Questions FM2 (51 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Edexcel FM2 2022 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1f39620e-c10f-4344-89f1-626fff36d187-16_567_602_260_735} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The uniform plane lamina shown in Figure 3 is formed from two squares, \(A B C O\) and \(O D E F\), and a sector \(O D C\) of a circle with centre \(O\). Both squares have sides of length \(3 a\) and \(A O\) is perpendicular to \(O F\). The radius of the sector is \(3 a\)
[0pt] [In part (a) you may use, without proof, any of the centre of mass formulae given in the formulae booklet.]
  1. Show that the distance of the centre of mass of the sector \(O D C\) from \(O C\) is \(\frac { 4 a } { \pi }\)
  2. Find the distance of the centre of mass of the lamina from \(F C\) The lamina is freely suspended from \(F\) and hangs in equilibrium with \(F C\) at an angle \(\theta ^ { \circ }\) to the downward vertical.
  3. Find the value of \(\theta\)
Edexcel FM2 2022 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1f39620e-c10f-4344-89f1-626fff36d187-20_369_815_255_632} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} The shaded region shown in Figure 4 is bounded by the \(x\)-axis, the line with equation \(x = 9\) and the line with equation \(y = \frac { 1 } { 3 } x\). This shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution. This solid of revolution is used to model a solid right circular cone of height 9 cm and base radius 3 cm . The cone is non-uniform and the mass per unit volume of the cone at the point ( \(x , y , z\) ) is \(\lambda x \mathrm {~kg} \mathrm {~cm} ^ { - 3 }\), where \(0 \leqslant x \leqslant 9\) and \(\lambda\) is constant.
  1. Find the distance of the centre of mass of the cone from its vertex. A toy is made by joining the circular plane face of the cone to the circular plane face of a uniform solid hemisphere of radius 3 cm , so that the centres of the two plane surfaces coincide. The weight of the cone is \(W\) newtons and the weight of the hemisphere is \(k W\) newtons.
    When the toy is placed on a smooth horizontal plane with any point of the curved surface of the hemisphere in contact with the plane, the toy will remain at rest.
  2. Find the value of \(k\)
Edexcel FM2 2022 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1f39620e-c10f-4344-89f1-626fff36d187-24_639_593_246_737} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A package \(P\) of mass \(m\) is attached to one end of a string of length \(\frac { 2 a } { 5 }\). The other end of the string is attached to a fixed point \(O\). The package hangs at rest vertically below \(O\) with the string taut and is then projected horizontally with speed \(u\), as shown in Figure 5. When \(O P\) has turned through an angle \(\theta\) and the string is still taut, the tension in the string is \(T\) The package is modelled as a particle and the string as being light and inextensible.
  1. Show that \(T = 3 m g \cos \theta - 2 m g + \frac { 5 m u ^ { 2 } } { 2 a }\) Given that \(P\) moves in a complete vertical circle with centre \(O\)
  2. find, in terms of \(a\) and \(g\), the minimum possible value of \(u\) Given that \(u = 2 \sqrt { a g }\)
  3. find, in terms of \(g\), the magnitude of the acceleration of \(P\) at the instant when \(O P\) is horizontal.
  4. Apart from including air resistance, suggest one way in which the model could be refined to make it more realistic.
Edexcel FM2 2022 June Q8
  1. Throughout this question, use \(\boldsymbol { g } = \mathbf { 1 0 m ~ s } ^ { \mathbf { - 2 } }\)
A light elastic string has natural length 1.25 m and modulus of elasticity 25 N .
A particle \(P\) of mass 0.5 kg 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 with \(P\) vertically below \(A\) The particle is then pulled vertically down to a point \(B\) and released from rest.
  1. Show that, while the string is taut, \(P\) moves with simple harmonic motion with period \(\frac { \pi } { \sqrt { 10 } }\) seconds. The maximum kinetic energy of \(P\) during the subsequent motion is 2.5 J .
  2. Show that \(A B = 2 \mathrm {~m}\) The particle returns to \(B\) for the first time \(T\) seconds after it was released from rest at \(B\)
  3. Find the value of \(T\)
Edexcel FM2 2023 June Q1
  1. Three particles of masses \(3 m , 4 m\) and \(k m\) are positioned at the points with coordinates ( \(2 a , 3 a\) ), ( \(a , 5 a\) ) and ( \(2 \mu a , \mu a\) ) respectively, where \(k\) and \(\mu\) are constants.
