Questions — Edexcel (9685 questions)

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Edexcel M3 2022 January Q6
15 marks Challenging +1.2
  1. A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length 2l. The other end of the spring is attached to a fixed point \(A\). The particle \(P\) hangs in equilibrium vertically below \(A\), at the point \(E\) where \(A E = 6 l\). The particle \(P\) is then raised a vertical distance \(2 l\) and released from rest.
Air resistance is modelled as being negligible.
  1. Show that \(P\) moves with simple harmonic motion of period \(T\) where $$T = 4 \pi \sqrt { \frac { l } { g } }$$
  2. Find, in terms of \(m , l\) and \(g\), the kinetic energy of \(P\) as it passes through \(E\)
  3. Find, in terms of \(T\), the exact time from the instant when \(P\) is released to the instant when \(P\) has moved a distance 31 .
Edexcel M3 2022 January Q7
15 marks Challenging +1.2
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a1365c54-4910-449b-b270-c56c1bc5a751-24_396_992_246_539} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity 2 mg . The other end of the string is attached to a fixed point \(O\) on a rough plane which is inclined to the horizontal at an angle \(\alpha\) The string lies along a line of greatest slope of the plane.
The particle \(P\) is held at rest on the plane at the point \(A\), where \(O A = a\), as shown in Figure 5. The particle \(P\) is released from \(A\) and slides down the plane, coming to rest at the point \(B\). The coefficient of friction between \(P\) and the plane is \(\mu\), where \(\mu < \tan \alpha\) Air resistance is modelled as being negligible.
  1. Show that \(A B = a ( \sin \alpha - \mu \cos \alpha )\). Given that \(\tan \alpha = \frac { 3 } { 4 }\) and \(\mu = \frac { 1 } { 2 }\)
  2. find, in terms of \(a\) and \(g\), the maximum speed of \(P\) as it moves from \(A\) to \(B\)
  3. Describe the motion of \(P\) after it reaches \(B\), justifying your answer.
Edexcel M3 2022 January Q1
7 marks Standard +0.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bd1e2b07-4a87-49d6-addd-c9f67467ef2f-02_472_750_255_660} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform lamina is in the shape of the region \(R\).
Region \(R\) is bounded by the curve with equation \(y = x ( x + a )\) where \(a\) is a positive constant, the positive \(x\)-axis and the line with equation \(x = a\), as shown shaded in Figure 1. Find the \(\boldsymbol { y }\) coordinate of the centre of mass of the lamina.
Edexcel M3 2022 January Q2
8 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bd1e2b07-4a87-49d6-addd-c9f67467ef2f-04_351_993_246_536} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(2 l\). The other end of the string is attached to a fixed point \(A\) above a smooth horizontal floor. The particle moves in a horizontal circle on the floor with the string taut. The centre \(O\) of the circle is vertically below \(A\) with \(O A = l\), as shown in Figure 2 . The particle moves with constant angular speed \(\omega\) and remains in contact with the floor.
Show that $$\omega \leqslant \sqrt { \frac { g } { l } }$$
Edexcel M3 2022 January Q3
8 marks Standard +0.8
  1. A particle \(P\) of mass \(m \mathrm {~kg}\) is initially held at rest at the point \(O\) on a smooth inclined plane. The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 2 } { 5 }\)
The particle is released from rest and slides down the plane against a force which acts towards \(O\). The force has magnitude \(\frac { 1 } { 3 } m x ^ { 2 } \mathrm {~N}\), where \(x\) metres is the distance of \(P\) from \(O\).
  1. Find the speed of \(P\) when \(x = 2\) The particle first comes to instantaneous rest at the point \(A\).
  2. Find the distance \(O A\).
Edexcel M3 2022 January Q4
8 marks Standard +0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bd1e2b07-4a87-49d6-addd-c9f67467ef2f-12_659_513_246_774} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A thin uniform right hollow cylinder, of radius \(2 a\) and height \(k a\), has a base but no top. A thin uniform hemispherical shell, also of radius \(2 a\), is made of the same material as the cylinder. The hemispherical shell is attached to the end of the cylinder forming a container \(C\). The open circular rim of the cylinder coincides with the rim of the hemispherical shell. The centre of the base of \(C\) is \(O\), as shown in Figure 3.
