Questions — Edexcel M3 (510 questions)

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Edexcel M3 2010 January Q6
14 marks Standard +0.3
6. A bend of a race track is modelled as an arc of a horizontal circle of radius 120 m . The track is not banked at the bend. The maximum speed at which a motorcycle can be ridden round the bend without slipping sideways is \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The motorcycle and its rider are modelled as a particle and air resistance is assumed to be negligible.
  1. Show that the coefficient of friction between the motorcycle and the track is \(\frac { 2 } { 3 }\). The bend is now reconstructed so that the track is banked at an angle \(\alpha\) to the horizontal. The maximum speed at which the motorcycle can now be ridden round the bend without slipping sideways is \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The radius of the bend and the coefficient of friction between the motorcycle and the track are unchanged.
  2. Find the value of \(\tan \alpha\).
Edexcel M3 2010 January Q7
14 marks Standard +0.3
7. A light elastic string has natural length \(a\) and modulus of elasticity \(\frac { 3 } { 2 } m g\). A particle \(P\) of mass \(m\) is attached to one end of the string. The other end of the string is attached to a fixed point \(A\). The particle is released from rest at \(A\) and falls vertically. When \(P\) has fallen a distance \(a + x\), where \(x > 0\), the speed of \(P\) is \(v\).
  1. Show that \(v ^ { 2 } = 2 g ( a + x ) - \frac { 3 g x ^ { 2 } } { 2 a }\).
  2. Find the greatest speed attained by \(P\) as it falls. After release, \(P\) next comes to instantaneous rest at a point \(D\).
  3. Find the magnitude of the acceleration of \(P\) at \(D\).
Edexcel M3 2012 January Q1
4 marks Standard +0.3
  1. A particle of mass 0.8 kg is attached to one end of a light elastic string of natural length 0.6 m . The other end of the string is attached to a fixed point \(A\). The particle is released from rest at \(A\) and comes to instantaneous rest 1.1 m below \(A\).
Find the modulus of elasticity of the string.
Edexcel M3 2012 January Q2
8 marks Standard +0.3
2. A particle \(P\) is moving in a straight line with simple harmonic motion. The centre of the oscillation is the fixed point \(C\), the amplitude of the oscillation is 0.5 m and the time to complete one oscillation is \(\frac { 2 \pi } { 3 }\) seconds. The point \(A\) is on the path of \(P\) and 0.2 m from \(C\). Find
  1. the magnitude and direction of the acceleration of \(P\) when it passes through \(A\),
  2. the speed of \(P\) when it passes through \(A\),
  3. the time \(P\) takes to move directly from \(C\) to \(A\).
Edexcel M3 2012 January Q3
10 marks Standard +0.8
3. A particle \(P\) is moving in a straight line. At time \(t\) seconds, \(P\) is at a distance \(x\) metres from a fixed point \(O\) on the line and is moving away from \(O\) with speed \(\frac { 10 } { x + 6 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the acceleration of \(P\) when \(x = 14\) Given that \(x = 2\) when \(t = 1\),
  2. find the value of \(t\) when \(x = 14\)
Edexcel M3 2012 January Q4
10 marks Standard +0.3
4. A light elastic string \(A B\) has natural length 0.8 m and modulus of elasticity 19.6 N . The end \(A\) is attached to a fixed point. A particle of mass 0.5 kg is attached to the end \(B\). The particle is moving with constant angular speed \(\omega\) rad s \(^ { - 1 }\) in a horizontal circle whose centre is vertically below \(A\). The string is inclined at \(60 ^ { \circ }\) to the vertical.
  1. Show that the extension of the string is 0.4 m .
  2. Find the value of \(\omega\).
Edexcel M3 2012 January Q5
12 marks Standard +0.8
5. Above the Earth's surface, the magnitude of the gravitational force on a particle due to the Earth is inversely proportional to the square of the distance of the particle from the centre of the Earth. The Earth is modelled as a sphere of radius \(R\) and the acceleration due to gravity at the Earth's surface is \(g\). A particle \(P\) of mass \(m\) is at a height \(x\) above the surface of the Earth.
