Questions — Edexcel M3 (469 questions)

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Edexcel M3 2017 June Q6
6. The path followed by a motorcycle round a circular race track is modelled as a horizontal circle of radius 50 m . The track is banked at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\). The motorcycle travels round the track at constant speed. The motorcycle is modelled as a particle and air resistance can be ignored. In an initial model it is assumed that there is no sideways friction between the motorcycle tyres and the track.
  1. Find the speed, in \(\mathrm { m } \mathrm { s } ^ { - 1 }\), of the motorcycle. In a refined model it is assumed that there is sideways friction. The coefficient of friction between the motorcycle tyres and the track is \(\frac { 1 } { 4 }\). It is still assumed that air resistance can be ignored and that the motorcycle is modelled as a particle. The motorcycle's path is unchanged. Using this model,
  2. find the maximum speed, in \(\mathrm { m } \mathrm { s } ^ { - 1 }\), at which the motorcycle can travel without slipping sideways.
Edexcel M3 2017 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a67e3644-13fa-4196-a2ef-ea1e26f5726c-20_442_967_283_486} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A light inextensible string of length \(a\) has one end attached to a fixed point \(O\) on a horizontal plane. A particle \(P\) is attached to the other end of the string. The particle is held at the point \(A\), where \(A\) is vertically above \(O\) and \(O A = a\). The particle is then projected horizontally with speed \(\sqrt { 10 a g }\), as shown in Figure 2. The particle strikes the plane at the point \(B\). After rebounding from the plane, \(P\) passes through \(A\). The coefficient of restitution between the plane and \(P\) is \(e\).
  1. Show that \(e \geqslant \frac { 1 } { 2 }\) The point \(C\) is above the horizontal plane such that \(O C = a\) and angle \(C O B = 120 ^ { \circ }\) As the particle reaches \(C\), the string breaks. The particle now moves freely under gravity and strikes the plane at the point \(D\).
    Given that \(e = \frac { \sqrt { 3 } } { 2 }\)
  2. find the size of the angle between the horizontal and the direction of motion of \(P\) at \(D\).
Edexcel M3 2018 June Q1
  1. A light elastic string of modulus of elasticity 29.4 N has one end attached to a fixed point \(A\). A particle \(P\) of mass 1.5 kg is attached to the other end of the string and \(P\) hangs freely in equilibrium 0.5 m vertically below \(A\). Find the natural length of the string.
Edexcel M3 2018 June Q2
2. A particle \(P\) is moving in a straight line with simple harmonic motion about the fixed point \(O\) as centre. When \(P\) is a distance 0.02 m from \(O\), the speed of \(P\) is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the magnitude of the acceleration of \(P\) is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
  1. Find the period of the motion. The amplitude of the motion is \(a\) metres. Find
  2. the value of \(a\),
  3. the total length of time during each complete oscillation for which \(P\) is within \(\frac { 1 } { 2 } a\)
    metres of \(O\). metres of \(O\).
Edexcel M3 2018 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2273ca38-1e16-44ab-ae84-f4c576cbb8f9-08_583_549_210_760} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A light inextensible string of length \(7 l\) has one end attached to a fixed point \(A\) and the other end attached to a fixed point \(B\), where \(A\) is vertically above \(B\) and \(A B = 5\) l. A particle of mass \(m\) is attached to the string at the point \(C\) where \(A C = 4 l\), as shown in Figure 1. The particle moves in a horizontal circle with constant angular speed \(\omega\). Both parts of the string are taut.
  1. Find, in terms of \(m , g , l\) and \(\omega\),
    1. the tension in \(A C\),
    2. the tension in \(B C\). The time taken by the particle to complete one revolution is \(R\).
      Given that \(R \leqslant k \pi \sqrt { \frac { l } { 5 g } }\)
  2. find the least possible value of \(k\).
Edexcel M3 2018 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2273ca38-1e16-44ab-ae84-f4c576cbb8f9-12_469_844_269_552} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a light elastic string, of modulus of elasticity \(\lambda\) newtons and natural length 0.6 m . One end of the string is attached to a fixed point \(A\) on a rough plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle \(P\) of mass 0.5 kg . The string lies along a line of greatest slope of the plane. The particle is held at rest on the plane at the point \(B\), where \(B\) is lower than \(A\) and \(A B = 1.2 \mathrm {~m}\). The particle then receives an impulse of magnitude 1.5 N s in the direction parallel to the string, causing \(P\) to move up the plane towards \(A\). The coefficient of friction between \(P\) and the plane is 0.7 . Given that \(P\) comes to rest at the instant when the string becomes slack, find the value of \(\lambda\).
