Questions — Edexcel (10514 questions)

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Edexcel M3 2013 January Q3
10 marks Standard +0.8
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.
Edexcel M3 2005 June Q3
9 marks Standard +0.3
3. A light elastic string has natural length \(2 l\) and modulus of elasticity \(4 m g\). One end of the string is attached to a fixed point \(A\) and the other end to a fixed point \(B\), where \(A\) and \(B\) lie on a smooth horizontal table, with \(A B = 4 l\). A particle \(P\) of mass \(m\) is attached to the mid-point of the string. The particle is released from rest at the point of the line \(A B\) which is \(\frac { 5 l } { 3 }\) from \(B\). The speed of \(P\) at the mid-point of \(A B\) is \(V\).
  1. Find \(V\) in terms of \(g\) and \(L\).
  2. Explain why \(V\) is the maximum speed of \(P\).
    (Total 9 marks)
Edexcel M3 2005 June Q4
10 marks Standard +0.3
4. A particle \(P\) of mass \(m\) moves on the smooth inner surface of a spherical bowl of internal radius \(r\). The particle moves with constant angular speed in a horizontal circle, which is at a depth \(\frac { 1 } { 2 } r\) below the centre of the bowl.
  1. Find the normal reaction of the bowl on \(P\).
  2. Find the time for \(P\) to complete one revolution of its circular path.
    (6)
    (Total 10 marks)
Edexcel M3 2005 June Q5
13 marks Standard +0.3
5. A smooth solid sphere, with centre \(O\) and radius \(a\), is fixed to the upper surface of a horizontal table. A particle \(P\) is placed on the surface of the sphere at a point \(A\), where \(O A\) makes an angle \(\alpha\) with the upward vertical, and \(0 < \alpha < \frac { \pi } { 2 }\). The particle is released from rest. When \(O P\) makes an angle \(\theta\) with the upward vertical, and \(P\) is still on the surface of the sphere, the speed of \(P\) is \(v\).
  1. Show that \(v ^ { 2 } = 2 g a ( \cos \alpha - \cos \theta )\). Given that \(\cos \alpha = \frac { 3 } { 4 }\), find
  2. the value of \(\theta\) when \(P\) loses contact with the sphere,
  3. the speed of \(P\) as it hits the table.
    (Total 13 marks)
Edexcel M3 2005 June Q6
14 marks Standard +0.3
6. The rise and fall of the water level in a harbour is modelled as simple harmonic motion. On a particular day the maximum and minimum depths of water in the harbour are 10 m and 4 m and these occur at 1100 hours and 1700 hours respectively.
  1. Find the speed, in \(\mathrm { m } \mathrm { h } ^ { - 1 }\), at which the water level in the harbour is falling at 1600 hours on this particular day.
  2. Find the total time, between 1100 hours and 2300 hours on this particular day, for which the depth of water in the harbour is less than 5.5 m .
    (Total 14 marks)
Edexcel M3 2005 June Q7
14 marks Standard +0.8
7. A particle \(P\) of mass \(\frac { 1 } { 3 } \mathrm {~kg}\) moves along the positive \(x\)-axis under the action of a single force. The force is directed towards the origin \(O\) and has magnitude \(\frac { k } { ( x + 1 ) ^ { 2 } } \mathrm {~N}\), where \(O P = x\) metres and \(k\) is a constant. Initially \(P\) is moving away from \(O\). At \(x = 1\) the speed of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and at \(x = 8\) the speed of \(P\) is \(\sqrt { } 2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the value of \(k\).
  2. Find the distance of \(P\) from \(O\) when \(P\) first comes to instantaneous rest.
    (Total 14 marks)
Edexcel M3 2008 June Q1
9 marks Standard +0.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-02_259_659_283_642} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A light elastic spring, of natural length \(L\) and modulus of elasticity \(\lambda\), has a particle \(P\) of mass \(m\) attached to one end. The other end of the spring is fixed to a point \(O\) on the closed end of a fixed smooth hollow tube of length \(L\). The tube is placed horizontally and \(P\) is held inside the tube with \(O P = \frac { 1 } { 2 } L\), as shown
in Figure 1. The particle \(P\) is released and passes through the open end of the tube with speed \(\sqrt { } ( 2 g L )\).