  1. Show that the magnitude of the gravitational force acting on \(P\) is $$\frac { m g R ^ { 2 } } { ( R + x ) ^ { 2 } }$$ A rocket is fired vertically upwards from the surface of the Earth. When the rocket is at height \(2 R\) above the surface of the Earth its speed is \(\sqrt { } \left( \frac { g R } { 2 } \right)\). You may assume that air resistance can be ignored and that the engine of the rocket is switched off before the rocket reaches height \(R\). Modelling the rocket as a particle,
  2. find the speed of the rocket when it was at height \(R\) above the surface of the Earth.
Edexcel M3 2012 January Q6
15 marks Standard +0.8
  1. A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(O\). The particle is hanging in equilibrium at the point \(A\), vertically below \(O\), when it is set in motion with a horizontal speed \(\frac { 1 } { 2 } \sqrt { } ( 11 g l )\). When the string has turned through an angle \(\theta\) and the string is still taut, the tension in the string is \(T\).
    1. Show that \(T = 3 m g \left( \cos \theta + \frac { 1 } { 4 } \right)\).
    At the instant when \(P\) reaches the point \(B\), the string becomes slack. Find
  2. the speed of \(P\) at \(B\),
  3. the maximum height above \(B\) reached by \(P\) before it starts to fall.
Edexcel M3 2012 January Q7
16 marks Challenging +1.2
7. Diagram NOT accurately drawn \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bbd531ab-05f8-48ff-8a68-ec6f33ac0a2f-12_444_768_253_603} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The shaded region \(R\) is bounded by the curve with equation \(y = \frac { 1 } { 2 } x ( 6 - x )\), the \(x\)-axis and the line \(x = 2\), as shown in Figure 1. The unit of length on both axes is 1 cm . A uniform solid \(P\) is formed by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis.
  1. Show that the centre of mass of \(P\) is, to 3 significant figures, 1.42 cm from its plane face. The uniform solid \(P\) is placed with its plane face on an inclined plane which makes an angle \(\theta\) with the horizontal. Given that the plane is sufficiently rough to prevent \(P\) from sliding and that \(P\) is on the point of toppling when \(\theta = \alpha\),
  2. find the angle \(\alpha\). Given instead that \(P\) is on the point of sliding down the plane when \(\theta = \beta\) and that the coefficient of friction between \(P\) and the plane is 0.3 ,
  3. find the angle \(\beta\).
Edexcel M3 2013 January Q1
4 marks Moderate -0.3
  1. A particle \(P\) is moving along the positive \(x\)-axis. When the displacement of \(P\) from the origin is \(x\) metres, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the acceleration of \(P\) is \(9 x \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
When \(x = 2 , v = 6\) Show that \(v ^ { 2 } = 9 x ^ { 2 }\).
(4)
Edexcel M3 2013 January Q2
9 marks Standard +0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d19c7390-0332-4cab-82e5-72976bd499a2-03_636_529_322_662} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform solid consists of a right circular cone of radius \(r\) and height \(k r\), where \(k > \sqrt { } 3\), fixed to a hemisphere of radius \(r\). The centre of the plane face of the hemisphere is \(O\) and this plane face coincides with the base of the cone, as shown in Figure 1.
