Questions M1 (1912 questions)

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OCR MEI M1 Q4
4 A small box has weight \(\mathbf { W } \mathrm { N }\) and is held in equilibrium by two strings with tensions \(\mathbf { T } _ { 1 } \mathrm {~N}\) and \(\mathbf { T } _ { 2 } \mathrm {~N}\). This situation is shown in Fig. 2 which also shows the standard unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) that are horizontal and vertically upwards, respectively. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b80eced6-2fea-4b95-9104-d13339643df0-2_252_631_414_803} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} The tension \(\mathbf { T } _ { 1 }\) is \(10 \mathbf { i } + 24 \mathbf { j }\).
  1. Calculate the magnitude of \(\mathbf { T } _ { 1 }\) and the angle between \(\mathbf { T } _ { 1 }\) and the vertical. The magnitude of the weight is \(w \mathrm {~N}\).
  2. Write down the vector \(\mathbf { W }\) in terms of \(w\) and \(\mathbf { j }\). The tension \(\mathbf { T } _ { 2 }\) is \(k \mathbf { i } + 10 \mathbf { j }\), where \(k\) is a scalar.
  3. Find the values of \(k\) and of \(w\).
OCR MEI M1 Q5
5 A particle has a position vector \(\mathbf { r }\), where \(\mathbf { r } = 4 \mathbf { i } - 5 \mathbf { j }\) and \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the directions east and north respectively.
  1. Sketch \(\mathbf { r }\) on a diagram showing \(\mathbf { i }\) and \(\mathbf { j }\) and the origin O .
  2. Calculate the magnitude of \(\mathbf { r }\) and its direction as a bearing.
  3. Write down the vector that has the same direction as \(\mathbf { r }\) and three times its magnitude.
OCR MEI M1 Q6
6 Force \(\mathbf { F } _ { 1 }\) is \(\binom { 6 } { 13 } \mathrm {~N}\) and force \(\mathbf { F } _ { 2 }\) is \(\binom { 3 } { 5 }\), where \(\left. \int _ { 0 } \right] _ { \text {and } } \binom { 0 } { 1 }\) are vectors east and north respectively.
  1. Calculate the magnitude of \(\mathbf { F } _ { 1 }\), correct to three significant figures.
  2. Calculate the direction of the force \(\mathbf { F } _ { 1 } - \mathbf { F } _ { 2 }\) as a bearing. Force \(\mathbf { F } _ { 2 }\) is the resultant of all the forces acting on an object of mass 5 kg .
  3. Calculate the acceleration of the object and the change in its velocity after 10 seconds.
OCR MEI M1 Q1
1 A particle rests on a smooth, horizontal plane. Horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) lie in this plane. The particle is in equilibrium under the action of the three forces \(( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { N }\) and \(( 21 \mathbf { i } - 7 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { R N }\).
  1. Write down an expression for \(\mathbf { R }\) in terms of \(\mathbf { i }\) and \(\mathbf { j }\).
  2. Find the magnitude of \(\mathbf { R }\) and the angle between \(\mathbf { R }\) and the \(\mathbf { i }\) direction.
OCR MEI M1 Q2
2 The position vector of a particle at time \(t\) is given by $$\mathbf { r } = \frac { 1 } { 2 } t \mathbf { i } + \left( t ^ { 2 } - 1 \right) \mathbf { j } .$$ referred to an origin \(O\) where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors in the directions of the cartesian axes Ox and Oy respectively.
  1. Write down the value of \(t\) for which the \(x\)-coordinate of the position of the particle is 2 . Find the \(y\)-coordinate at this time.
