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Edexcel M3 2016 January Q6
14 marks Challenging +1.2
6.
  1. Use algebraic integration to show that the centre of mass of a uniform solid hemisphere of radius \(r\) is at a distance \(\frac { 3 } { 8 } r\) from the centre of its plane face.
    [0pt] [You may assume that the volume of a sphere of radius \(r\) is \(\frac { 4 } { 3 } \pi r ^ { 3 }\) ]
    (5) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ffe0bc72-3136-48d9-9d5b-4a364d134070-09_351_597_598_678} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} A uniform solid hemisphere of mass \(m\) and radius \(r\) is joined to a uniform solid right circular cone to form a solid \(S\). The cone has mass \(M\), base radius \(r\) and height \(4 r\). The vertex of the cone is \(O\). The plane face of the cone coincides with the plane face of the hemisphere, as shown in Figure 3.
  2. Find the distance of the centre of mass of \(S\) from \(O\). The point \(A\) lies on the circumference of the base of the cone. The solid is placed on a horizontal table with \(O A\) in contact with the table. The solid remains in equilibrium in this position.
  3. Show that \(M \geqslant \frac { 1 } { 10 } m\)
Edexcel M3 2016 January Q7
15 marks Challenging +1.2
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ffe0bc72-3136-48d9-9d5b-4a364d134070-11_581_641_262_678} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A particle 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 turn freely in a vertical plane about a horizontal axis through \(O\). The particle is projected with speed \(u\) from a point \(A\), where \(O A\) makes an angle \(\alpha\) with the upward vertical through \(O\), as shown in Figure 4. The particle moves in complete vertical circles. Given that \(\cos \alpha = \frac { 4 } { 5 }\)
  1. show that \(u > \sqrt { \frac { 2 g l } { 5 } }\) As the rod rotates, the least tension in the rod is \(T\) and the greatest tension is \(4 T\).
  2. Show that \(u = \sqrt { \frac { 17 } { 5 } g l }\)
    \includegraphics[max width=\textwidth, alt={}]{ffe0bc72-3136-48d9-9d5b-4a364d134070-12_2639_1830_121_121}
Edexcel M3 2017 January Q1
7 marks Standard +0.8
1. \begin{figure}[h]
[diagram]
\captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The shaded region \(R\) is bounded by the curve with equation \(y ^ { 2 } = 9 ( 4 - x )\), the positive \(x\)-axis and the positive \(y\)-axis, as shown in Figure 1. A uniform solid \(S\) is formed by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis.
Use algebraic integration to find the \(x\) coordinate of the centre of mass of \(S\).
Edexcel M3 2017 January Q2
7 marks Standard +0.3
2. A particle \(P\) of mass 0.6 kg is moving along the positive \(x\)-axis in the positive direction. The only force acting on \(P\) acts in the direction of \(x\) increasing and has magnitude \(\left( 3 t + \frac { 1 } { 2 } \right) \mathrm { N }\), where \(t\) seconds is the time after \(P\) leaves the origin \(O\). When \(t = 0 , P\) is at rest at \(O\).
  1. Find an expression, in terms of \(t\), for the velocity of \(P\) at time \(t\) seconds. The particle passes through the point \(A\) with speed \(\frac { 10 } { 3 } \mathrm {~ms} ^ { - 1 }\).
  2. Find the distance \(O A\).
Edexcel M3 2017 January Q3
6 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{85d8fc7d-8863-419e-8eef-8751a6fb6315-04_647_684_260_635} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A uniform right circular solid cylinder has radius \(4 a\) and height \(6 a\). A solid hemisphere of radius \(3 a\) is removed from the cylinder forming a solid \(S\). The upper plane face of the cylinder coincides with the plane face of the hemisphere. The centre of the upper plane face of the cylinder is \(O\) and this is also the centre of the plane face of the hemisphere, as shown in Figure 2. Find the distance from \(O\) to the centre of mass of \(S\).