Edexcel M3 2018 June Q5
  1. A particle \(P\) of mass 0.8 kg moves along the \(x\)-axis in the positive \(x\) direction under the action of a resultant force. This force acts in the direction of \(x\) increasing. At time \(t\) seconds, \(t \geqslant 0 , P\) is \(x\) metres from the origin \(O , P\) is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the force has magnitude \(\frac { 4 } { ( x + 1 ) ^ { 3 } } \mathrm {~N}\).
When \(t = 0 , P\) is at rest at \(O\).
  1. Show that \(v ^ { 2 } = 5 \left( \frac { ( x + 1 ) ^ { 2 } - 1 } { ( x + 1 ) ^ { 2 } } \right)\) When \(t = 2 , P\) is at the point \(A\). When \(t = 4 , P\) is at the point \(B\).
  2. Using algebraic integration, find the distance \(A B\).
Edexcel M3 2018 June Q6
6. A uniform solid right circular cone has base radius \(r\) and height \(h\).
  1. Use algebraic integration to show that the distance of the centre of mass of the cone from its vertex is \(\frac { 3 } { 4 } h\).
    [0pt] [You may assume that the volume of a cone of base radius \(r\) and height \(h\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) ] \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2273ca38-1e16-44ab-ae84-f4c576cbb8f9-20_394_716_632_621} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} A solid \(S\) is formed by joining a uniform right circular solid cone of mass \(5 m\) to a uniform solid hemisphere, of radius \(r\) and mass \(k m\) where \(k < 20\). The cone has base radius \(r\) and height \(6 r\). The plane face of the cone coincides with the plane face of the hemisphere. The centre of the plane face of the cone is \(O\) and the point \(A\) is on the circular edge of this plane face, as shown in Figure 3.
  2. Find the distance from \(O\) to the centre of mass of \(S\). The solid is suspended from \(A\) and hangs freely in equilibrium. The angle between the axis of the cone and the horizontal is \(30 ^ { \circ }\).
  3. Find, to the nearest whole number, the value of \(k\).
Edexcel M3 2018 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2273ca38-1e16-44ab-ae84-f4c576cbb8f9-24_575_821_214_566} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A smooth solid sphere, with centre \(O\) and radius \(r\), is fixed with its lowest point on a horizontal plane. A particle is placed on the surface of the sphere at the highest point of the sphere. The particle is then projected horizontally with speed \(u\) and starts to move on the surface of the sphere. The particle leaves the surface of the sphere at the point \(A\) where \(O A\) makes an angle \(\alpha , \alpha > 0\), with the upward vertical, as shown in Figure 4.
  1. Show that \(\cos \alpha = \frac { 1 } { 3 g r } \left( u ^ { 2 } + 2 g r \right)\)
  2. Show that \(u < \sqrt { g r }\) After leaving the surface of the sphere, the particle strikes the plane with speed \(3 \sqrt { \frac { g r } { 2 } }\)
  3. Find the value of \(\cos \alpha\).
Edexcel M3 2020 June Q1
1.
VILV SIHI NI JIIIM IONOOVIIN SIHI NI JIIIM IONOOVARV SIHI NI JIIIM ION OC
\includegraphics[max width=\textwidth, alt={}, center]{ace84823-db30-463e-b24b-f0cd7df73746-03_62_37_2659_1914}
Edexcel M3 2020 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ace84823-db30-463e-b24b-f0cd7df73746-04_542_831_301_552} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A smooth bead of weight 12 N is threaded onto a light elastic string of natural length 3 m . The points \(A\) and \(B\) are on a horizontal ceiling, with \(A B = 3 \mathrm {~m}\). One end of the string is attached to \(A\) and the other end of the string is attached to \(B\). The bead hangs freely in equilibrium, 2 m below the ceiling, as shown in Figure 2.