  1. Show that the distance of the centre of mass of the solid from \(O\) is $$\frac { \left( k ^ { 2 } - 3 \right) r } { 4 ( k + 2 ) }$$ The point \(A\) lies on the circumference of the base of the cone. The solid is suspended by a string attached at \(A\) and hangs freely in equilibrium. The angle between \(A O\) and the vertical is \(\theta\), where \(\tan \theta = \frac { 11 } { 14 }\)
  2. Find the value of \(k\).
Edexcel M3 2013 January Q3
10 marks Standard +0.8
  1. A particle \(P\) of mass 0.6 kg is moving along the \(x\)-axis in the positive direction. At time \(t = 0 , P\) passes through the origin \(O\) with speed \(15 \mathrm {~ms} ^ { - 1 }\). At time \(t\) seconds the distance \(O P\) is \(x\) metres, the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the resultant force acting on \(P\) has magnitude \(\frac { 12 } { ( t + 2 ) ^ { 2 } }\) newtons. The resultant force is directed towards \(O\).
    1. Show that \(v = 5 \left( \frac { 4 } { t + 2 } + 1 \right)\).
    2. Find the value of \(x\) when \(t = 5\)
Edexcel M3 2013 January Q4
11 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d19c7390-0332-4cab-82e5-72976bd499a2-07_503_618_242_646} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(2 a\) and modulus of elasticity \(6 m g\). The other end of the string is attached to a fixed point \(A\). The particle moves with constant speed \(v\) in a horizontal circle with centre \(O\), where \(O\) is vertically below \(A\) and \(O A = 2 a\), as shown in Figure 2 .
  1. Show that the extension in the string is \(\frac { 2 } { 5 } a\).
  2. Find \(v ^ { 2 }\) in terms of \(a\) and \(g\).
Edexcel M3 2013 January Q5
12 marks Challenging +1.2
5. A particle \(P\) is moving in a straight line with simple harmonic motion on a smooth horizontal floor. The particle comes to instantaneous rest at points \(A\) and \(B\) where \(A B\) is 0.5 m . The mid-point of \(A B\) is \(O\). The mid-point of \(O A\) is \(C\). The mid-point of \(O B\) is \(D\). The particle takes 0.2 s to travel directly from \(C\) to \(D\). At time \(t = 0 , P\) is moving through \(O\) towards \(A\).
  1. Show that the period of the motion is \(\frac { 6 } { 5 } \mathrm {~s}\).
  2. Find the distance of \(P\) from \(B\) when \(t = 2 \mathrm {~s}\).
  3. Find the maximum magnitude of the acceleration of \(P\).
  4. Find the maximum speed of \(P\).
Edexcel M3 2013 January Q6
14 marks Challenging +1.2
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d19c7390-0332-4cab-82e5-72976bd499a2-11_412_533_258_685} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A smooth hollow cylinder of internal radius \(a\) is fixed with its axis horizontal. A particle \(P\) moves on the inner surface of the cylinder in a vertical circle with radius \(a\) and centre \(O\), where \(O\) lies on the axis of the cylinder. The particle is projected vertically downwards with speed \(u\) from point \(A\) on the circle, where \(O A\) is horizontal. The particle first loses contact with the cylinder at the point \(B\), where \(\angle A O B = 150 ^ { \circ }\), as shown in Figure 3. Given that air resistance can be ignored,
  1. show that the speed of \(P\) at \(B\) is \(\sqrt { } \left( \frac { a g } { 2 } \right)\),
  2. find \(u\) in terms of \(a\) and \(g\). After losing contact with the cylinder, \(P\) crosses the diameter through \(A\) at the point \(D\). At \(D\) the velocity of \(P\) makes an angle \(\theta ^ { \circ }\) with the horizontal.
  3. Find the value of \(\theta\).
Edexcel M3 2013 January Q7
15 marks Standard +0.8
7. A particle \(P\) of mass 1.5 kg is attached to the mid-point of a light elastic string of natural length 0.30 m and modulus of elasticity \(\lambda\) newtons. The ends of the string are attached to two fixed points \(A\) and \(B\), where \(A B\) is horizontal and \(A B = 0.48 \mathrm {~m}\). Initially \(P\) is held at rest at the mid-point, \(M\), of the line \(A B\) and the tension in the string is 240 N .