The centre of mass of the three particles is at the point with coordinates \(( 2 a , 4 a )\).
Find (i) the value of \(k\)
(ii) the value of \(\mu\)
Edexcel FM2 2023 June Q2
  1. A particle of mass 2 kg is moving in a straight line on a smooth horizontal surface under the action of a horizontal force of magnitude \(F\) newtons.
At time \(t\) seconds \(( t > 0 )\),
  • the particle is moving with speed \(v \mathrm {~ms} ^ { - 1 }\)
  • \(F = 2 + v\)
The time taken for the speed of the particle to increase from \(5 \mathrm {~ms} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is \(T\) seconds.
  1. Show that \(T = 2 \ln \frac { 12 } { 7 }\) The distance moved by the particle as its speed increases from \(5 \mathrm {~ms} ^ { - 1 }\) to \(10 \mathrm {~ms} ^ { - 1 }\) is \(D\) metres.
  2. Find the exact value of \(D\).
Edexcel FM2 2023 June Q3
  1. \hspace{0pt} [In this question you may quote, without proof, the formula for the distance of the centre of mass of a uniform circular arc from its centre.]
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b070338-1de4-4c33-be29-d37ac06c9fed-08_816_483_338_790} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Five pieces of a uniform wire are joined together to form the rigid framework \(O A B C O\) shown in Figure 1, where
  • \(O A , O B\) and \(B C\) are straight, with \(O A = O B = B C = r\)
  • arc \(A B\) is one quarter of a circle with centre \(O\) and radius \(r\)
  • arc \(O C\) is one quarter of a circle of radius \(r\)
  • all five pieces of wire lie in the same plane
    1. Show that the centre of mass of arc \(A B\) is a distance \(\frac { 2 r } { \pi }\) from \(O A\).
Given that the distance of the centre of mass of the framework from \(O A\) is \(d\),
  • show that \(\mathrm { d } = \frac { 7 r } { 2 ( 3 + ) }\) The framework is freely pivoted at \(A\).
    The framework is held in equilibrium, with \(A O\) vertical, by a horizontal force of magnitude \(F\) which is applied to the framework at \(C\). Given that the weight of the framework is \(W\)
  • find \(F\) in terms of \(W\)
    •
    Edexcel FM2 2023 June Q4
    4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3b070338-1de4-4c33-be29-d37ac06c9fed-12_490_1177_219_507} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} A smooth hemisphere of radius \(a\) is fixed on a horizontal surface with its plane face in contact with the surface. The centre of the plane face of the hemisphere is \(O\). A particle \(P\) of mass \(M\) is disturbed from rest at the highest point of the hemisphere.
    When \(P\) is still on the surface of the hemisphere and the radius from \(O\) to \(P\) is at an angle \(\theta\) to the vertical,
    • the speed of \(P\) is \(v\)
    • the normal reaction between the hemisphere and the particle is \(R\), as shown in Figure 2.
      1. Show that \(\mathrm { R } = \mathrm { Mg } ( 3 \cos \theta - 2 )\)
      2. Find, in terms of \(a\) and \(g\), the speed of the particle at the instant when the particle leaves the surface of the hemisphere.
    Edexcel FM2 2023 June Q5
    5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3b070338-1de4-4c33-be29-d37ac06c9fed-16_730_442_223_877} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} A uniform lamina \(O A B\) is modelled by the finite region bounded by the \(x\)-axis, the \(y\)-axis and the curve with equation \(y = 9 - x ^ { 2 }\), for \(x \geqslant 0\), as shown shaded in Figure 3. The unit of length on both axes is 1 m . The area of the lamina is \(18 \mathrm {~m} ^ { 2 }\)
    1. Show that the centre of mass of the lamina is 3.6 m from \(\boldsymbol { O B }\).
      [0pt] [ Solutions relying on calculator technology are not acceptable.] A light string has one end attached to the lamina at \(O\) and the other end attached to the ceiling. A second light string has one end attached to the lamina at \(A\) and the other end attached to the ceiling.