  1. Show that the distance from \(O\) to the centre of mass of \(C\) is $$\frac { \left( k ^ { 2 } + 4 k + 4 \right) } { 2 ( k + 3 ) } a$$ The container is placed with its circular base on a plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The plane is sufficiently rough to prevent \(C\) from sliding. The container is on the point of toppling.
  2. Find the value of \(k\).
Edexcel M3 2022 January Q5
12 marks Moderate -0.3
  1. A particle \(P\) is moving along the \(x\)-axis. At time \(t\) seconds the displacement of \(P\) from the origin \(O\) is \(x\) metres, where \(x = 4 \cos \left( \frac { 1 } { 5 } \pi t \right)\)
    1. Prove that \(P\) is moving with simple harmonic motion.
    2. Find the period of the motion.
    3. State the amplitude of the motion.
    4. Find, in terms of \(\pi\), the maximum speed of \(P\)
    The points \(A\) and \(B\) lie on the \(x\)-axis, on opposite sides of \(O\), with \(O A = 1.5 \mathrm {~m}\) and \(O B = 2.5 \mathrm {~m}\).
  2. Find the time taken by \(P\) to move directly from \(A\) to \(B\).
Edexcel M3 2022 January Q6
15 marks Standard +0.8
  1. A particle \(P\) of mass 1.2 kg is attached to the midpoint of a light elastic string of natural length 0.5 m and modulus of elasticity \(\lambda\) newtons.
The fixed points \(A\) and \(B\) are 0.8 m apart on a horizontal ceiling. One end of the string is attached to \(A\) and the other end of the string is attached to \(B\). Initially \(P\) is held at rest at the midpoint \(M\) of the line \(A B\) and the tension in the string is 30 N .
  1. Show that \(\lambda = 50\) The particle is now held at rest at the point \(C\), where \(C\) is 0.3 m vertically below \(M\). The particle is released from rest.
  2. Find the magnitude of the initial acceleration of \(P\)
  3. Find the speed of \(P\) at the instant immediately before it hits the ceiling.
Edexcel M3 2022 January Q7
17 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bd1e2b07-4a87-49d6-addd-c9f67467ef2f-24_518_538_264_753} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light rod of length \(l\). The other end of the rod is attached to a fixed point \(O\). The rod can rotate freely in a vertical plane about \(O\). The particle is projected with speed \(u\) from a point \(A\). The line \(O A\) makes an angle \(\alpha\) with the upward vertical through \(O\), where \(\alpha < \frac { \pi } { 2 }\) When \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\), the speed of \(P\) is \(v\), as shown in Figure 4.
  1. Show that \(v ^ { 2 } = u ^ { 2 } - 2 g l ( \cos \theta - \cos \alpha )\) Given that \(\cos \alpha = \frac { 2 } { 5 }\) and that \(u = \sqrt { 3 g l }\)
  2. show that \(P\) moves in a complete vertical circle. As the rod rotates, the least tension in the rod is \(T\) and the greatest tension is \(k T\)
  3. Find the exact value of \(k\)
Edexcel M3 2023 January Q1
8 marks Standard +0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-02_703_561_280_753} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The shaded region R is bounded by the x -axis, the line with equation \(\mathrm { x } = 1\), the curve with equation \(y = 1 + \sqrt { x }\) and the y-axis, as shown in Figure 1. The unit of length on both of the axes is 1 m . The region R is rotated through \(2 \pi\) radians about the x-axis to form a solid of revolution which is used to model a uniform solid \(S\). Show, using the model and algebraic integration, that
  1. the volume of \(S\) is \(\frac { 17 \pi } { 6 } \mathrm {~m} ^ { 3 }\)
  2. the centre of mass of \(S\) is \(\frac { 49 } { 85 } \mathrm {~m}\) from 0 . \includegraphics[max width=\textwidth, alt={}, center]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-02_2264_41_314_1987}
Edexcel M3 2023 January Q2
6 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-04_252_842_285_609} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A light elastic string AB has natural length I and modulus of elasticity 2 mg .