(6)
Edexcel M3 2017 January Q4
12 marks Standard +0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{85d8fc7d-8863-419e-8eef-8751a6fb6315-05_654_515_267_712} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A light inextensible string has its ends attached to two fixed points \(A\) and \(B\). The point \(A\) is vertically above \(B\) and \(A B = 7 a\). A particle \(P\) of mass \(m\) is fixed to the string and moves with constant angular speed \(\omega\) in a horizontal circle of radius \(4 a\). The centre of the circle is \(C\), where \(C\) lies on \(A B\) and \(A C = 3 a\), as shown in Figure 3. Both parts of the string are taut.
  1. Show that the tension in \(A P\) is \(\frac { 5 } { 7 } m \left( 4 a \omega ^ { 2 } + g \right)\).
  2. Find the tension in \(B P\).
  3. Deduce that \(\omega \geqslant \sqrt { \frac { g } { k a } }\), stating the value of \(k\).
Edexcel M3 2017 January Q5
9 marks Standard +0.8
A particle \(P\) of mass \(4 m\) is attached to one end of a light elastic string of natural length \(l\) and modulus of elasticity 3 mg . The other end of the string is attached to a fixed point \(O\) on a rough horizontal table. The particle lies at rest at the point \(A\) on the table, where \(O A = \frac { 4 } { 3 } l\). The coefficient of friction between \(P\) and the table is \(\mu\).
  1. Show that \(\mu \geqslant \frac { 1 } { 4 }\) The particle is now moved along the table to the point \(B\), where \(O B = 2 l\), and released from rest. Given that \(\mu = \frac { 2 } { 5 }\)
  2. show that \(P\) comes to rest before the string becomes slack.
    (5)
Edexcel M3 2017 January Q6
17 marks Challenging +1.2
6. One end of a light elastic string, of natural length 5 l and modulus of elasticity 20 mg , is attached to a fixed point \(A\). A particle \(P\) of mass \(2 m\) is attached to the free end of the string and \(P\) hangs freely in equilibrium at the point \(B\).
  1. Find the distance \(A B\).
    (3) The particle is now pulled vertically downwards from \(B\) to the point \(C\) and released from rest. In the subsequent motion the string does not become slack.
  2. Show that \(P\) moves with simple harmonic motion with centre \(B\).
  3. Find the period of this motion. The greatest speed of \(P\) during this motion is \(\frac { 1 } { 5 } \sqrt { g l }\)
  4. Find the amplitude of this motion. The point \(D\) is the midpoint of \(B C\) and the point \(E\) is the highest point reached by \(P\).
  5. Find the time taken by \(P\) to move directly from \(D\) to \(E\).
Edexcel M3 2017 January Q7
17 marks Challenging +1.2
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{85d8fc7d-8863-419e-8eef-8751a6fb6315-11_412_1054_260_447} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A hollow sphere has internal radius \(r\) and centre \(O\). A bowl with a plane circular rim is formed by removing part of the sphere. The bowl is fixed to a horizontal floor with the rim uppermost and horizontal. The point \(B\) is the lowest point of the inner surface of the bowl. The point \(A\), where angle \(A O B = 120 ^ { \circ }\), lies on the rim of the bowl, as shown in Figure 4. A particle \(P\) of mass \(m\) is projected from \(A\), with speed \(U\) at \(90 ^ { \circ }\) to \(O A\), and moves on the smooth inner surface of the bowl. The motion of \(P\) takes place in the vertical plane \(O A B\).
  1. Find, in terms of \(m , g , U\) and \(r\), the magnitude of the force exerted on \(P\) by the bowl at the instant when \(P\) passes through \(B\).
  2. Find, in terms of \(g , U\) and \(r\), the greatest height above the floor reached by \(P\). Given that \(U > \sqrt { 2 g r }\)
  3. show that, after leaving the surface of the bowl, \(P\) does not fall back into the bowl.
Edexcel M3 2018 January Q1
5 marks Standard +0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d93ae982-9395-4311-9972-be727b3ce954-02_333_890_264_529} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform solid \(S\) consists of a right solid circular cone of base radius \(r\) and a right solid cylinder, also of radius \(r\). The cone has height \(4 h\) and the centre of the plane face of the cone is \(O\). The cylinder has height \(3 h\). The cone and cylinder are joined so that the plane face of the cone coincides with one of the plane faces of the cylinder, as shown in Figure 1. Find the distance from \(O\) to the centre of mass of \(S\).
