Questions M3 (796 questions)

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Edexcel M3 Q6
13 marks Challenging +1.2
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8b7133ed-3748-46cb-99d2-570ee33c7393-4_526_620_196_598} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a bowl formed by removing from a solid hemisphere of radius \(\frac { 3 } { 2 } r\) a smaller hemisphere of radius \(r\) having the same axis of symmetry and the same plane face.
  1. Show that the centre of mass of the bowl is a distance of \(\frac { 195 } { 304 } r\) from its plane face.
    (7 marks)
    The bowl has mass \(M\) and is placed with its curved surface on a smooth horizontal plane. A stud of mass \(\frac { 1 } { 2 } M\) is attached to the outer rim of the bowl. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8b7133ed-3748-46cb-99d2-570ee33c7393-4_517_729_1318_539} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} When the bowl is in equilibrium its plane surface is inclined at an angle \(\alpha\) to the horizontal as shown in Figure 2.
  2. Find tan \(\alpha\).
    (6 marks)
Edexcel M3 Q7
18 marks Standard +0.8
7. A cyclist is travelling round a circular bend of radius 25 m on a track which is banked at an angle of \(35 ^ { \circ }\) to the horizontal. In a model of the situation, the cyclist and her bicycle are represented by a particle of mass 60 kg and air resistance and friction are ignored. Using this model and assuming that the cyclist is not slipping,
  1. find, correct to 3 significant figures, the speed at which she is travelling. In tests it is found that the cyclist must travel at a minimum speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to prevent the bicycle from slipping down the slope. A more refined model is now used with a coefficient of friction between the bicycle and the track of \(\mu\). Using this model,
  2. show that \(\mu = 0.227\), correct to 3 significant figures,
  3. find, correct to 2 significant figures, the maximum speed at which the cyclist can travel without slipping up the slope. END
Edexcel M3 Q1
7 marks Standard +0.3
  1. The velocity, \(\mathbf { v ~ c m ~ s } { } ^ { - 1 }\), at time \(t\) seconds, of a radio-controlled toy is modelled by the formula
$$\mathbf { v } = \mathrm { e } ^ { 2 t } \mathbf { i } + 2 t \mathbf { j } ,$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors.
  1. Find the acceleration of the toy in terms of \(t\).
  2. Find, correct to 2 significant figures, the time at which the acceleration of the toy is parallel to the vector \(( 4 \mathbf { i } + \mathbf { j } )\).
  3. Explain why this model is unlikely to be realistic for large values of \(t\).
Edexcel M3 Q2
8 marks Standard +0.3
2. A particle \(P\) of mass 0.4 kg is moving in a straight line through a fixed point \(O\). At time \(t\) seconds after it passes through \(O\), the distance \(O P\) is \(x\) metres and the resultant force acting on \(P\) is of magnitude ( \(5 + 4 \mathrm { e } ^ { - x }\) ) N in the direction \(O P\). When \(x = 1 , P\) is at the point \(A\).
  1. Find, correct to 3 significant figures, the work done in moving \(P\) from \(O\) to \(A\). Given that \(P\) passes through \(O\) with speed \(2 \mathrm {~ms} ^ { - 1 }\),
  2. find, correct to 3 significant figures, the speed of \(P\) as it passes through \(A\).
Edexcel M3 Q3
8 marks Moderate -0.5
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00776cc0-0214-4029-8ef1-c1cba89f4b87-2_382_796_1640_479} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A popular racket game involves a tennis ball of mass 0.1 kg which is attached to one end of a light inextensible string. The other end of the string is attached to the top of a fixed rigid pole. A boy strikes the ball such that it moves in a horizontal circle with angular speed \(4 \mathrm { rad } \mathrm { s } ^ { - 1 }\) and the string makes an angle of \(60 ^ { \circ }\) with the downward vertical as shown in Figure 1.
  1. Find the tension in the string.
  2. Find the length of the string.
Edexcel M3 Q4
9 marks Standard +0.3
4. A particle moves with simple harmonic motion along a straight line. When the particle is 3 cm from its centre of motion it has a speed of \(8 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\) and an acceleration of magnitude \(12 \mathrm {~cm} \mathrm {~s} ^ { - 2 }\).