  1. Find the tension in the string.
  2. Show that the modulus of elasticity of the string is 11.25 N . The bead is now pulled down to a point vertically below its equilibrium position and released from rest.
  3. Find the elastic energy stored in the string at the instant when the bead is moving at its maximum speed.
Edexcel M3 2020 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ace84823-db30-463e-b24b-f0cd7df73746-06_615_1134_290_409} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A particle \(P\) of mass 2 kg is attached to one end of a light elastic spring, of natural length 0.8 m and modulus of elasticity 12 N . The other end of the spring is attached to a fixed point \(A\) on a rough plane. The plane is inclined at \(30 ^ { \circ }\) to the horizontal. Initially \(P\) is held at rest on the plane at the point \(B\), where \(B\) is below \(A\), with \(A B = 0.3 \mathrm {~m}\) and \(A B\) lies along a line of greatest slope of the plane. The point \(C\) lies on the plane with \(A C = 1 \mathrm {~m}\), as shown in Figure 3. The coefficient of friction between \(P\) and the plane is 0.3 After being released \(P\) passes through the point \(C\). Find the speed of \(P\) at the instant it passes through \(C\).
Edexcel M3 2020 June Q4
4. (a) Use algebraic integration to show that the centre of mass of a uniform solid hemisphere of radius \(a\) is a distance \(\frac { 3 } { 8 } a\) from the centre of its plane face.
[0pt] [You may assume that the volume of a sphere of radius \(r\) is \(\frac { 4 } { 3 } \pi r ^ { 3 }\) ] \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ace84823-db30-463e-b24b-f0cd7df73746-08_444_764_539_591} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A uniform solid hemisphere has mass \(m\) and radius \(a\). A particle of mass \(k m\) is attached to a point \(A\) on the circumference of the plane face of the hemisphere to form the loaded solid \(S\). The centre of the plane face of the hemisphere is the point \(O\), as shown in Figure 4. The loaded solid \(S\) is placed on a horizontal plane. The curved surface of \(S\) is in contact with the plane and \(S\) rests in equilibrium with \(O A\) making an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \sqrt { 3 }\)
(b) Find the exact value of \(k\).
\includegraphics[max width=\textwidth, alt={}, center]{ace84823-db30-463e-b24b-f0cd7df73746-09_2255_50_314_34}
VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel M3 2020 June Q5
5. A particle \(P\) of mass 0.5 kg moves along the positive \(x\)-axis in the positive \(x\) direction. At time \(t\) seconds, \(t \geqslant 1 , P\) is \(x\) metres from the origin \(O\) and is moving with speed \(v \mathrm {~ms} ^ { - 1 }\). The resultant force acting on \(P\) has magnitude \(\frac { 2 } { x ^ { 3 } } \mathrm {~N}\) and is directed towards \(O\). When \(t = 1 , x = 1\) and \(v = 3\)
Show that
  1. \(v ^ { 2 } = \frac { 4 } { x ^ { 2 } } + 5\)
  2. \(t = \frac { a + \sqrt { b x ^ { 2 } + c } } { d }\), where \(a , b , c\) and \(d\) are integers to be found.
    \includegraphics[max width=\textwidth, alt={}, center]{ace84823-db30-463e-b24b-f0cd7df73746-13_2255_50_314_34}
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel M3 2020 June Q6
6. A light elastic string has natural length \(a\) and modulus of elasticity \(\frac { 3 } { 4 } \mathrm { mg }\). 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 \(A\). Particle \(P\) hangs freely in equilibrium at the point \(O\), vertically below \(A\).
  1. Find the distance \(O A\). The particle \(P\) is now pulled vertically down to a point \(B\), where \(A B = 3 a\), and released from rest.
  2. Show that, throughout the subsequent motion, \(P\) performs only simple harmonic motion, justifying your answer. The point \(C\) is vertically below \(A\), where \(A C = 2 a\).