  2. Show that the cartesian equation of the path of the particle is \(y = 4 x ^ { 2 } - 1\).
  3. Find the coordinates of the point where the particle is moving at \(45 ^ { \circ }\) to both Ox and Oy .
OCR MEI M1 Q3
3 The vectors \(\mathbf { p }\) and \(\mathbf { q }\) are given by $$\mathbf { p } = 8 \mathbf { i } + \mathbf { j } \text { and } \mathbf { q } = 4 \mathbf { i } - 7 \mathbf { j } .$$
  1. Show that \(\mathbf { p }\) and \(\mathbf { q }\) are equal in magnitude.
  2. Show that \(\mathbf { p } + \mathbf { q }\) is parallel to \(2 \mathbf { i } - \mathbf { j }\).
  3. Draw \(\mathbf { p } + \mathbf { q }\) and \(\mathbf { p } - \mathbf { q }\) on the grid. Write down the angle between these two vectors.
OCR MEI M1 Q4
4 In this question, \(\mathbf { i }\) is a horizontal unit vector and \(\mathbf { j }\) is a unit vector pointing vertically upwards.
A force \(\mathbf { F }\) is \(- \mathbf { i } + 5 \mathbf { j }\).
  1. Calculate the magnitude of \(\mathbf { F }\). Calculate also the angle between \(\mathbf { F }\) and the upward vertical. Force \(\mathbf { G }\) is \(2 a \mathbf { i } + a \mathbf { j }\) and force \(\mathbf { H }\) is \(- 2 \mathbf { i } + 3 b \mathbf { j }\), where \(a\) and \(b\) are constants. The force \(\mathbf { H }\) is the resultant of forces \(4 \mathbf { F }\) and \(\mathbf { G }\).
  2. Find \(\mathbf { G }\) and \(\mathbf { H }\).
OCR MEI M1 Q5
5 The resultant of the force \(\binom { - 4 } { 8 } \mathrm {~N}\) and the force \(\mathbf { F }\) gives an object of mass 6 kg an acceleration of \(\binom { 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Calculate \(\mathbf { F }\).
  2. Calculate the angle between \(\mathbf { F }\) and the vector \(\binom { 0 } { 1 }\).
OCR MEI M1 Q6
6 The force acting on a particle of mass 1.5 kg is given by the vector \(\binom { 6 } { 9 } \mathrm {~N}\).
  1. Give the acceleration of the particle as a vector.
  2. Calculate the angle that the acceleration makes with the direction \(\binom { 1 } { 0 }\).
  3. At a certain point of its motion, the particle has a velocity of \(\binom { - 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate the displacement of the particle over the next two seconds.
OCR MEI M1 Q7
7 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 \(( 7 \mathbf { i } + 24 \mathbf { j } ) \mathrm { N }\). 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 \(\mathbf { F }\).
CAIE M1 2022 June Q4
  1. In the case where \(F = 20\), find the tensions in each of the strings.
  2. Find the greatest value of \(F\) for which the block remains in equilibrium in the position shown.
CAIE M1 2022 June Q6
  1. It is given that the plane \(B C\) is smooth and that the particles are released from rest. Find the tension in the string and the magnitude of the acceleration of the particles.
  2. It is given instead that the plane \(B C\) is rough. A force of magnitude 3 N is applied to \(Q\) directly up the plane along a line of greatest slope of the plane. Find the least value of the coefficient of friction between \(Q\) and the plane \(B C\) for which the particles remain at rest.
CAIE M1 2023 June Q4
  1. Given that the forces are in equilibrium, find the value of \(F\) and the value of \(\theta\).
  2. Given instead that \(F = 10 \sqrt { 2 }\) and \(\theta = 45\), find the direction and the exact magnitude the resultant force.
    \includegraphics[max width=\textwidth, alt={}, center]{f9e3d562-ae3c-49cc-bc92-56956d939252-10_518_627_264_756} Two particles \(P\) and \(Q\), of masses 0.2 kg and 0.1 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley at \(B\) which is attached to two inclined planes. Particle \(P\) lies on a smooth plane \(A B\) which is inclined at \(60 ^ { \circ }\) to the horizontal. Particle \(Q\) lies on a plane \(B C\) which is inclined at an angle of \(\theta ^ { \circ }\) to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes (see diagram).