      The lamina hangs in equilibrium with the strings vertical and \(O A\) horizontal.
      The weight of the lamina is \(W\)
      The tension in the string attached to the lamina at \(A\) is \(\lambda W\)
    2. Find the value of \(\lambda\)
    Edexcel FM2 2023 June Q6
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3b070338-1de4-4c33-be29-d37ac06c9fed-20_611_782_210_660} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} A hollow right circular cone, of internal base radius 0.6 m and height 0.8 m , is fixed with its axis vertical and its vertex \(V\) pointing downwards, as shown in Figure 4. A particle \(P\) of mass \(m \mathrm {~kg}\) moves in a horizontal circle of radius 0.5 m on the rough inner surface of the cone. The particle \(P\) moves with constant angular speed \(\omega\) rads \(^ { - 1 }\)
    The coefficient of friction between the particle \(P\) and the inner surface of the cone is 0.25 Find the greatest possible value of \(\omega\)
    Edexcel FM2 2023 June Q7
    7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3b070338-1de4-4c33-be29-d37ac06c9fed-24_590_469_292_484} \captionsetup{labelformat=empty} \caption{Figure 5}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3b070338-1de4-4c33-be29-d37ac06c9fed-24_415_554_383_1025} \captionsetup{labelformat=empty} \caption{Figure 6}
    \end{figure} The shaded region shown in Figure 5 is bounded by the line with equation \(x = a\) and the curve with equation \(x ^ { 2 } + y ^ { 2 } = 4 a ^ { 2 }\) This shaded region is rotated through \(180 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution. This solid is used to model a dome with height \(a\) metres and base radius \(\sqrt { 3 } a\) metres.
    The dome is modelled as being non-uniform with the mass per unit volume of the dome at the point \(( x , y , z )\) equal to \(\frac { \lambda } { x ^ { 2 } } \mathrm {~kg} \mathrm {~m} ^ { - 3 }\), where \(a \leqslant x \leqslant 2 a\) and \(\lambda\) is a constant.
    1. Show that the distance of the centre of mass of the dome from the centre of its plane face is \(\left( 4 \ln 2 - \frac { 5 } { 2 } \right) a\) metres. A solid uniform right circular cone has base radius \(\sqrt { 3 } a\) metres and perpendicular height \(4 a\) metres. A toy is formed by attaching the plane surface of the dome to the plane surface of the cone, as shown in Figure 6. The weight of the cone is \(k W\) and the weight of the dome is \(2 W\)
      The centre of mass of the toy is a distance \(d\) metres from the plane face of the dome.
    2. Show that \(d = \frac { | k + 5 - 8 \ln 2 | } { 2 + k } a\) The toy is suspended from a point on the circumference of the plane face of the dome and hangs freely in equilibrium with the plane face of the dome at an angle \(\alpha\) to the downward vertical.
      Given that \(\tan \alpha = \frac { 1 } { 2 \sqrt { 3 } }\)
    3. find the exact value of \(k\).
    Edexcel FM2 2023 June Q8
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3b070338-1de4-4c33-be29-d37ac06c9fed-28_200_1086_214_552} \captionsetup{labelformat=empty} \caption{Figure 7}
    \end{figure} The fixed points \(A\) and \(B\) lie on a smooth horizontal surface with \(A B = 6 \mathrm {~m}\).
    A particle \(P\) has mass 0.3 kg .
    One end of a light elastic string, of natural length 2 m and modulus of elasticity 20 N , is attached to \(P\), and the other end is attached to \(A\). One end of another light elastic string, of natural length 2 m and modulus of elasticity 40 N , is attached to \(P\) and the other end is attached to \(B\). The particle \(P\) is at rest in equilibrium at the point \(E\) on the surface, as shown in Figure 7.
    1. Show that \(E B = \frac { 8 } { 3 } \mathrm {~m}\). The particle \(P\) is now held at the midpoint of \(A B\) and released from rest.
    2. Show that \(P\) oscillates with simple harmonic motion about the point \(E\). The time between the instant when \(P\) is released and the instant when it first returns to the point \(E\) is \(S\) seconds.