The end A of the elastic string is attached to a fixed point. The other end B is attached to a particle of mass m . The particle is held in equilibrium, with the elastic string taut and horizontal, by a force of magnitude F . The line of action of the force and the elastic string lie in the same vertical plane. The direction of the force makes an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), with the upward vertical, as shown in Figure 2.
Find, in terms of I , the length AB . \includegraphics[max width=\textwidth, alt={}, center]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-04_2264_53_311_1981}
Edexcel M3 2023 January Q3
10 marks Challenging +1.2
3.
\includegraphics[max width=\textwidth, alt={}]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-06_908_1367_269_349}
A square ABCD of side 4a is made from thin uniform cardboard. The centre of the square is 0 . A circle with centre 0 and radius \(\frac { 7 a } { 4 }\) is then removed from the square to form a template T, shown shaded in Figure 3.
A right conical shell, with no base, has radius \(\frac { 7 a } { 4 }\) and perpendicular height \(6 a\).
The shell is made of the same thin uniform cardboard as T.
The shell is attached to T so that the circumference of the end of the shell coincides with the circumference of the circle centre 0 , to form the hat H , shown in Figure 4.
[0pt] [The surface area of a right conical shell of radius r and slant height I is \(\pi r l\).]
  1. Show that the exact distance of the centre of mass of H from O is $$\frac { 175 \pi a } { ( 63 \pi + 128 ) }$$ A fixed rough plane is inclined to the horizontal at an angle \(\alpha\). The hat H is placed on the plane, with ABCD in contact with the plane, and AB parallel to a line of greatest slope of the plane. The plane is sufficiently rough to prevent the hat from sliding down the plane. Given that the hat is on the point of toppling,
  2. find the exact value of \(\tan \alpha\), giving your answer in simplest form.
Edexcel M3 2023 January Q4
10 marks Challenging +1.2
  1. In this question you must show all stages in your working. Solutions relying entirely on calculator technology are not acceptable.
A particle \(P\) is moving along the \(x\)-axis.
At time \(t\) seconds, where \(0 \leqslant t \leqslant \frac { 2 } { 3 } , P\) is \(x\) metres from the origin 0 and is moving with velocity \(\mathrm { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) in the positive x direction where $$v = ( 2 x + 1 ) ^ { \frac { 3 } { 2 } }$$ When \(\mathrm { t } = 0 , \mathrm { P }\) passes through 0 .
  1. Find the value of x when the acceleration of P is \(243 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
  2. Find v in terms of t .
Edexcel M3 2023 January Q5
14 marks Standard +0.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-12_535_674_283_699} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(\mathrm { a } \sqrt { 3 }\). The other end of the string is attached to a fixed point A . The particle P is also attached to one end of a second light inextensible string of length a. The other end of this string is attached to a fixed point B , where B is vertically below A , with \(\mathrm { AB } = \mathrm { a }\). The particle \(P\) moves in a horizontal circle with centre 0 , where 0 is vertically below \(B\).
The particle P moves with constant angular speed \(\omega\), with both strings taut, as shown in Figure 5.
  1. Show that the upper string makes an angle of \(30 ^ { \circ }\) with the downward vertical and the lower string makes an angle of \(60 ^ { \circ }\) with the downward vertical.
  2. Show that the tension in the upper string is \(\frac { 1 } { 2 } m \sqrt { 3 } \left( 2 g - a \omega ^ { 2 } \right)\).