    Find, in terms of \(a\) and \(g\),
  3. the speed of \(P\) at the instant that it passes through \(C\),
  4. the time taken for \(P\) to move directly from \(B\) to \(C\). \includegraphics[max width=\textwidth, alt={}, center]{ace84823-db30-463e-b24b-f0cd7df73746-17_2255_50_314_34}
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
Edexcel M3 2020 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ace84823-db30-463e-b24b-f0cd7df73746-20_808_542_264_703} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A particle of mass \(m\) is attached to one end of a light inextensible string of length 8a. The other end of the string is fixed to the point \(O\) on the smooth horizontal surface of a desk. The point \(E\) is on the edge of the desk, where \(O E = 5 a\) and \(O E\) is perpendicular to the edge of the desk. The particle is held at the point \(A\), vertically above \(O\), with the string taut. The particle is projected horizontally from \(A\) with speed \(\sqrt { 8 a g }\) in the direction \(O E\), as shown in Figure 5. When the particle is above the level of \(O E\) the particle is moving in a vertical circle with radius \(8 a\). Given that, when the string makes an angle \(\theta\) with the upward vertical through \(O\), the tension in the string is \(T\),
  1. show that \(T = 3 m g ( 1 - \cos \theta )\) At the instant when the string is horizontal, the particle passes through the point \(B\).
  2. Find the instantaneous change in the tension in the string as the particle passes through \(B\). The particle hits the vertical side \(E F\) of the desk and rebounds. As a result of the impact, the particle loses one third of the kinetic energy it had immediately before the impact. In the subsequent motion the string becomes slack when it makes an angle \(\alpha\) with the upward vertical through \(O\).
  3. Show that \(\cos \alpha = \frac { 7 } { 12 }\) DO NOT WRITEIN THIS AREA
    VIXV SIHIANI III IM IONOOVIAV SIHI NI JYHAM ION OOVI4V SIHI NI JLIYM ION OO
    \includegraphics[max width=\textwidth, alt={}, center]{ace84823-db30-463e-b24b-f0cd7df73746-23_2255_50_314_34}
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    Q7

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Edexcel M3 2021 June Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b99b3eb0-9bca-42e3-bea9-3b0454a872db-02_622_730_251_694} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A hollow toy is formed by joining a uniform right circular conical shell \(C\), with radius \(4 a\) and height \(3 a\), to a uniform hemispherical shell \(H\), with radius \(4 a\). The circular edge of \(C\) coincides with the circular edge of \(H\), as shown in Figure 1. The mass per unit area of \(C\) is \(\lambda\) and the mass per unit area of \(H\) is \(k \lambda\) where \(k\) is a constant.
Given that the centre of mass of the toy is a distance \(4 a\) from the vertex of the cone, find the value of \(k\).
Edexcel M3 2021 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b99b3eb0-9bca-42e3-bea9-3b0454a872db-04_374_1084_246_493} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a fairground ride that consists of a chair of mass \(m\) attached to one end of a rigid arm of length \(\frac { 5 a } { 4 }\). The other end of the arm is freely hinged to the rim of a thin horizontal circular disc of radius \(a\). The disc rotates with constant angular speed \(\omega\) about a vertical axis through the centre of the disc. As the ride rotates the arm remains in a vertical plane through the centre of the disc. The arm makes a constant angle \(\theta\) with the vertical, where \(\tan \theta = \frac { 3 } { 4 }\) The chair is modelled as a particle and the arm is modelled as a light rod.
  1. Find the tension in the arm in terms of \(m\) and \(g\)
  2. Find \(\omega\) in terms of \(a\) and \(g\)
Edexcel M3 2021 June Q3
  1. The finite region enclosed by the curve with equation \(y = 3 - \sqrt { x }\) and the lines \(x = 0\) and \(y = 0\) is rotated through \(2 \pi\) radians about the \(x\)-axis, to form a uniform solid \(S\).
Use algebraic integration to
  1. show that the volume of \(S\) is \(\frac { 27 } { 2 } \pi\)
  2. find the \(x\) coordinate of the centre of mass of \(S\).
Edexcel M3 2021 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b99b3eb0-9bca-42e3-bea9-3b0454a872db-12_483_848_306_589} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A circus performer has mass \(m\). She is attached to one end of a cable of length \(l\). The other end of the cable is attached to a fixed point \(O\) Initially she is held at rest at point \(A\) with the cable taut and at an angle of \(30 ^ { \circ }\) below the horizontal, as shown in Figure 3. The circus performer is released from \(A\) and she moves on a vertical circular path with centre \(O\) The circus performer is modelled as a particle and the cable is modelled as light and inextensible.