  1. Show that \(\lambda = 400\) The particle is now held at rest at the point \(C\), where \(C\) is 0.07 m vertically below \(M\). The particle is released from rest at \(C\).
  2. Find the magnitude of the initial acceleration of \(P\).
  3. Find the speed of \(P\) as it passes through \(M\).
Edexcel M3 2004 June Q1
3 marks Easy -1.2
  1. A circular flywheel of diameter 7 cm is rotating about the axis through its centre and perpendicular to its plane with constant angular speed 1000 revolutions per minute.
Find, in \(\mathrm { m } \mathrm { s } ^ { - 1 }\) to 3 significant figures, the speed of a point on the rim of the flywheel.
Edexcel M3 2004 June Q2
7 marks Standard +0.8
2. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{b9e9b91c-7e6d-4b84-9f0e-180b626887c2-2_460_549_651_792}
\end{figure} Two light elastic strings each have natural length \(a\) and modulus of elasticity \(\lambda\). A particle \(P\) of mass \(m\) is attached to one end of each string. The other ends of the strings are attached to points \(A\) and \(B\), where \(A B\) is horizontal and \(A B = 2 a\). The particle is held at the mid-point of \(A B\) and released from rest. It comes to rest for the first time in the subsequent motion when \(P A\) and \(P B\) make angles \(\alpha\) with \(A B\), where \(\tan \alpha = \frac { 4 } { 3 }\), as shown in Fig. 1. Find \(\lambda\) in terms of \(m\) and \(g\).
Edexcel M3 2004 June Q3
10 marks Challenging +1.2
3. A particle \(P\) of mass \(m \mathrm {~kg}\) slides from rest down a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. When \(P\) has moved a distance \(x\) metres down the plane, the resistance to the motion of \(P\) from non-gravitational forces has magnitude \(m x ^ { 2 }\) newtons. Find
  1. the speed of \(P\) when \(x = 2\),
  2. the distance \(P\) has moved when it comes to rest for the first time.
Edexcel M3 2004 June Q4
11 marks Standard +0.3
4. A rough disc rotates in a horizontal plane with constant angular velocity \(\omega\) about a fixed vertical axis. A particle \(P\) of mass \(m\) lies on the disc at a distance \(\frac { 4 } { 3 } a\) from the axis. The coefficient of friction between \(P\) and the disc is \(\frac { 3 } { 5 }\). Given that \(P\) remains at rest relative to the disc,
  1. prove that \(\omega ^ { 2 } \leqslant \frac { 9 g } { 20 a }\). The particle is now connected to the axis by a horizontal light elastic string of natural length \(a\) and modulus of elasticity 2 mg . The disc again rotates with constant angular velocity \(\omega\) about the axis and \(P\) remains at rest relative to the disc at a distance \(\frac { 4 } { 3 } a\) from the axis.
  2. Find the greatest and least possible values of \(\omega ^ { 2 }\).
Edexcel M3 2004 June Q5
13 marks Standard +0.3
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{b9e9b91c-7e6d-4b84-9f0e-180b626887c2-3_522_659_1043_701}
\end{figure} A toy is formed by joining a uniform solid right circular cone, of base radius \(3 r\) and height \(4 r\), to a uniform solid cylinder, also of radius \(3 r\) and height \(4 r\). The cone and the cylinder are made from the same material, and the plane face of the cone coincides with a plane face of the cylinder, as shown in Fig. 2. The centre of this plane face is \(O\).
  1. Find the distance of the centre of mass of the toy from \(O\). The point \(A\) lies on the edge of the plane face of the cylinder which forms the base of the toy. The toy is suspended from \(A\) and hangs in equilibrium.
  2. Find, in degrees to one decimal place, the angle between the axis of symmetry of the toy and the vertical. The toy is placed with the curved surface of the cone on horizontal ground.