  1. Show that \(\lambda = 8 \mathrm { mg }\). The tube is now fixed vertically and \(P\) is held inside the tube with \(O P = \frac { 1 } { 2 } L\) and \(P\) above \(O\). The particle \(P\) is released and passes through the open top of the tube with speed \(u\).
  2. Find \(u\).
Edexcel M3 2008 June Q2
11 marks Standard +0.3
2. A particle \(P\) moves with simple harmonic motion and comes to rest at two points \(A\) and \(B\) which are 0.24 m apart on a horizontal line. The time for \(P\) to travel from \(A\) to \(B\) is 1.5 s . The midpoint of \(A B\) is \(O\). At time \(t = 0 , P\) is moving through \(O\), towards \(A\), with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the value of \(u\).
  2. Find the distance of \(P\) from \(B\) when \(t = 2 \mathrm {~s}\).
  3. Find the speed of \(P\) when \(t = 2 \mathrm {~s}\).
Edexcel M3 2008 June Q3
13 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-05_495_972_239_484} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a particle \(B\), of mass \(m\), attached to one end of a light elastic string. The other end of the string is attached to a fixed point \(A\), at a distance \(h\) vertically above a smooth horizontal table. The particle moves on the table in a horizontal circle with centre \(O\), where \(O\) is vertically below \(A\). The string makes a constant angle \(\theta\) with the downward vertical and \(B\) moves with constant angular speed \(\omega\) about \(O A\).
  1. Show that \(\omega ^ { 2 } \leqslant \frac { g } { h }\). The elastic string has natural length \(h\) and modulus of elasticity \(2 m g\).
    Given that \(\tan \theta = \frac { 3 } { 4 }\),
  2. find \(\omega\) in terms of \(g\) and \(h\).
Edexcel M3 2008 June Q4
13 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-07_454_614_239_662} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A uniform solid hemisphere, of radius \(6 a\) and centre \(O\), has a solid hemisphere of radius \(2 a\), and centre \(O\), removed to form a bowl \(B\) as shown in Figure 3.
  1. Show that the centre of mass of \(B\) is \(\frac { 30 } { 13 } a\) from \(O\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-07_735_614_1126_662} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} The bowl \(B\) is fixed to a plane face of a uniform solid cylinder made from the same material as \(B\). The cylinder has radius \(2 a\) and height \(6 a\) and the combined solid \(S\) has an axis of symmetry which passes through \(O\), as shown in Figure 4.
  2. Show that the centre of mass of \(S\) is \(\frac { 201 } { 61 } a\) from \(O\). The plane surface of the cylindrical base of \(S\) is placed on a rough plane inclined at \(12 ^ { \circ }\) to the horizontal. The plane is sufficiently rough to prevent slipping.
  3. Determine whether or not \(S\) will topple. \section*{
    \includegraphics[max width=\textwidth, alt={}]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-08_56_366_251_178}
    }
Edexcel M3 2008 June Q5
15 marks Standard +0.8
A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is released from rest with the string taut and \(O P\) horizontal.
  1. Find the tension in the string when \(O P\) makes an angle of \(60 ^ { \circ }\) with the downward vertical. A particle \(Q\) of mass \(3 m\) is at rest at a distance \(a\) vertically below \(O\). When \(P\) strikes \(Q\) the particles join together and the combined particle of mass \(4 m\) starts to move in a vertical circle with initial speed \(u\).
  2. Show that \(u = \sqrt { } \left( \frac { g a } { 8 } \right)\). The combined particle comes to instantaneous rest at \(A\).
  3. Find
    1. the angle that the string makes with the downward vertical when the combined particle is at \(A\),
    2. the tension in the string when the combined particle is at \(A\).
      \section*{LU \(\_\_\_\_\)}
Edexcel M3 2008 June Q6
14 marks Standard +0.8
A particle \(P\) of mass 0.5 kg moves along the positive \(x\)-axis. It moves away from the origin \(O\) under the action of a single force directed away from \(O\). When \(O P = x\) metres, the magnitude of the force is \(\frac { 3 } { ( x + 1 ) ^ { 3 } } \mathrm {~N}\) and the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Initially \(P\) is at rest at \(O\).
  1. Show that \(v ^ { 2 } = 6 \left( 1 - \frac { 1 } { ( x + 1 ) ^ { 2 } } \right)\).
  2. Show that the speed of \(P\) never reaches \(\sqrt { } 6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  3. Find \(x\) when \(P\) has been moving for 2 seconds.
    \section*{LL \(\_\_\_\_\)}