    3. Find the exact value of \(S\).
    4. Find the length of time during one oscillation for which the speed of \(P\) is more than \(2 \mathrm {~ms} ^ { - 1 }\)
    Edexcel FM2 2024 June Q1
    1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
    A particle \(P\) moves along a straight line. Initially \(P\) is at rest at the point \(O\) on the line. At time \(t\) seconds, where \(t \geqslant 0\)
    • the displacement of \(P\) from \(O\) is \(x\) metres
    • the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction
    • the acceleration of \(P\) is \(\frac { 96 } { ( 3 t + 5 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }\) in the positive \(x\) direction
      1. Show that, at time \(t\) seconds, \(v = p - \frac { q } { ( 3 t + 5 ) ^ { 2 } }\), where \(p\) and \(q\) are constants to be determined.
      2. Find the limiting value of \(v\) as \(t\) increases.
      3. Find the value of \(x\) when \(t = 2\)
    Edexcel FM2 2024 June Q2
    2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c14975b7-6afa-44ce-beab-1cba2e82b249-06_373_847_251_609} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A uniform rod of length \(28 a\) is cut into seven identical rods each of length \(4 a\). These rods are joined together to form the rigid framework \(A B C D E A\) shown in Figure 1. All seven rods lie in the same plane.
    The distance of the centre of mass of the framework from \(E D\) is \(d\).
    1. Show that \(d = \frac { 8 \sqrt { 3 } } { 7 } a\) The weight of the framework is \(W\).
      The framework is freely pivoted about a horizontal axis through \(C\).
      The framework is held in equilibrium in a vertical plane, with \(A C\) vertical and \(A\) below \(C\), by a horizontal force that is applied to the framework at \(A\). The force acts in the same vertical plane as the framework and has magnitude \(F\).
    2. Find \(F\) in terms of \(W\).
    Edexcel FM2 2024 June Q3
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c14975b7-6afa-44ce-beab-1cba2e82b249-10_433_753_246_657} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a hemispherical bowl of internal radius \(10 d\) that is fixed with its circular rim horizontal. The centre of the circular rim is at the point \(O\).
    A particle \(P\) moves with constant angular speed on the smooth inner surface of the bowl. The particle \(P\) moves in a horizontal circle with radius \(8 d\) and centre \(C\).
    1. Find, in terms of \(g\), the exact magnitude of the acceleration of \(P\). The time for \(P\) to complete one revolution is \(T\).
    2. Find \(T\) in terms of \(d\) and \(g\).
    Edexcel FM2 2024 June Q4
    4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c14975b7-6afa-44ce-beab-1cba2e82b249-14_675_528_242_772} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} A uniform lamina \(O A B\) is in the shape of the region \(R\).
    Region \(R\) lies in the first quadrant and is bounded by the curve with equation \(\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 36 } = 1\), the \(x\)-axis, and the \(y\)-axis, as shown shaded in Figure 3. The point \(A\) is the point of intersection of the curve and the \(x\)-axis.
    The point \(B\) is the point of intersection of the curve and the \(y\)-axis.
    One unit on each axis represents 1 m .
    The area of \(R\) is \(6 \pi\)
    The centre of mass of \(R\) lies at the point with coordinates \(( \bar { x } , \bar { y } )\)
    1. Use algebraic integration to show that \(\bar { x } = \frac { 16 } { 3 \pi }\)
    2. Use algebraic integration to find the exact value of \(\bar { y }\) The lamina is freely suspended from \(A\) and hangs in equilibrium with \(O A\) at angle \(\theta ^ { \circ }\) to the downward vertical.
    3. Find the value of \(\theta\)
    Edexcel FM2 2024 June Q5
    1. A particle \(P\) moves in a straight line with simple harmonic motion about a fixed point \(O\). The magnitude of the greatest acceleration of \(P\) is \(18 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
    When \(P\) is 0.3 m from \(O\), the speed of \(P\) is \(2.4 \mathrm {~ms} ^ { - 1 }\)
    The amplitude of the motion is \(a\) metres.
    1. Show that \(a = 0.5\)
    2. Find the greatest speed of \(P\). During one oscillation, the speed of \(P\) is at least \(2 \mathrm {~ms} ^ { - 1 }\) for \(S\) seconds.