  3. Show that \(\frac { 2 g } { 3 a } < \omega ^ { 2 } < \frac { 2 g } { a }\)
    \(\_\_\_\_\) VIAV SIHI NI JIIHM ION OC
    VILU SIHIL NI GLIUM ION OC
    VEYV SIHI NI III HM ION OC
Edexcel M3 2023 January Q6
14 marks Challenging +1.2
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-16_574_506_283_776} \captionsetup{labelformat=empty} \caption{Figure 6}
\end{figure} A small smooth ring \(R\) of mass \(m\) is threaded on to a smooth wire in the shape of a circle with centre 0 and radius \(I\). The wire is fixed in a vertical plane. The ring \(R\) is attached to one end of a light elastic string of natural length I and modulus of elasticity mg . The other end of the elastic string is attached to A , the lowest point of the wire. The point B is on the wire and \(O B\) is horizontal. The ring \(R\) is at rest at the highest point of the wire, as shown in Figure 6.
The ring \(R\) is slightly disturbed from rest and slides along the wire.
At the instant when \(R\) reaches the point \(B\), the speed of \(R\) is \(v\) and the magnitude of the force exerted on R by the wire is N .
  1. Show that $$v ^ { 2 } = 2 g l \sqrt { 2 }$$
  2. Show that $$N = \frac { 1 } { 2 } m g ( 5 \sqrt { 2 } - 2 )$$
    \(\_\_\_\_\) VIAV SIHI NI JIIHM ION OC
    VILU SIHIL NI GLIUM ION OC
    VEYV SIHI NI ELIUM ION OC
Edexcel M3 2023 January Q7
13 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-20_358_1161_278_452} \captionsetup{labelformat=empty} \caption{Figure 7}
\end{figure} Two points \(A\) and \(B\) lie on a smooth horizontal table where \(A B = 41\).
A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length I and modulus of elasticity 2 mg . The other end of the spring is attached to A . The particle P is also attached to one end of another light elastic spring of natural length I and modulus of elasticity mg . The other end of the spring is attached to B.
The particle \(P\) rests in equilibrium on the table at the point 0 , where \(A 0 = \frac { 5 } { 3 } I\), as shown in Figure 7.
The particle \(P\) is moved a distance \(\frac { 1 } { 2 } \mathrm { I }\) along the table, from 0 towards \(A\), and released from rest.
  1. Show that P moves with simple harmonic motion of period T , where $$\mathrm { T } = 2 \pi \sqrt { \frac { l } { 3 g } }$$
  2. Find, in terms of I and g , the speed of P as it passes through 0 .
  3. Find, in terms of g , the maximum acceleration of P .
  4. Find the exact time, in terms of I and g , from the instant when P is released from rest to the instant when P is first moving with speed \(\frac { 3 } { 4 } \sqrt { g l }\) \includegraphics[max width=\textwidth, alt={}, center]{631b78c4-2763-4a1e-9d30-2f301fe3af2e-20_2269_56_311_1978} \(\_\_\_\_\) VIAV SIHI NI JIIHM ION OC
    VILU SIHIL NI GLIUM ION OC
    VEYV SIHI NI ELIUM ION OC
Edexcel M3 2024 January Q1
6 marks Standard +0.3
  1. A spacecraft \(S\) of mass \(m\) moves in a straight line towards the centre, \(O\), of a planet.
The planet is modelled as a fixed sphere of radius \(R\).
The spacecraft \(S\) is modelled as a particle.
The gravitational force of the planet is the only force acting on \(S\).
When \(S\) is a distance \(x ( x \geqslant R )\) from \(O\)
  • the gravitational force is directed towards \(O\) and has magnitude \(\frac { m g R ^ { 2 } } { 2 x ^ { 2 } }\)
  • the speed of \(S\) is \(v\)
    1. Show that
$$v ^ { 2 } = \frac { g R ^ { 2 } } { x } + C$$ where \(C\) is a constant. When \(x = 3 R , v = \sqrt { 3 g R }\)
  • Find, in terms of \(g\) and \(R\), the speed of \(S\) as it hits the surface of the planet.