Edexcel M3 2018 January Q2
5 marks Standard +0.3
  1. A particle of mass 0.9 kg is attached to one end of a light elastic string, of natural length 1.2 m and modulus of elasticity 29.4 N . The other end of the string is attached to a fixed point \(A\) on a ceiling.
The particle is held at \(A\) and then released from rest. The particle first comes to instantaneous rest at the point \(B\). Find the distance \(A B\).
(5)
Edexcel M3 2018 January Q3
10 marks Standard +0.3
A particle \(P\) of mass 0.4 kg moves along the \(x\)-axis in the positive direction. At time \(t = 0 , P\) passes through the origin \(O\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t\) seconds \(P\) is \(x\) metres from \(O\) and the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resultant force acting on \(P\) has magnitude \(\frac { 8 } { ( t + 4 ) ^ { 2 } } \mathrm {~N}\) and is directed towards \(O\).
  1. Show that \(v = \frac { 20 } { t + 4 } + 5\) When \(v = 6 , x = a + b \ln 5\), where \(a\) and \(b\) are integers.
  2. Using algebraic integration, find the value of \(a\) and the value of \(b\).
Edexcel M3 2018 January Q4
12 marks Challenging +1.2
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d93ae982-9395-4311-9972-be727b3ce954-10_547_841_244_555} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A small smooth bead \(P\) is threaded on a light inextensible string of length \(8 a\). One end of the string is attached to a fixed point \(A\) on a smooth horizontal table. The other end of the string is attached to the fixed point \(B\), where \(B\) is vertically above \(A\) and \(A B = 4 a\), as shown in Figure 2. The bead moves with constant angular speed, in a horizontal circle, centre \(A\), with \(A P\) horizontal. The bead remains in contact with the table and both parts of the string, \(A P\) and \(B P\), are taut. The time for \(P\) to complete one revolution is \(S\). Show that \(\quad S \geqslant \pi \sqrt { \frac { 6 a } { g } }\)
Edexcel M3 2018 January Q5
11 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d93ae982-9395-4311-9972-be727b3ce954-14_510_723_269_607} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows the finite region \(R\) which is bounded by part of the curve with equation \(y = \sin x\), the \(x\)-axis and the line with equation \(x = \frac { \pi } { 2 }\). A uniform solid \(S\) is formed by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis. Using algebraic integration,
  1. show that the volume of \(S\) is \(\frac { \pi ^ { 2 } } { 4 }\)
  2. find, in terms of \(\pi\), the \(x\) coordinate of the centre of mass of \(S\).
Edexcel M3 2018 January Q6
15 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d93ae982-9395-4311-9972-be727b3ce954-18_483_730_242_609} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(O\). The particle is held at the point \(A\), where \(O A = l\) and \(O A\) is horizontal. The particle is then projected vertically downwards from \(A\) with speed \(\sqrt { 2 g l }\), as shown in Figure 4 . When the string makes an angle \(\theta\) with the downward vertical through \(O\) and the string is still taut, the tension in the string is \(T\).
  1. Show that \(T = m g ( 3 \cos \theta + 2 )\) At the instant when the particle reaches the point \(B\), the string becomes slack.
  2. Find the speed of \(P\) at \(B\).
  3. Find the greatest height above \(O\) reached by \(P\) in the subsequent motion.
Edexcel M3 2018 January Q7
17 marks Challenging +1.2
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d93ae982-9395-4311-9972-be727b3ce954-22_197_945_251_497} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} The fixed points \(A\) and \(B\) are 4.2 m apart on a smooth horizontal floor. One end of a light elastic spring, of natural length 1.8 m and modulus of elasticity 20 N , is attached to a particle \(P\) and the other end is attached to \(A\). One end of another light elastic spring, of natural length 0.9 m and modulus of elasticity 15 N , 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 5.
  1. Show that \(A O = 2.7 \mathrm {~m}\). The particle \(P\) now receives an impulse acting in the direction \(O B\) and moves away from \(O\) towards \(B\). In the subsequent motion \(P\) does not reach \(B\).