  1. Show that the period of the motion is \(\pi\) seconds.
  2. Find the amplitude of the motion.
  3. Hence, find the greatest speed of the particle.
Edexcel M3 Q5
10 marks Standard +0.3
5. A physics student is set the task of finding the mass of an object without using a set of scales. She decides to use a light elastic string of natural length 2 m and modulus of elasticity 280 N attached to two points \(A\) and \(B\) which are on the same horizontal level and 2.4 m apart. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00776cc0-0214-4029-8ef1-c1cba89f4b87-3_307_1072_993_438} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} She attaches the object to the midpoint of the string so that it hangs in equilibrium 0.35 m below \(A B\) as shown in Figure 2.
  1. Explain why it is reasonable to assume that the tensions in each half of the string are equal.
  2. Find the mass of the object.
  3. Find the elastic potential energy of the string when the object is suspended from it.
Edexcel M3 Q6
13 marks Challenging +1.2
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00776cc0-0214-4029-8ef1-c1cba89f4b87-4_455_540_201_660} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 shows part of the curve \(y = x ^ { 2 } + 1\). The shaded region enclosed by the curve, the coordinate axes and the line \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
  1. Find the coordinates of the centre of mass of the solid obtained. The solid is suspended from a point on its larger circular rim and hangs in equilibrium.
  2. Find, correct to the nearest degree, the acute angle which the plane surfaces of the solid make with the vertical.
    (3 marks)
Edexcel M3 Q7
20 marks Standard +0.8
7. A particle of mass 0.5 kg is hanging vertically at one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a fixed point. The particle is given an initial horizontal speed of \(u \mathrm {~ms} ^ { - 1 }\).
  1. Show that the particle will perform complete circles if \(u \geq \sqrt { 3 g }\). Given that \(u = 5\),
  2. find, correct to the nearest degree, the angle through which the string turns before it becomes slack,
  3. find, correct to the nearest centimetre, the greatest height the particle reaches above its position when the string becomes slack.
Edexcel M3 Q1
8 marks Moderate -0.3
  1. A particle \(P\) of mass 1.5 kg moves from rest at the origin such that at time \(t\) seconds it is subject to a single force of magnitude \(( 4 t + 3 ) \mathrm { N }\) in the direction of the positive \(x\)-axis.
    1. Find the magnitude of the impulse exerted by the force during the interval \(1 \leq t \leq 4\).
    Given that at time \(T\) seconds, \(P\) has a speed of \(22 \mathrm {~ms} ^ { - 1 }\),
  2. find the value of \(T\) correct to 3 significant figures.
Edexcel M3 Q2
8 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0668f31-4b72-4dfd-9cf7-470acef0bfdb-2_469_465_776_680} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A particle \(P\) of mass 0.5 kg is at rest at the highest point \(A\) of a smooth sphere, centre \(O\), of radius 1.25 m which is fixed to a horizontal surface. When \(P\) is slightly disturbed it slides along the surface of the sphere. Whilst \(P\) is in contact with the sphere it has speed \(v \mathrm {~ms} ^ { - 1 }\) when \(\angle A O P = \theta\) as shown in Figure 1.
  1. Show that \(v ^ { 2 } = 24.5 ( 1 - \cos \theta )\).
  2. Find the value of \(\cos \theta\) when \(P\) leaves the surface of the sphere.
Edexcel M3 Q3
12 marks Standard +0.8
3. A car starts from rest at the point \(O\) and moves along a straight line. The car accelerates to a maximum velocity, \(V \mathrm {~ms} ^ { - 1 }\), before decelerating and coming to rest again at the point \(A\). The acceleration of the car during this journey, \(a \mathrm {~ms} ^ { - 2 }\), is modelled by the formula $$a = \frac { 500 - k x } { 150 }$$ where \(x\) is the distance in metres of the car from \(O\).
Using this model and given that the car is travelling at \(16 \mathrm {~ms} ^ { - 1 }\) when it is 40 m from \(O\),
  1. find \(k\),
  2. show that \(V = 41\), correct to 2 significant figures,
  3. find the distance \(O A\).
Edexcel M3 Q4
12 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0668f31-4b72-4dfd-9cf7-470acef0bfdb-3_316_536_1087_639} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A particle \(P\) of mass 2 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity \(\lambda\). The other end of the string is fixed to a point \(A\) on a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal as shown in Figure 2. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 6 } \sqrt { 3 }\). \(P\) is held at rest at \(A\) and then released. It first comes to instantaneous rest at the point \(B , 2.2 \mathrm {~m}\) from \(A\). For the motion of \(P\) from \(A\) to \(B\),
  1. show that the work done against friction is 10.78 J ,
  2. find the change in the gravitational potential energy of \(P\). By using the work-energy principle, or otherwise,
  3. find \(\lambda\).