    3. Find the value of \(S\).
    Edexcel FM2 2024 June Q6
    6. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c14975b7-6afa-44ce-beab-1cba2e82b249-20_245_435_356_817} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} The shaded region, shown in Figure 4, is bounded by the \(x\)-axis, the line with equation \(x = 6\), the line with equation \(y = 2\) and the \(y\)-axis. This region is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { x }\)-axis to form a solid of revolution. This solid is used to model a non-uniform cylinder of height 6 cm and radius 2 cm . The mass per unit volume of the cylinder at the point \(( x , y , z )\) is \(\lambda ( x + 2 ) \mathrm { kg } \mathrm { cm } ^ { - 3 }\), where \(0 \leqslant x \leqslant 6\) and \(\lambda\) is a constant.
    1. Show that the mass of the cylinder is \(120 \lambda \pi \mathrm {~kg}\).
    2. Show that the centre of mass of the cylinder is 3.6 cm from \(O\). The point \(O\) is the centre of one end of the cylinder. The point \(A\) is the centre of the other end of the cylinder. A uniform solid hemisphere of radius 3 cm has density \(\lambda \mathrm { kg } \mathrm { cm } ^ { - 3 }\). The hemisphere is attached to the cylinder with the centre of its circular face in contact with the point \(A\) on the cylinder to form the model shown in Figure 5. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{c14975b7-6afa-44ce-beab-1cba2e82b249-20_309_673_1713_696} \captionsetup{labelformat=empty} \caption{Figure 5}
      \end{figure} The model is placed with the end containing \(O\) on a rough inclined plane which is inclined at angle \(\alpha ^ { \circ }\) to the horizontal. The plane is sufficiently rough to prevent the model from sliding. The model is on the point of toppling.
    3. Find the value of \(\alpha\).
    Edexcel FM2 2024 June Q7
    7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{c14975b7-6afa-44ce-beab-1cba2e82b249-24_419_935_251_566} \captionsetup{labelformat=empty} \caption{Figure 6}
    \end{figure} A smooth solid hemisphere has radius \(r\) and the centre of its plane face is \(O\).
    The hemisphere is fixed with its plane face in contact with horizontal ground, as shown in Figure 6.
    A small stone is at the point \(A\), the highest point on the surface of the hemisphere. The stone is projected horizontally from \(A\) with speed \(U\).
    The stone is still in contact with the hemisphere at the point \(B\), where \(O B\) makes an angle \(\theta\) with the upward vertical.
    The speed of the stone at the instant it reaches \(B\) is \(v\).
    The stone is modelled as a particle \(P\) and air resistance is modelled as being negligible.
    1. Use the model to find \(v ^ { 2 }\) in terms of \(U , r , g\) and \(\theta\) When \(P\) leaves the surface of the hemisphere, the speed of \(P\) is \(W\).
      Given that \(U = \sqrt { \frac { 2 r g } { 3 } }\)
    2. show that \(W ^ { 2 } = \frac { 8 } { 9 } r g\) After leaving the surface of the hemisphere, \(P\) moves freely under gravity until it hits the ground.
    3. Find the speed of \(P\) as it hits the ground, giving your answer in terms of \(r\) and \(g\). At the instant when \(P\) hits the ground it is travelling at \(\alpha ^ { \circ }\) to the horizontal.
    4. Find the value of \(\alpha\).
    Edexcel FM2 Specimen Q1
    1. A flag pole is 15 m long.
    The flag pole is non-uniform so that, at a distance \(x\) metres from its base, the mass per unit length of the flag pole, \(m \mathrm {~kg} \mathrm {~m} ^ { - 1 }\) is given by the formula \(m = 10 \left( 1 - \frac { x } { 25 } \right)\). The flag pole is modelled as a rod.
    1. Show that the mass of the flag pole is 105 kg .
    2. Find the distance of the centre of mass of the flag pole from its base.
    Edexcel FM2 Specimen Q2
    2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f06704e5-454c-41c1-9577-b1210f60480d-04_655_643_207_639} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A hollow right circular cone, of base diameter \(4 a\) and height \(4 a\) is fixed with its axis vertical and vertex \(V\) downwards, as shown in Figure 1. A particle of mass \(m\) moves in a horizontal circle with centre \(C\) on the rough inner surface of the cone with constant angular speed \(\omega\). The height of \(C\) above \(V\) is \(3 a\).