  2. Show that \(P\) moves with simple harmonic motion about centre \(O\). The mass of \(P\) is 10 kg and the magnitude of the impulse is \(J \mathrm { Ns }\). Given that \(P\) first comes to instantaneous rest at the point \(C\) where \(A C = 2.9 \mathrm {~m}\),
    1. find the value of \(J\),
    2. find the time taken by \(P\) to travel a total distance of 0.5 m from when it first leaves \(O\).
Edexcel M3 2019 January Q1
6 marks Standard +0.8
  1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the displacement of \(P\) from the origin \(O\) is \(x\) metres and the acceleration of \(P\) is \(\left( \frac { 7 } { 2 } - 2 x \right) \mathrm { m } \mathrm { s } ^ { - 2 }\), measured in the positive \(x\) direction. At time \(t = 0 , P\) passes through \(O\) moving with speed \(3 \mathrm {~ms} ^ { - 1 }\) in the positive \(x\) direction. Find the distance of \(P\) from \(O\) when \(P\) first comes to instantaneous rest.
    (6)
Edexcel M3 2019 January Q2
12 marks Challenging +1.2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ae189c40-0071-4a6b-91eb-8ffebe082a04-04_573_456_264_712} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A small ball \(P\) of mass \(m\) is attached to the midpoint of a light inextensible string of length \(2 a\). The ends of the string are attached to fixed points \(A\) and \(B\), where \(A\) is vertically above \(B\) and \(A B = a\), as shown in Figure 1. The system rotates about the line \(A B\) with constant angular speed \(\omega\). The ball moves in a horizontal circle with both parts of the string taut. The tension in the string must be less than \(3 m g\) otherwise the string will break. Given that the time taken by the ball to complete one revolution is \(S\), show that $$\pi \sqrt { \frac { a } { g } } < S < \pi \sqrt { \frac { k a } { g } }$$ stating the value of the constant \(k\).
Edexcel M3 2019 January Q3
12 marks Standard +0.3
3. A particle \(P\) is moving in a straight line with simple harmonic motion between two points \(A\) and \(B\), where \(A B\) is \(2 a\) metres. The point \(C\) lies on the line \(A B\) and \(A C = \frac { 1 } { 2 } a\) metres. The particle passes through \(C\) with speed \(\frac { 3 a \sqrt { 3 } } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the period of the motion. The maximum magnitude of the acceleration of \(P\) is \(45 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find
  2. the value of \(a\),
  3. the maximum speed of \(P\). The point \(D\) lies on \(A B\) and \(P\) takes a quarter of one period to travel directly from \(C\) to \(D\).
  4. Find the distance CD.
Edexcel M3 2019 January Q4
13 marks Challenging +1.2
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ae189c40-0071-4a6b-91eb-8ffebe082a04-12_364_718_278_612} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The ends of a light elastic string, of natural length \(4 l\) and modulus of elasticity \(\lambda\), are attached to two fixed points \(A\) and \(B\), where \(A B\) is horizontal and \(A B = 4 l\). A particle \(P\) of mass \(2 m\) is attached to the midpoint of the string. The particle hangs freely in equilibrium at a distance \(\frac { 3 } { 2 } l\) vertically below the midpoint of \(A B\), as shown in Figure 2.
  1. Show that \(\lambda = \frac { 20 } { 3 } m g\). The particle is pulled vertically downwards from its equilibrium position until the total length of the string is 6l. The particle is then released from rest.
  2. Show that \(P\) comes to instantaneous rest before reaching the line \(A B\).
Edexcel M3 2019 January Q5
16 marks Standard +0.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ae189c40-0071-4a6b-91eb-8ffebe082a04-16_492_442_237_744} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The region \(R\), shown shaded in Figure 3, is bounded by the circle with centre \(O\) and radius \(r\), the line with equation \(x = \frac { 3 } { 5 } r\) and the \(x\)-axis. The region is rotated through one complete revolution about the \(x\)-axis to form a uniform solid \(S\).