Edexcel M3 Q5
16 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e0668f31-4b72-4dfd-9cf7-470acef0bfdb-4_693_554_196_717} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} A flask is modelled as a uniform solid formed by removing a cylinder of radius \(r\) and height \(h\) from a cylinder of radius \(\frac { 4 } { 3 } r\) and height \(\frac { 3 } { 2 } h\) with the same axis of symmetry and a common plane as shown in Figure 3.
  1. Show that the centre of mass of the flask is a distance of \(\frac { 9 } { 10 } h\) from the open end of the flask. The flask is made from a material of density \(\rho\) and is filled to the level of the open plane face with a liquid of density \(k \rho\). Given that the centre of mass of the flask and liquid together is a distance of \(\frac { 15 } { 22 } h\) from the open end of the flask,
  2. find the value of \(k\).
  3. Explain why it may be advantageous to make the base of the flask from a more dense material.
    (2 marks)
Edexcel M3 Q6
19 marks Standard +0.8
6. A particle \(P\) of mass 2.5 kg is moving with simple harmonic motion in a straight line between two points \(A\) and \(B\) on a smooth horizontal table. When \(P\) is 3 m from \(O\), the centre of the oscillations, its speed is \(6 \mathrm {~ms} ^ { - 1 }\). When \(P\) is 2.25 m from \(O\), its speed is \(8 \mathrm {~ms} ^ { - 1 }\).
  1. Show that \(A B = 7.5 \mathrm {~m}\).
  2. Find the period of the motion.
  3. Find the kinetic energy of \(P\) when it is 2.7 m from \(A\).
  4. Show that the time taken by \(P\) to travel directly from \(A\) to the midpoint of \(O B\) is \(\frac { \pi } { 4 }\) seconds.
Edexcel M3 2003 January Q3
10 marks Challenging +1.2
  1. Show that the distance \(d\) of the centre of mass of the toy from its lowest point \(O\) is given by $$d = \frac { h ^ { 2 } + 2 h r + 5 r ^ { 2 } } { 2 ( h + 4 r ) } .$$ When the toy is placed with any point of the curved surface of the hemisphere resting on the plane it will remain in equilibrium.
  2. Find \(h\) in terms of \(r\).
    (3)
OCR M3 2009 January Q4
10 marks Standard +0.8
  1. Show that \(v ^ { 2 } = 9 + 9.8 \sin \theta\).
  2. Find, in terms of \(\theta\), the radial and tangential components of the acceleration of \(P\).
  3. Show that the tension in the string is \(( 3.6 + 5.88 \sin \theta ) \mathrm { N }\) and hence find the value of \(\theta\) at the instant when the string becomes slack, giving your answer correct to 1 decimal place.
OCR M3 2010 January Q6
13 marks Challenging +1.2
  1. By considering the total energy of the system, obtain an expression for \(v ^ { 2 }\) in terms of \(\theta\).
  2. Show that the magnitude of the force exerted on \(P\) by the cylinder is \(( 7.12 \sin \theta - 4.64 \theta ) \mathrm { N }\).
  3. Given that \(P\) leaves the surface of the cylinder when \(\theta = \alpha\), show that \(1.53 < \alpha < 1.54\).
OCR M3 2007 June Q6
13 marks Challenging +1.8
  1. Show that, when \(P\) is in equilibrium, \(O P = 7.25 \mathrm {~m}\).
  2. Verify that \(P\) and \(Q\) together just reach the safety net.
  3. At the lowest point of their motion \(P\) releases \(Q\). Prove that \(P\) subsequently just reaches \(O\).
  4. State two additional modelling assumptions made when answering this question.
OCR M3 2006 January Q10
Standard +0.8
10 JANUARY 2006 Afternoon
1 hour 30 minutes
Additional materials:
8 page answer booklet
Graph paper
List of Formulae (MF1) TIME
1 hour 30 minutes
  • Write your name, centre number and candidate number in the spaces provided on the answer booklet.
  • Answer all the questions.
  • Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy is specified in the question or is clearly appropriate.
  • The acceleration due to gravity is denoted by \(\mathrm { g } \mathrm { m } \mathrm { s } ^ { - 2 }\). Unless otherwise instructed, when a numerical value is needed, use \(g = 9.8\).