  3. It is given that \(\theta = 60\), the plane \(B C\) is rough and the coefficient of friction between \(Q\) and the plane \(B C\) is 0.7 . The particles are released from rest. Determine whether the particles move.
  4. It is given instead that the plane \(B C\) is smooth. The particles are released from rest and in the subsequent motion the tension in the string is \(( \sqrt { 3 } - 1 ) \mathrm { N }\). Find the magnitude of the acceleration of \(P\) as it moves on the plane, and find the value of \(\theta\).
CAIE M1 2023 March Q5
  1. Find the magnitude of the force in each of the struts \(A D\) and \(B D\).
    A horizontal force of magnitude \(F \mathrm {~N}\) is applied to the block in a direction parallel to \(A B\).
  2. Find the value of \(F\) for which the magnitude of the force in the strut \(A D\) is zero.
    \includegraphics[max width=\textwidth, alt={}, center]{b2cd1b68-523f-40c3-8a51-acb2b55ae8c0-08_456_782_260_687} A block \(B\), of mass 2 kg , lies on a rough inclined plane sloping at \(30 ^ { \circ }\) to the horizontal. A light rope, inclined at an angle of \(20 ^ { \circ }\) above a line of greatest slope, is attached to \(B\). The tension in the rope is \(T \mathrm {~N}\). There is a friction force of \(F \mathrm {~N}\) acting on \(B\) (see diagram). The coefficient of friction between \(B\) and the plane is \(\mu\).
  3. It is given that \(F = 5\) and that the acceleration of \(B\) up the plane is \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    1. Find the value of \(T\).
    2. Find the value of \(\mu\).
  4. It is given instead that \(\mu = 0.8\) and \(T = 15\). Determine whether \(B\) will move up the plane.
CAIE M1 2011 June Q4
  1. Make a rough copy of the diagram and shade the region whose area represents the displacement of \(P\) from \(X\) at the instant when \(Q\) starts. It is given that \(P\) has travelled 70 m at the instant when \(Q\) starts.
  2. Find the value of \(T\).
  3. Find the distance between \(P\) and \(Q\) when \(Q\) 's speed reaches \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  4. Sketch a single diagram showing the displacement-time graphs for both \(P\) and \(Q\), with values shown on the \(t\)-axis at which the speed of either particle changes.
CAIE M1 2015 June Q6
  1. Find the value of \(h\).
  2. Find the value of \(m\), and find also the tension in the string while \(Q\) is moving.
  3. The string is slack while \(Q\) is at rest on the ground. Find the total time from the instant that \(P\) is released until the string becomes taut again.
CAIE M1 2019 June Q4
  1. Show that, before the string breaks, the magnitude of the acceleration of each particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and find the tension in the string.
  2. Find the difference in the times that it takes the particles to hit the ground.
CAIE M1 2007 November Q5
  1. The normal and frictional components of the contact force exerted on the ring by the rod are \(R \mathrm {~N}\) and \(F\) N respectively. Find \(R\) and \(F\) in terms of \(T\).
  2. The coefficient of friction between the rod and the ring is 0.7 . Find the value of \(T\) for which the ring is about to slip.
  3. A man walks in a straight line from \(A\) to \(B\) with constant acceleration \(0.004 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). His speed at \(A\) is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and his speed at \(B\) is \(2.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the time taken for the man to walk from \(A\) to \(B\), and find the distance \(A B\).
  4. A woman cyclist leaves \(A\) at the same instant as the man. She starts from rest and travels in a straight line to \(B\), reaching \(B\) at the same instant as the man. At time \(t \mathrm {~s}\) after leaving \(A\) the cyclist's speed is \(k \left( 200 t - t ^ { 2 } \right) \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a constant. Find
    (a) the value of \(k\),
    (b) the cyclist's speed at \(B\).