  1. Use algebraic integration to show that the \(x\) coordinate of the centre of mass of \(S\) is \(\frac { 48 } { 65 } r\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{ae189c40-0071-4a6b-91eb-8ffebe082a04-16_394_643_1311_653} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} A bowl is made from a uniform solid hemisphere of radius 6 cm by removing a hemisphere of radius 5 cm . Both hemispheres have the same centre \(A\) and the same axis of symmetry. The bowl is fixed with its open plane face uppermost and horizontal. Liquid is poured into the bowl. The depth of the liquid is 2 cm , as shown in Figure 4. The mass of the empty bowl is \(5 M \mathrm {~kg}\) and the mass of the liquid is \(2 M \mathrm {~kg}\).
  2. Find, to 3 significant figures, the distance from \(A\) to the centre of mass of the bowl with its liquid.
Edexcel M3 2019 January Q6
16 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ae189c40-0071-4a6b-91eb-8ffebe082a04-20_497_643_237_653} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a hollow sphere, with centre \(O\) and internal radius \(a\), which is fixed to a horizontal surface. A particle \(P\) of mass \(m\) is projected horizontally with speed \(\sqrt { \frac { 7 a g } { 2 } }\) from the lowest point \(A\) of the inner surface of the sphere. The particle moves in a vertical circle with centre \(O\) on the smooth inner surface of the sphere. The particle passes through the point \(B\), on the inner surface of the sphere, where \(O B\) is horizontal.
  1. Find, in terms of \(m\) and \(g\), the normal reaction exerted on \(P\) by the surface of the sphere when \(P\) is at \(B\). The particle leaves the inner surface of the sphere at the point \(C\), where \(O C\) makes an angle \(\theta , \theta > 0\), with the upward vertical.
  2. Show that, after leaving the surface of the sphere at \(C\), the particle is next in contact with the surface at \(A\).
    END
Edexcel M3 2021 January Q1
8 marks Standard +0.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-02_469_758_251_593} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The region \(R\), shown shaded in Figure 1, is bounded by the curve with equation \(y = \frac { 1 } { x }\), the line with equation \(x = 1\), the positive \(x\)-axis and the line with equation \(x = a\) where \(a > 1\) A uniform solid \(S\) is formed by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis.
  1. Show that the volume of \(S\) is $$\pi \left( 1 - \frac { 1 } { a } \right)$$
  2. Find the \(x\) coordinate of the centre of mass of \(S\).
Edexcel M3 2021 January Q2
10 marks Standard +0.8
2. A particle \(P\) of mass \(m\) is at a distance \(x\) above the surface of the Earth. The Earth exerts a gravitational force on \(P\). This force is directed towards the centre of the Earth. The magnitude of this force is inversely proportional to the square of the distance of \(P\) from the centre of the Earth. At the surface of the Earth the acceleration due to gravity is \(g\). The Earth is modelled as a fixed sphere of radius \(R\).
  1. Show that the magnitude of the gravitational force on \(P\) is \(\frac { m g R ^ { 2 } } { ( x + R ) ^ { 2 } }\) A particle is released from rest from a point above the surface of the Earth. When the particle is at a distance \(R\) above the surface of the Earth, the particle has speed \(U\). Air resistance is modelled as being negligible.
  2. Find, in terms of \(U , g\) and \(R\), the speed of the particle when it strikes the surface of the Earth.
    VIAV SIHI NI III IM I ON OCVIAV SIMI NI III M M O N OOVIUV SIMI NI JIIYM ION OC
Edexcel M3 2021 January Q3
11 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8a687d17-ec7e-463f-84dd-605f5c230db1-08_506_527_251_712} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A fairground ride consists of a cabin \(C\) that travels in a horizontal circle with a constant angular speed about a fixed vertical central axis. The cabin is attached to one end of each of two rigid arms, each of length 5 m . The other end of the top arm is attached to the fixed point \(A\) at the top of the central axis of the ride. The other end of the lower arm is attached to the fixed point \(B\) on the central axis, where \(A B\) is 8 m , as shown in Figure 2. Both arms are free to rotate about the central axis. The arms are modelled as light inextensible rods. The cabin, together with the people inside, is modelled as a particle. The cabin completes one revolution every 2 seconds. Given that the combined mass of the cabin and the people is 600 kg ,
  1. find
    1. the tension in the upper arm of the ride,
    2. the tension in the lower arm of the ride. In a refined model, it is assumed that both arms stretch to a length of 5.1 m .
  2. State how this would affect the sum of the tensions in the two arms, justifying your answer.