coming to rest. It deposits its waste and then moves forwards in a straight line accelerating to a maximum speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It maintains this speed for 4 s and then decelerates, coming to rest at the weighbridge.
  1. Calculate the distance from the weighbridge to the point where the lorry deposits the waste.
  2. Calculate the time which elapses between the lorry leaving the weighbridge and returning to it.
  3. Given that the acceleration of the lorry when it is moving forwards is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), calculate its final deceleration.
OCR M1 2010 June Q6
13 marks Standard +0.3
6 A block \(B\) of mass 0.85 kg lies on a smooth slope inclined at \(30 ^ { \circ }\) to the horizontal. \(B\) is attached to one end of a light inextensible string which is parallel to the slope. At the top of the slope, the string passes over a smooth pulley. The other end of the string hangs vertically and is attached to a particle \(P\) of mass 0.55 kg . The string is taut at the instant when \(P\) is projected vertically downwards.
  1. Calculate
    1. the acceleration of \(B\) and the tension in the string,
    2. the magnitude of the force exerted by the string on the pulley. The initial speed of \(P\) is \(1.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and after moving \(1.5 \mathrm {~m} P\) reaches the ground, where it remains at rest. \(B\) continues to move up the slope and does not reach the pulley.
    3. Calculate the total distance \(B\) moves up the slope before coming instantaneously to rest.
OCR M1 2010 June Q7
14 marks Standard +0.8
7 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4b703cf9-b3d3-4210-b57b-89136595f8a5-04_305_748_260_699} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A rectangular block \(B\) of weight 12 N lies in limiting equilibrium on a horizontal surface. A horizontal force of 4 N and a coplanar force of 5 N inclined at \(60 ^ { \circ }\) to the vertical act on \(B\) (see Fig. 1).
  1. Find the coefficient of friction between \(B\) and the surface. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4b703cf9-b3d3-4210-b57b-89136595f8a5-04_307_751_1000_696} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} \(B\) is now cut horizontally into two smaller blocks. The upper block has weight 9 N and the lower block has weight 3 N . The 5 N force now acts on the upper block and the 4 N force now acts on the lower block (see Fig. 2). The coefficient of friction between the two blocks is \(\mu\).
  2. Given that the upper block is in limiting equilibrium, find \(\mu\).
  3. Given instead that \(\mu = 0.1\), find the accelerations of the two blocks.
OCR M1 2010 June Q8
Moderate -0.8
8 {}
6 (ii)
{}
OCR M1 2010 June Q10
Moderate -0.8
10
7
  • 7 (ii)(continued)
    \multirow[t]{26}{*}{7 (iii)}
    \section*{PLEASE DO NOT WRITE ON THIS PAGE} RECOGNISING ACHIEVEMENT
    OCR MEI M1 2005 January Q1
    7 marks Moderate -0.8
    1 The position vector, \(\mathbf { r }\), of a particle of mass 4 kg at time \(t\) is given by $$\mathbf { r } = t ^ { 2 } \mathbf { i } + \left( 5 t - 2 t ^ { 2 } \right) \mathbf { j } ,$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors, lengths are in metres and time is in seconds.
    1. Find an expression for the acceleration of the particle. The particle is subject to a force \(\mathbf { F }\) and a force \(12 \mathbf { j } \mathbf { N }\).
    2. Find \(\mathbf { F }\).
    OCR MEI M1 2005 January Q2
    8 marks Standard +0.3
    2 Particles of mass 2 kg and 4 kg are attached to the ends \(X\) and \(Y\) of a light, inextensible string. The string passes round fixed, smooth pulleys at \(\mathrm { P } , \mathrm { Q }\) and R , as shown in Fig. 2. The system is released from rest with the string taut. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c84a748a-a6f4-48c5-b864-fe543569bdf5-2_478_397_1211_872} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure}
    1. State what information in the question tells you that
      (A) the tension is the same throughout the string,
      (B) the magnitudes of the accelerations of the particles at X and Y are the same. The tension in the string is \(T \mathrm {~N}\) and the magnitude of the acceleration of the particles is \(a \mathrm {~ms} ^ { - 2 }\).
    2. Draw a diagram showing the forces acting at X and a diagram showing the forces acting at Y .
    3. Write down equations of motion for the particles at X and at Y . Hence calculate the values of \(T\) and \(a\).
    OCR MEI M1 2005 January Q3
    6 marks Moderate -0.8
    3 A particle is in equilibrium when acted on by the forces \(\left( \begin{array} { r } x \\ - 7 \\ z \end{array} \right) , \left( \begin{array} { r } 4 \\ y \\ - 5 \end{array} \right)\) and \(\left( \begin{array} { r } 5 \\ 4 \\ - 7 \end{array} \right)\), where the units are newtons.