    •
    Edexcel M3 2024 January Q2
    9 marks Standard +0.3
    2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{92131234-bfc1-4e0e-87d4-db9335fbf343-04_401_1031_287_516} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A light elastic spring has natural length \(l\) and modulus of elasticity \(\lambda\) One end of the spring is attached to a point \(A\) on a smooth plane.
    The plane is inclined at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\) A particle \(P\) of mass \(m\) is attached to the other end of the spring. Initially \(P\) is held at the point \(B\) on the plane, where \(A B\) is a line of greatest slope of the plane. The point \(B\) is lower than \(A\) and \(A B = 2 l\), as shown in Figure 1 .
    The particle is released from rest at \(B\) and first comes to instantaneous rest at the point \(C\) on \(A B\), where \(A C = 0.7 l\)
    1. Use the principle of conservation of mechanical energy to show that $$\lambda = \frac { 100 } { 91 } m g$$
    2. Find the acceleration of \(P\) when it is released from rest at \(B\).
    Edexcel M3 2024 January Q3
    8 marks Challenging +1.2
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{92131234-bfc1-4e0e-87d4-db9335fbf343-08_246_734_296_667} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} The shaded region in Figure 2 is bounded by the \(x\)-axis, the line with equation \(x = 2\) and the curve with equation \(y = \frac { 1 } { 4 } x ( 3 - x )\).
    This region is rotated through \(2 \pi\) radians about the \(x\)-axis, to form a solid of revolution which is used to model a uniform solid \(S\). The volume of \(S\) is \(\frac { 2 } { 5 } \pi\)
    1. Use the model and algebraic integration to show that the \(x\) coordinate of the centre of mass of \(S\) is \(\frac { 31 } { 24 }\) The solid \(S\) is placed with its circular face on a rough plane which is inclined at \(\alpha ^ { \circ }\) to the horizontal. The plane is sufficiently rough to prevent \(S\) from sliding. The solid \(S\) is on the point of toppling.
    2. Find the value of \(\alpha\)
    Edexcel M3 2024 January Q4
    9 marks Standard +0.8
    4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{92131234-bfc1-4e0e-87d4-db9335fbf343-12_760_1212_294_429} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows a thin hollow right circular cone fixed with its circular rim horizontal.
    The centre of the circular rim is \(O\). The vertex \(V\) of the cone is vertically below \(O\).
    The radius of the circular rim is \(4 a\) and \(O V = 3 a\).
    A particle \(P\) of mass \(m\) moves in a horizontal circle of radius \(r ( 0 < r < 4 a )\) on the inner surface of the cone. The coefficient of friction between \(P\) and the inner surface of the cone is \(\frac { 1 } { 4 }\) The particle moves with a constant angular speed.
    Show that the maximum possible angular speed is \(\sqrt { \frac { 16 g } { 13 r } }\)
    Edexcel M3 2024 January Q5
    16 marks Challenging +1.2
    1. (a) Use algebraic integration to show that the centre of mass of a uniform semicircular disc of radius \(r\) and centre \(O\) is at a distance \(\frac { 4 r } { 3 \pi }\) from the diameter through \(O\) [You may assume, without proof, that the area of a circle of radius \(r\) is \(\pi r ^ { 2 }\) ]
    A uniform lamina L is in the shape of a semicircle with centre \(B\) and diameter \(A C = 8 a\). The semicircle with diameter \(A B\) is removed from \(L\) and attached to the straight edge \(B C\) to form the template \(T\), shown shaded in Figure 4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{92131234-bfc1-4e0e-87d4-db9335fbf343-16_419_1273_680_397} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} The distance of the centre of mass of \(T\) from \(A C\) is \(d\).
    (b) Show that \(d = \frac { 4 a } { \pi }\) The template \(T\) is freely suspended from \(A\) and hangs in equilibrium with \(A C\) at an angle \(\theta\) to the downward vertical.
