Questions M1 (1912 questions)

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CAIE M1 2004 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{38ece0f6-1c29-4e7a-9d66-16c3e2b695f9-3_330_572_1037_788} Two identical boxes, each of mass 400 kg , are at rest, with one on top of the other, on horizontal ground. A horizontal force of magnitude \(P\) newtons is applied to the lower box (see diagram). The coefficient of friction between the lower box and the ground is 0.75 and the coefficient of friction between the two boxes is 0.4 .
  1. Show that the boxes will remain at rest if \(P \leqslant 6000\). The boxes start to move with acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Given that no sliding takes place between the boxes, show that \(a \leqslant 4\) and deduce the maximum possible value of \(P\).
CAIE M1 2004 November Q7
7 A particle starts from rest at the point \(A\) and travels in a straight line until it reaches the point \(B\). The velocity of the particle \(t\) seconds after leaving \(A\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 0.009 t ^ { 2 } - 0.0001 t ^ { 3 }\). Given that the velocity of the particle when it reaches \(B\) is zero, find
  1. the time taken for the particle to travel from \(A\) to \(B\),
  2. the distance \(A B\),
  3. the maximum velocity of the particle.
CAIE M1 2005 November Q1
1 A car travels in a straight line with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). It passes the points \(A , B\) and \(C\), in this order, with speeds \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 } , 7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The distances \(A B\) and \(B C\) are \(d _ { 1 } \mathrm {~m}\) and \(d _ { 2 } \mathrm {~m}\) respectively.
  1. Write down an equation connecting
    (a) \(d _ { 1 }\) and \(a\),
    (b) \(d _ { 2 }\) and \(a\).
  2. Hence find \(d _ { 1 }\) in terms of \(d _ { 2 }\).
CAIE M1 2005 November Q2
2 A crate of mass 50 kg is dragged along a horizontal floor by a constant force of magnitude 400 N acting at an angle \(\alpha ^ { \circ }\) upwards from the horizontal. The total resistance to motion of the crate has constant magnitude 250 N . The crate starts from rest at the point \(O\) and passes the point \(P\) with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The distance \(O P\) is 20 m . For the crate's motion from \(O\) to \(P\), find
  1. the increase in kinetic energy of the crate,
  2. the work done against the resistance to the motion of the crate,
  3. the value of \(\alpha\).
CAIE M1 2005 November Q3
6 marks
3
\includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-2_479_771_1356_687} Each of three light strings has a particle attached to one of its ends. The other ends of the strings are tied together at a point \(A\). The strings are in equilibrium with two of them passing over fixed smooth horizontal pegs, and with the particles hanging freely. The weights of the particles, and the angles between the sloping parts of the strings and the vertical, are as shown in the diagram. Find the values of \(W _ { 1 }\) and \(W _ { 2 }\).
[0pt] [6]
CAIE M1 2005 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-3_276_570_264_790} A stone slab of mass 320 kg rests in equilibrium on rough horizontal ground. A force of magnitude \(X \mathrm {~N}\) acts upwards on the slab at an angle of \(\theta\) to the vertical, where \(\tan \theta = \frac { 7 } { 24 }\) (see diagram).
  1. Find, in terms of \(X\), the normal component of the force exerted on the slab by the ground.
  2. Given that the coefficient of friction between the slab and the ground is \(\frac { 3 } { 8 }\), find the value of \(X\) for which the slab is about to slip.
CAIE M1 2005 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-3_917_1451_1059_347} The diagram shows the displacement-time graph for a car's journey. The graph consists of two curved parts \(A B\) and \(C D\), and a straight line \(B C\). The line \(B C\) is a tangent to the curve \(A B\) at \(B\) and a tangent to the curve \(C D\) at \(C\). The gradient of the curves at \(t = 0\) and \(t = 600\) is zero, and the acceleration of the car is constant for \(0 < t < 80\) and for \(560 < t < 600\). The displacement of the car is 400 m when \(t = 80\).
  1. Sketch the velocity-time graph for the journey.
  2. Find the velocity at \(t = 80\).
  3. Find the total distance for the journey.
  4. Find the acceleration of the car for \(0 < t < 80\).
CAIE M1 2005 November Q6
6 A particle \(P\) starts from rest at \(O\) and travels in a straight line. Its velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) is given by \(v = 8 t - 2 t ^ { 2 }\) for \(0 \leqslant t \leqslant 3\), and \(v = \frac { 54 } { t ^ { 2 } }\) for \(t > 3\). Find
  1. the distance travelled by \(P\) in the first 3 seconds,
  2. an expression in terms of \(t\) for the displacement of \(P\) from \(O\), valid for \(t > 3\),
  3. the value of \(v\) when the displacement of \(P\) from \(O\) is 27 m .