    1. Find the values of \(x , y\) and \(z\).
    2. Calculate the magnitude of \(\left( \begin{array} { r } 5 \\ 4 \\ - 7 \end{array} \right)\).
    OCR MEI M1 2005 January Q4
    8 marks Standard +0.3
    4 A particle is projected vertically upwards from a point O at \(21 \mathrm {~ms} ^ { - 1 }\).
    1. Calculate the greatest height reached by the particle. When this particle is at its highest point, a second particle is projected vertically upwards from \(O\) at \(15 \mathrm {~ms} ^ { - 1 }\).
    2. Show that the particles collide 1.5 seconds later and determine the height above O at which the collision takes place.
    OCR MEI M1 2005 January Q5
    7 marks Moderate -0.8
    5 A small box B of weight 400 N is held in equilibrium by two light strings AB and BC . The string BC is fixed at C . The end A of string AB is fixed so that AB is at an angle \(\alpha\) to the vertical where \(\alpha < 60 ^ { \circ }\). String BC is at \(60 ^ { \circ }\) to the vertical. This information is shown in Fig. 5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c84a748a-a6f4-48c5-b864-fe543569bdf5-3_424_472_1599_774} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure}
    1. Draw a labelled diagram showing all the forces acting on the box.
    2. In one situation string AB is fixed so that \(\alpha = 30 ^ { \circ }\). By drawing a triangle of forces, or otherwise, calculate the tension in the string BC and the tension in the string AB .
    3. Show carefully, but briefly, that the box cannot be in equilibrium if \(\alpha = 60 ^ { \circ }\) and BC remains at \(60 ^ { \circ }\) to the vertical.
    OCR MEI M1 2005 January Q7
    17 marks Standard +0.3
    7 The trajectory ABCD of a small stone moving with negligible air resistance is shown in Fig. 7. AD is horizontal and BC is parallel to AD . The stone is projected from A with speed \(40 \mathrm {~ms} ^ { - 1 }\) at \(50 ^ { \circ }\) to the horizontal. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c84a748a-a6f4-48c5-b864-fe543569bdf5-4_341_1107_484_498} \captionsetup{labelformat=empty} \caption{Fig. 7}
    \end{figure}
    1. Write down an expression for the horizontal displacement from A of the stone \(t\) seconds after projection. Write down also an expression for the vertical displacement at time \(t\).
    2. Show that the stone takes 6.253 seconds (to three decimal places) to travel from A to D . Calculate the range of the stone. You are given that \(X = 30\).
    3. Calculate the time it takes the stone to reach B . Hence determine the time for it to travel from A to C.
    4. Calculate the direction of the motion of the stone at \(\mathbf { C }\). Section B (36 marks)
    OCR MEI M1 2006 January Q1
    6 marks Easy -1.3
    1 A particle travels in a straight line during the time interval \(0 \leqslant t \leqslant 12\), where \(t\) is the time in seconds. Fig. 1 is the velocity-time graph for the motion. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{19d42df9-e752-4d33-85e1-4ec59b32135a-2_455_874_484_593} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure}
    1. Calculate the acceleration of the particle in the interval \(0 < t < 6\).
    2. Calculate the distance travelled by the particle from \(t = 0\) to \(t = 4\).
    3. When \(t = 0\) the particle is at A . Calculate how close the particle gets to A during the interval \(4 \leqslant t \leqslant 12\).
    OCR MEI M1 2006 January Q2
    5 marks Moderate -0.8
    2 Fig. 2 shows a light string with an object of mass 4 kg attached at end A . The string passes over a smooth pulley and its other end B is attached to two light strings BC and BD of the same length. The strings BC and BD are attached to horizontal ground and are each inclined at \(20 ^ { \circ }\) to the vertical. The system is in equilibrium. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{19d42df9-e752-4d33-85e1-4ec59b32135a-2_588_451_1749_806} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure}
    1. What information in the question tells you that the tension is the same throughout the string AB ?
    2. What is the tension in the string AB ?
    3. Calculate the tension in the strings BC and BD .
    OCR MEI M1 2006 January Q3
    7 marks Moderate -0.8
    3 A force \(\mathbf { F }\) is given by \(\mathbf { F } = ( 3.5 \mathbf { i } + 12 \mathbf { j } ) \mathrm { N }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors east and north respectively.
    1. Calculate the magnitude of \(\mathbf { F }\) and also its direction as a bearing.
    2. \(\mathbf { G }\) is the force \(( 7 \mathbf { i } + 24 \mathbf { j } )\) N. Show that \(\mathbf { G }\) and \(\mathbf { F }\) are in the same direction and compare their magnitudes.
    3. Force \(\mathbf { F } _ { 1 }\) is \(( 9 \mathbf { i } - 18 \mathbf { j } ) \mathrm { N }\) and force \(\mathbf { F } _ { 2 }\) is \(( 12 \mathbf { i } + q \mathbf { j } ) \mathrm { N }\). Find \(q\) so that the sum \(\mathbf { F } _ { 1 } + \mathbf { F } _ { 2 }\) is in the direction of \(\mathbf { F }\).