    The coefficient of friction between the particle and the inner surface of the cone is \(\frac { 1 } { 4 }\). Find, in terms of \(a\) and \(g\), the greatest possible value of \(\omega\).
    Edexcel FM2 Specimen Q3
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f06704e5-454c-41c1-9577-b1210f60480d-06_608_924_226_541} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} A uniform solid cylinder has radius \(2 a\) and height \(h ( h > a )\).
    A solid hemisphere of radius \(a\) is removed from the cylinder to form the vessel \(V\).
    The plane face of the hemisphere coincides with the upper plane face of the cylinder.
    The centre \(O\) of the hemisphere is also the centre of the upper plane face of the cylinder, as shown in Figure 2.
    1. Show that the centre of mass of \(V\) is \(\frac { 3 \left( 8 h ^ { 2 } - a ^ { 2 } \right) } { 8 ( 6 h - a ) }\) from \(O\). The vessel \(V\) is placed on a rough plane which is inclined at an angle \(\phi\) to the horizontal. The lower plane circular face of \(V\) is in contact with the inclined plane. Given that \(h = 5 a\), the plane is sufficiently rough to prevent \(V\) from slipping and \(V\) is on the point of toppling,
    2. find, to three significant figures, the size of the angle \(\phi\).
    Edexcel FM2 Specimen Q4
    1. A car of mass 500 kg moves along a straight horizontal road.
    The engine of the car produces a constant driving force of 1800 N .
    The car accelerates from rest from the fixed point \(O\) at time \(t = 0\) and at time \(t\) seconds the car is \(x\) metres from \(O\), moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the speed of the car is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the resistance to the motion of the car has magnitude \(2 v ^ { 2 } \mathrm {~N}\). At time \(T\) seconds, the car is at the point \(A\), moving with speed \(10 \mathrm {~ms} ^ { - 1 }\).
    1. Show that \(T = \frac { 25 } { 6 } \ln 2\)
    2. Show that the distance from \(O\) to \(A\) is \(125 \ln \frac { 9 } { 8 } \mathrm {~m}\).
    Edexcel FM2 Specimen Q5
    5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f06704e5-454c-41c1-9577-b1210f60480d-12_693_515_210_781} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} A shop sign is modelled as a uniform rectangular lamina \(A B C D\) with a semicircular lamina removed. The semicircle has radius \(a , B C = 4 a\) and \(C D = 2 a\).
    The centre of the semicircle is at the point \(E\) on \(A D\) such that \(A E = d\), as shown in Figure 3.
    1. Show that the centre of mass of the sign is \(\frac { 44 a } { 3 ( 16 - \pi ) }\) from \(A D\). The sign is suspended using vertical ropes attached to the sign at \(A\) and at \(B\) and hangs in equilibrium with \(A B\) horizontal. The weight of the sign is \(W\) and the ropes are modelled as light inextensible strings.
    2. Find, in terms of \(W\) and \(\pi\), the tension in the rope attached at \(B\). The rope attached at \(B\) breaks and the sign hangs freely in equilibrium suspended from \(A\), with \(A D\) at an angle \(\alpha\) to the downward vertical. Given that \(\tan \alpha = \frac { 11 } { 18 }\)
    3. find \(d\) in terms of \(a\) and \(\pi\).
    Edexcel FM2 Specimen Q6
    1. A small bead \(B\) of mass \(m\) is threaded on a circular hoop.
    The hoop has centre \(O\) and radius \(a\) and is fixed in a vertical plane.
    The bead is projected with speed \(\sqrt { \frac { 7 } { 2 } g a }\) from the lowest point of the hoop.
    The hoop is modelled as being smooth.
    When the angle between \(O B\) and the downward vertical is \(\theta\), the speed of \(B\) is \(v\).
    1. Show that \(v ^ { 2 } = g a \left( \frac { 3 } { 2 } + 2 \cos \theta \right)\)
    2. Find the size of \(\theta\) at the instant when the contact force between \(B\) and the hoop is first zero.
    3. Give a reason why your answer to part (b) is not likely to be the actual value of \(\theta\).
    4. Find the magnitude and direction of the acceleration of \(B\) at the instant when \(B\) is first at instantaneous rest.