CAIE M1 2005 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-4_601_515_699_815} Two particles \(A\) and \(B\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. Particle \(B\) is held on the horizontal floor and particle \(A\) hangs in equilibrium. Particle \(B\) is released and each particle starts to move vertically with constant acceleration of magnitude \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Find the value of \(a\). Particle \(A\) hits the floor 1.2 s after it starts to move, and does not rebound upwards.
  2. Show that \(A\) hits the floor with a speed of \(2.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Find the gain in gravitational potential energy by \(B\), from leaving the floor until reaching its greatest height. \footnotetext{Every reasonable effort has been made to trace all copyright holders where the publishers (i.e. UCLES) are aware that third-party material has been reproduced.
    The publishers would be pleased to hear from anyone whose rights they have unwittingly infringed.
    University of Cambridge International Examinations is part of the University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE M1 2006 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{d0fa61eb-f320-427e-8883-de224d293933-2_421_1223_267_461} A box of mass 8 kg is pulled, at constant speed, up a straight path which is inclined at an angle of \(15 ^ { \circ }\) to the horizontal. The pulling force is constant, of magnitude 30 N , and acts upwards at an angle of \(10 ^ { \circ }\) from the path (see diagram). The box passes through the points \(A\) and \(B\), where \(A B = 20 \mathrm {~m}\) and \(B\) is above the level of \(A\). For the motion from \(A\) to \(B\), find
  1. the work done by the pulling force,
  2. the gain in potential energy of the box,
  3. the work done against the resistance to motion of the box.
CAIE M1 2006 November Q2
2
\includegraphics[max width=\textwidth, alt={}, center]{d0fa61eb-f320-427e-8883-de224d293933-2_701_323_1244_913} A small ring of mass 0.6 kg is threaded on a rough rod which is fixed vertically. The ring is in equilibrium, acted on by a force of magnitude 5 N pulling upwards at \(30 ^ { \circ }\) to the vertical (see diagram).
  1. Show that the frictional force acting on the ring has magnitude 1.67 N , correct to 3 significant figures.
  2. The ring is on the point of sliding down the rod. Find the coefficient of friction between the ring and the rod.
CAIE M1 2006 November Q3
3 A cyclist travels along a straight road working at a constant rate of 420 W . The total mass of the cyclist and her cycle is 75 kg . Ignoring any resistance to motion, find the acceleration of the cyclist at an instant when she is travelling at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
  1. given that the road is horizontal,
  2. given instead that the road is inclined at \(1.5 ^ { \circ }\) to the horizontal and the cyclist is travelling up the slope.
CAIE M1 2006 November Q4
4 The velocity of a particle \(t \mathrm {~s}\) after it starts from rest is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 1.25 t - 0.05 t ^ { 2 }\). Find
  1. the initial acceleration of the particle,
  2. the displacement of the particle from its starting point at the instant when its acceleration is \(0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
CAIE M1 2006 November Q5
5
\includegraphics[max width=\textwidth, alt={}, center]{d0fa61eb-f320-427e-8883-de224d293933-3_515_789_995_676} The diagram shows the vertical cross-section \(L M N\) of a fixed smooth surface. \(M\) is the lowest point of the cross-section. \(L\) is 2.45 m above the level of \(M\), and \(N\) is 1.2 m above the level of \(M\). A particle of mass 0.5 kg is released from rest at \(L\) and moves on the surface until it leaves it at \(N\). Find
  1. the greatest speed of the particle,
  2. the kinetic energy of the particle at \(N\). The particle is now projected from \(N\), with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), along the surface towards \(M\).
  3. Find the least value of \(v\) for which the particle will reach \(L\).
CAIE M1 2006 November Q6
6
\includegraphics[max width=\textwidth, alt={}, center]{d0fa61eb-f320-427e-8883-de224d293933-4_474_831_269_657} Forces of magnitudes \(P \mathrm {~N}\) and 25 N act at right angles to each other. The resultant of the two forces has magnitude \(R \mathrm {~N}\) and makes an angle of \(\theta ^ { \circ }\) with the \(x\)-axis (see diagram). The force of magnitude \(P \mathrm {~N}\) has components - 2.8 N and 9.6 N in the \(x\)-direction and the \(y\)-direction respectively, and makes an angle of \(\alpha ^ { \circ }\) with the negative \(x\)-axis.
  1. Find the values of \(P\) and \(R\).
  2. Find the value of \(\alpha\), and hence find the components of the force of magnitude 25 N in
    (a) the \(x\)-direction,
    (b) the \(y\)-direction.
  3. Find the value of \(\theta\).
CAIE M1 2006 November Q7
7 A particle of mass \(m \mathrm {~kg}\) moves up a line of greatest slope of a rough plane inclined at \(21 ^ { \circ }\) to the horizontal. The frictional and normal components of the contact force on the particle have magnitudes \(F \mathrm {~N}\) and \(R \mathrm {~N}\) respectively. The particle passes through the point \(P\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and 2 s later it reaches its highest point on the plane.