  3. Determine whether the toy will topple.
    (4)
Edexcel M3 2004 June Q6
15 marks Standard +0.8
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{b9e9b91c-7e6d-4b84-9f0e-180b626887c2-4_460_799_301_657}
\end{figure} Figure 3 represents the path of a skier of mass 70 kg moving on a ski-slope \(A B C D\). The path lies in a vertical plane. From \(A\) to \(B\), the path is modelled as a straight line inclined at \(60 ^ { \circ }\) to the horizontal. From \(B\) to \(D\), the path is modelled as an arc of a vertical circle of radius 50 m . The lowest point of the \(\operatorname { arc } B D\) is \(C\). At \(B\), the skier is moving downwards with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At \(D\), the path is inclined at \(30 ^ { \circ }\) to the horizontal and the skier is moving upwards. By modelling the slope as smooth and the skier as a particle, find
  1. the speed of the skier at \(C\),
  2. the normal reaction of the slope on the skier at \(C\),
  3. the speed of the skier at \(D\),
  4. the change in the normal reaction of the slope on the skier as she passes \(B\). The model is refined to allow for the influence of friction on the motion of the skier.
  5. State briefly, with a reason, how the answer to part (b) would be affected by using such a model. (No further calculations are expected.)
Edexcel M3 2004 June Q7
16 marks Challenging +1.3
7. A particle \(P\) of mass 0.3 kg is attached to one end of a light elastic spring. The other end of the spring is attached to a fixed point \(O\) on a smooth horizontal table. The spring has natural length 2 m and modulus of elasticity 21.6 N . The particle \(P\) is placed on the table at the point \(A\), where \(O A = 2 \mathrm {~m}\). The particle \(P\) is now pulled away from \(O\) to the point \(B\), where \(O A B\) is a straight line with \(O B = 3.5 \mathrm {~m}\). It is then released from rest.
  1. Prove that \(P\) moves with simple harmonic motion of period \(\frac { \pi } { 3 } \mathrm {~s}\).
  2. Find the speed of \(P\) when it reaches \(A\). The point \(C\) is the mid-point of \(A B\).
  3. Find, in terms of \(\pi\), the time taken for \(P\) to reach \(C\) for the first time. Later in the motion, \(P\) collides with a particle \(Q\) of mass 0.2 kg which is at rest at \(A\).
    After the impact, \(P\) and \(Q\) coalesce to form a single particle \(R\).
  4. Show that \(R\) also moves with simple harmonic motion and find the amplitude of this motion. END
Edexcel M3 2005 June Q1
6 marks Standard +0.3
1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{fecee25b-e5d9-4669-89a1-6ae445090126-2_336_624_306_683}
\end{figure} A particle of mass 0.8 kg is attached to one end of a light elastic spring, of natural length 2 m and modulus of elasticity 20 N . The other end of the spring is attached to a fixed point \(O\) on a smooth plane which is inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 3 } { 4 }\). The particle is held on the plane at a point which is 1.6 m down a line of greatest slope of the plane from \(O\), as shown in Figure 1. The particle is then released from rest. Find the initial acceleration of the particle.
(Total 6 marks)
Edexcel M3 2005 June Q2
9 marks Standard +0.3
2. A closed container \(C\) consists of a thin uniform hollow hemispherical bowl of radius \(a\), together with a lid. The lid is a thin uniform circular disc, also of radius \(a\). The centre \(O\) of the disc coincides with the centre of the hemispherical bowl. The bowl and its lid are made of the same material.
  1. Show that the centre of mass of \(C\) is at a distance \(\frac { 1 } { 3 } a\) from \(O\). The container \(C\) has mass \(M\). A particle of mass \(\frac { 1 } { 2 } M\) is attached to the container at a point \(P\) on the circumference of the lid. The container is then placed with a point of its curved surface in contact with a horizontal plane. The container rests in equilibrium with \(P , O\) and the point of contact in the same vertical plane.
  2. Find, to the nearest degree, the angle made by the line \(P O\) with the horizontal.