  5. Sketch, using the same axes, the velocity-time graphs for the man's motion and the woman's motion from \(A\) to \(B\).
CAIE M1 2011 November Q5
  1. Show that \(\mu \geqslant \frac { 6 } { 17 }\). When the applied force acts upwards as in Fig. 2 the block slides along the floor.
  2. Find another inequality for \(\mu\).
CAIE M1 2012 November Q5
  1. Find the value of \(\theta\). At time 4.8 s after leaving \(A\), the particle comes to rest at \(C\).
  2. Find the coefficient of friction between \(P\) and the rough part of the plane.
CAIE M1 2014 November Q6
  1. the work done against the frictional force acting on \(B\),
  2. the loss of potential energy of the system,
  3. the gain in kinetic energy of the system. At the instant when \(B\) has moved 0.9 m the string breaks. \(A\) is at a height of 0.54 m above a horizontal floor at this instant.
    (ii) Find the speed with which \(A\) reaches the floor.
    \(6 \quad A B C\) is a line of greatest slope of a plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). The point \(A\) is at the top of the plane, the point \(C\) is at the bottom of the plane and the length of \(A C\) is 5 m . The part of the plane above the level of \(B\) is smooth and the part below the level of \(B\) is rough. A particle \(P\) is released from rest at \(A\) and reaches \(C\) with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of friction between \(P\) and the part of the plane below \(B\) is 0.5 . Find
    (i) the acceleration of \(P\) while moving
  4. from \(A\) to \(B\),
  5. from \(B\) to \(C\),
    (ii) the distance \(A B\),
    (iii) the time taken for \(P\) to move from \(A\) to \(C\).
CAIE M1 2017 November Q6
  1. Show that the coefficient of friction between \(P\) and the plane is \(\frac { 4 } { 3 }\).
    A force of magnitude 10 N is applied to \(P\), acting up a line of greatest slope of the plane, and \(P\) accelerates at \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. Find the value of \(m\).
CAIE M1 2019 November Q4
  1. Find the acceleration of the blocks and the tension in the string.
  2. At a particular instant, the speed of the blocks is \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the time, after this instant, that it takes for the blocks to travel 0.65 m .
    \includegraphics[max width=\textwidth, alt={}, center]{dd1828e1-5b90-4584-92de-f00f9c4f9657-08_574_895_260_625} A small ring \(P\) is threaded on a fixed smooth horizontal \(\operatorname { rod } A B\). Three horizontal forces of magnitudes \(4.5 \mathrm {~N} , 7.5 \mathrm {~N}\) and \(F \mathrm {~N}\) act on \(P\) (see diagram).
  3. Given that these three forces are in equilibrium, find the values of \(F\) and \(\theta\).
  4. It is given instead that the values of \(F\) and \(\theta\) are 9.5 and 30 respectively, and the acceleration of the ring is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the mass of the ring.
CAIE M1 Specimen Q4
  1. Find the values of \(F\) and \(R\).
  2. Initially the bead is at rest at \(A\). It reaches \(B\) with a speed of \(11.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the mass of the bead.
Edexcel M1 2013 June Q2
  1. the tension in the cable,
  2. the magnitude of the force exerted on the woman by the floor of the lift.
    \item \end{enumerate} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3c8dce6f-367a-42bb-be60-d03d0a23664f-04_616_780_118_584} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A box of mass 2 kg is held in equilibrium on a fixed rough inclined plane by a rope. The rope lies in a vertical plane containing a line of greatest slope of the inclined plane. The rope is inclined to the plane at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), and the plane is at an angle of \(30 ^ { \circ }\) to the horizontal, as shown in Figure 1. The coefficient of friction between the box and the inclined plane is \(\frac { 1 } { 3 }\) and the box is on the point of slipping up the plane. By modelling the box as a particle and the rope as a light inextensible string, find the tension in the rope.