  1. Show that \(R = 9.336 m\) and \(F = 1.416 m\), each correct to 4 significant figures.
  2. Find the coefficient of friction between the particle and the plane. After the particle reaches its highest point it starts to move down the plane.
  3. Find the speed with which the particle returns to \(P\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
    University of Cambridge International Examinations is part of the University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE M1 2007 November Q1
1 A car of mass 900 kg travels along a horizontal straight road with its engine working at a constant rate of \(P \mathrm {~kW}\). The resistance to motion of the car is 550 N . Given that the acceleration of the car is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at an instant when its speed is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the value of \(P\).
CAIE M1 2007 November Q2
2 A particle is projected vertically upwards from a point \(O\) with initial speed \(12.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At the same instant another particle is released from rest at a point 10 m vertically above \(O\). Find the height above \(O\) at which the particles meet.
CAIE M1 2007 November Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{6853f050-45ed-4a76-b450-648d8ac91468-2_429_826_721_662} A particle is in equilibrium on a smooth horizontal table when acted on by the three horizontal forces shown in the diagram.
  1. Find the values of \(F\) and \(\theta\).
  2. The force of magnitude 7 N is now removed. State the magnitude and direction of the resultant of the remaining two forces.
CAIE M1 2007 November Q4
4
\includegraphics[max width=\textwidth, alt={}, center]{6853f050-45ed-4a76-b450-648d8ac91468-2_560_591_1663_776} The diagram shows the vertical cross-section of a surface. \(A\) and \(B\) are two points on the cross-section, and \(A\) is 5 m higher than \(B\). A particle of mass 0.35 kg passes through \(A\) with speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), moving on the surface towards \(B\).
  1. Assuming that there is no resistance to motion, find the speed with which the particle reaches \(B\).
  2. Assuming instead that there is a resistance to motion, and that the particle reaches \(B\) with speed \(11 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the work done against this resistance as the particle moves from \(A\) to \(B\). A ring of mass 4 kg is threaded on a fixed rough vertical rod. A light string is attached to the ring, and is pulled with a force of magnitude \(T \mathrm {~N}\) acting at an angle of \(60 ^ { \circ }\) to the downward vertical (see diagram). The ring is in equilibrium.
CAIE M1 2007 November Q7
7
\includegraphics[max width=\textwidth, alt={}, center]{6853f050-45ed-4a76-b450-648d8ac91468-4_309_1209_269_468} A rough inclined plane of length 65 cm is fixed with one end at a height of 16 cm above the other end. Particles \(P\) and \(Q\), of masses 0.13 kg and 0.11 kg respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley at the top of the plane. Particle \(P\) is held at rest on the plane and particle \(Q\) hangs vertically below the pulley (see diagram). The system is released from rest and \(P\) starts to move up the plane.
  1. Draw a diagram showing the forces acting on \(P\) during its motion up the plane.
  2. Show that \(T - F > 0.32\), where \(T \mathrm {~N}\) is the tension in the string and \(F \mathrm {~N}\) is the magnitude of the frictional force on \(P\). The coefficient of friction between \(P\) and the plane is 0.6 .
  3. Find the acceleration of \(P\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE M1 2008 November Q1
1
\includegraphics[max width=\textwidth, alt={}, center]{a4cb105b-55d2-4793-95d2-3d791990a1f6-2_341_929_269_609} Forces of magnitudes 10 N and 8 N act in directions as shown in the diagram.
  1. Write down in terms of \(\theta\) the component of the resultant of the two forces
    (a) parallel to the force of magnitude 10 N ,
    (b) perpendicular to the force of magnitude 10 N .
  2. The resultant of the two forces has magnitude 8 N . Show that \(\cos \theta = \frac { 5 } { 8 }\).
CAIE M1 2008 November Q2
2 A block of mass 20 kg is at rest on a plane inclined at \(10 ^ { \circ }\) to the horizontal. A force acts on the block parallel to a line of greatest slope of the plane. The coefficient of friction between the block and the plane is 0.32 . Find the least magnitude of the force necessary to move the block,
  1. given that the force acts up the plane,
  2. given instead that the force acts down the plane.
CAIE M1 2008 November Q3
3 A car of mass 1200 kg is travelling on a horizontal straight road and passes through a point \(A\) with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The power of the car's engine is 18 kW and the resistance to the car's motion is 900 N .
  1. Find the deceleration of the car at \(A\).
  2. Show that the speed of the car does not fall below \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) while the car continues to move with the engine exerting a constant power of 18 kW .
CAIE M1 2008 November Q4
4 A load of mass 160 kg is lifted vertically by a crane, with constant acceleration. The load starts from rest at the point \(O\). After 7 s , it passes through the point \(A\) with speed \(0.5 \mathrm {~ms} ^ { - 1 }\). By considering energy, find the work done by the crane in moving the load from \(O\) to \(A\).