    (c) Find the exact value of \(\tan \theta\)
    Edexcel M3 2024 January Q6
    14 marks Challenging +1.2
    1. The fixed point \(A\) is vertically above the fixed point \(B\), with \(A B = 3 l\)
    A light elastic string has natural length \(l\) and modulus of elasticity \(4 m g\) One end of the string is attached to \(A\) and the other end is attached to a particle \(P\) of mass \(m\) A second light elastic string also has natural length \(l\) and modulus of elasticity \(4 m g\) One end of this string is attached to \(P\) and the other end is attached to \(B\). Initially \(P\) rests in equilibrium at the point \(E\), where \(A E B\) is a vertical straight line.
    1. Show that \(A E = \frac { 13 } { 8 } l\) The particle \(P\) is now held at the point that is a distance \(2 l\) vertically below \(A\) and released from rest. At time \(t\), the vertical displacement of \(P\) from \(E\) is \(x\), where \(x\) is measured vertically downwards.
    2. Show that \(\ddot { x } = - \frac { 8 g } { l } x\)
    3. Find, in terms of \(g\) and \(l\), the speed of \(P\) when it is \(\frac { 1 } { 8 } l\) below \(E\).
    4. Find the length of time, in each complete oscillation, for which \(P\) is more than \(1.5 l\) from \(A\), giving your answer in terms of \(g\) and \(l\)
    Edexcel M3 2024 January Q7
    13 marks Challenging +1.2
    7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{92131234-bfc1-4e0e-87d4-db9335fbf343-24_506_640_296_715} \captionsetup{labelformat=empty} \caption{Figure 5}
    \end{figure} A thin smooth hollow spherical shell has centre \(O\) and radius \(r\). Part of the shell is removed to form a bowl with a plane circular rim. The bowl is fixed with the circular rim uppermost and horizontal. The point \(A\) is the lowest point of the bowl, as shown in Figure 5. The point \(B\) is on the rim of the bowl, with \(O B\) at an angle \(\theta\) to the upward vertical, where \(\tan \theta = \frac { 12 } { 5 }\) A small ball is placed in the bowl at \(A\). The ball is projected from \(A\) with horizontal speed \(u\) and moves in the vertical plane \(A O B\). The ball stays in contact with the bowl until it reaches \(B\). At the instant when the ball reaches \(B\), the speed of the ball is \(v\).
    By modelling the ball as a particle and ignoring air resistance,
    1. use the principle of conservation of mechanical energy to show that $$v ^ { 2 } = u ^ { 2 } - \frac { 36 } { 13 } g r$$
    2. show that \(u ^ { 2 } \geqslant \frac { 41 } { 13 } g r\) The point \(C\) is such that \(B C\) is a diameter of the rim of the bowl.
      Given that \(u ^ { 2 } = 4 g r\)
    3. use the model to show that, after leaving the inner surface of the bowl at \(B\), the ball falls back into the bowl before reaching \(C\).
    Edexcel M3 2014 June Q1
    8 marks Standard +0.3
    1. A particle \(P\) moves in a straight line with simple harmonic motion. The period of the motion is \(\frac { \pi } { 4 }\) seconds. At time \(t = 0 , P\) is at rest at the point \(A\) and the acceleration of \(P\) has magnitude \(20 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    Find
    1. the amplitude of the motion,
    2. the greatest speed of \(P\) during the motion,
    3. the time \(P\) takes to travel a total distance of 1.5 m after it has first left \(A\).
    Edexcel M3 2014 June Q2
    6 marks Standard +0.8
    2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{e5b08946-7311-4cf7-9c5f-5f309a1feed7-03_522_654_223_646} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A uniform lamina \(L\) is in the shape of an equilateral triangle of side \(2 a\). The lamina is placed in the \(x y\)-plane with one vertex at the origin \(O\) and an axis of symmetry along the \(x\)-axis, as shown in Figure 1. Use algebraic integration to find the \(x\) coordinate of the centre of mass of \(L\).