  • You are permitted to use a graphical calculator in this paper.
  • 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 72.
  • Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper.
  • You are reminded of the need for clear presentation in your answers.
AQA M3 2009 June Q1
5 marks Standard +0.3
1 A ball of mass \(m\) is travelling vertically downwards with speed \(u\) when it hits a horizontal floor. The ball bounces vertically upwards to a height \(h\). It is thought that \(h\) depends on \(m , u\), the acceleration due to gravity \(g\), and a dimensionless constant \(k\), such that $$h = k m ^ { \alpha } u ^ { \beta } g ^ { \gamma }$$ where \(\alpha , \beta\) and \(\gamma\) are constants.
By using dimensional analysis, find the values of \(\alpha , \beta\) and \(\gamma\).
AQA M3 2009 June Q2
10 marks Standard +0.3
2 A particle is projected from a point \(O\) on a horizontal plane and has initial velocity components of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(10 \mathrm {~ms} ^ { - 1 }\) parallel to and perpendicular to the plane respectively. At time \(t\) seconds after projection, the horizontal and upward vertical distances of the particle from the point \(O\) are \(x\) metres and \(y\) metres respectively.
  1. Show that \(x\) and \(y\) satisfy the equation $$y = - \frac { g } { 8 } x ^ { 2 } + 5 x$$
  2. By using the equation in part (a), find the horizontal distance travelled by the particle whilst it is more than 1 metre above the plane.
  3. Hence find the time for which the particle is more than 1 metre above the plane.
AQA M3 2009 June Q3
14 marks Standard +0.8
3 A fishing boat is travelling between two ports, \(A\) and \(B\), on the shore of a lake. The bearing of \(B\) from \(A\) is \(130 ^ { \circ }\). The fishing boat leaves \(A\) and travels directly towards \(B\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A patrol boat on the lake is travelling with speed \(4 \mathrm {~ms} ^ { - 1 }\) on a bearing of \(040 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-3_713_1319_443_406}
  1. Find the velocity of the fishing boat relative to the patrol boat, giving your answer as a speed together with a bearing.
  2. When the patrol boat is 1500 m due west of the fishing boat, it changes direction in order to intercept the fishing boat in the shortest possible time.
    1. Find the bearing on which the patrol boat should travel in order to intercept the fishing boat.
    2. Given that the patrol boat intercepts the fishing boat before it reaches \(B\), find the time, in seconds, that it takes the patrol boat to intercept the fishing boat after changing direction.
    3. State a modelling assumption necessary for answering this question, other than the boats being particles.
AQA M3 2009 June Q4
10 marks Standard +0.3
4 A particle of mass 0.5 kg is initially at rest. The particle then moves in a straight line under the action of a single force. This force acts in a constant direction and has magnitude \(\left( t ^ { 3 } + t \right) \mathrm { N }\), where \(t\) is the time, in seconds, for which the force has been acting.
  1. Find the magnitude of the impulse exerted by the force on the particle between the times \(t = 0\) and \(t = 4\).
  2. Hence find the speed of the particle when \(t = 4\).
  3. Find the time taken for the particle to reach a speed of \(12 \mathrm {~ms} ^ { - 1 }\).
AQA M3 2009 June Q5
12 marks Challenging +1.2
5 Two smooth spheres, \(A\) and \(B\), of equal radii and different masses are moving on a smooth horizontal surface when they collide. Just before the collision, \(A\) is moving with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) to the line of centres of the spheres, and \(B\) is moving with speed \(3 \mathrm {~ms} ^ { - 1 }\) perpendicular to the line of centres, as shown in the diagram below. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{719b82f7-2ab5-48db-9b2a-98284096a78a-4_314_1100_593_392} \captionsetup{labelformat=empty} \caption{Before collision}
\end{figure} Immediately after the collision, \(A\) and \(B\) move with speeds \(u\) and \(v\) in directions which make angles of \(90 ^ { \circ }\) and \(40 ^ { \circ }\) respectively with the line of centres, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{719b82f7-2ab5-48db-9b2a-98284096a78a-4_392_1102_1155_392}
  1. Show that \(v = 4.67 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to three significant figures.
  2. Find the coefficient of restitution between the spheres.
  3. Given that the mass of \(A\) is 0.5 kg , show that the magnitude of the impulse exerted on \(A\) during the collision is 2.17 Ns , correct to three significant figures.
  4. Find the mass of \(B\).