  1. Find, in terms of \(m\) and \(g\), the tension in the cable at the instant immediately after the circus performer is released.
  2. Show that, during the motion following her release, the greatest tension in the cable is 4 times the least tension in the cable.
Edexcel M3 2021 June Q5
  1. A particle \(P\) of mass 0.5 kg moves on the \(x\)-axis under the action of a single force.
At time \(t\) seconds, \(t \geqslant 0\)
  • \(O P = x\) metres, \(0 \leqslant x < \frac { \pi } { 2 }\)
  • the force has magnitude \(\sin 2 x \mathrm {~N}\) and is directed towards the origin \(O\)
  • \(P\) is moving in the positive \(x\) direction with speed \(v \mathrm {~ms} ^ { - 1 }\)
At time \(t = 0 , P\) passes through the origin with speed \(2 \mathrm {~ms} ^ { - 1 }\)
  1. Show that \(v = 2 \cos x\)
  2. Show that \(t = \frac { 1 } { 2 } \ln ( \sqrt { 2 } + 1 )\) when \(x = \frac { \pi } { 4 }\)
Edexcel M3 2021 June Q6
  1. A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string, of natural length 0.8 m and modulus of elasticity 0.6 N . The other end of the string is fixed to a point \(A\) on a rough horizontal table. The coefficient of friction between \(P\) and the table is \(\frac { 1 } { 7 }\)
The particle \(P\) is projected from \(A\), with speed \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), along the surface of the table.
After travelling 0.8 m from \(A\), the particle passes through the point \(B\) on the table.
  1. Find the speed of \(P\) at the instant it passes through \(B\). The particle \(P\) comes to rest at the point \(C\) on the table, where \(A B C\) is a straight line.
  2. Find the total distance travelled by \(P\) as it moves directly from \(A\) to \(C\).
  3. Show that \(P\) remains at rest at \(C\).
Edexcel M3 2021 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b99b3eb0-9bca-42e3-bea9-3b0454a872db-24_177_876_260_593} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The fixed points \(A\) and \(B\) are 7 m apart on a smooth horizontal surface.
A light elastic string has natural length 2 m and modulus of elasticity 4 N . One end of the string is attached to a particle \(P\) of mass 2 kg and the other end is attached to \(A\) Another light elastic string has natural length 3 m and modulus of elasticity 2 N . One end of this string is attached to \(P\) and the other end is attached to \(B\) The particle \(P\) rests in equilibrium at the point \(O\), where \(A O B\) is a straight line, as shown in Figure 4.
  1. Show that \(O A = 2.5 \mathrm {~m}\). The particle \(P\) now receives an impulse of magnitude 6Ns in the direction \(O B\)
    1. Show that \(P\) initially moves with simple harmonic motion with centre \(O\)
    2. Determine the amplitude of this simple harmonic motion. The point \(C\) lies on \(O B\). As \(P\) passes through \(C\) the string attached to \(B\) becomes slack.
  3. Find the speed of \(P\) as it passes through \(C\)
  4. Find the time taken for \(P\) to travel directly from \(O\) to \(C\)
Edexcel M3 2022 June Q1
  1. A particle \(P\) moves in a straight line with simple harmonic motion between two fixed points \(A\) and \(B\). The particle performs 2 complete oscillations per second. The midpoint of \(A B\) is \(O\) and the midpoint of \(O A\) is \(C\)
The length of \(A B\) is 0.6 m .
  1. Find the maximum speed of \(P\)
  2. Find the time taken by \(P\) to move directly from \(O\) to \(C\)
Edexcel M3 2022 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2e837bb9-4ada-4f0f-8b21-2730611335f2-04_390_515_246_772} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A hemispherical bowl of internal radius \(6 r\) is fixed with its circular rim horizontal. The centre of the circular rim is \(O\) and the point \(A\) on the surface of the bowl is vertically below \(O\). A particle \(P\) moves in a horizontal circle, with centre \(C\), on the smooth inner surface of the bowl. The particle moves with constant angular speed \(\sqrt { \frac { g } { 4 r } }\). The point \(C\) lies on \(O A\), as shown in Figure 1. Find, in terms of \(r\), the distance \(O C\)