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Edexcel M3 2003 June Q7
16 marks Standard +0.3
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{c0a3336d-b0ca-4588-80d1-445e2a5e493c-5_530_628_221_730}
\end{figure} The shaded region \(R\) is bounded by part of the curve with equation \(y = \frac { 1 } { 2 } ( x - 2 ) ^ { 2 }\), the \(x\)-axis and the \(y\)-axis, as shown in Fig. 3. The unit of length on both axes is 1 cm . A uniform solid \(S\) is made by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis. Using integration,
  1. calculate the volume of the solid \(S\), leaving your answer in terms of \(\pi\),
  2. show that the centre of mass of \(S\) is \(\frac { 1 } { 3 } \mathrm {~cm}\) from its plane face. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{c0a3336d-b0ca-4588-80d1-445e2a5e493c-5_411_772_1357_568}
    \end{figure} A tool is modelled as having two components, a solid uniform cylinder \(C\) and the solid \(S\). The diameter of \(C\) is 4 cm and the length of \(C\) is 8 cm . One end of \(C\) coincides with the plane face of \(S\). The components are made of different materials. The weight of \(C\) is \(10 W\) newtons and the weight of \(S\) is \(2 W\) newtons. The tool lies in equilibrium with its axis of symmetry horizontal on two smooth supports \(A\) and \(B\), which are at the ends of the cylinder, as shown in Fig. 4.
  3. Find the magnitude of the force of the support \(A\) on the tool.
Edexcel M3 2004 June Q1
3 marks Easy -1.2
  1. A circular flywheel of diameter 7 cm is rotating about the axis through its centre and perpendicular to its plane with constant angular speed 1000 revolutions per minute.
Find, in \(\mathrm { m } \mathrm { s } ^ { - 1 }\) to 3 significant figures, the speed of a point on the rim of the flywheel.
Edexcel M3 2004 June Q2
7 marks Standard +0.8
2. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{b9e9b91c-7e6d-4b84-9f0e-180b626887c2-2_460_549_651_792}
\end{figure} Two light elastic strings each have natural length \(a\) and modulus of elasticity \(\lambda\). A particle \(P\) of mass \(m\) is attached to one end of each string. The other ends of the strings are attached to points \(A\) and \(B\), where \(A B\) is horizontal and \(A B = 2 a\). The particle is held at the mid-point of \(A B\) and released from rest. It comes to rest for the first time in the subsequent motion when \(P A\) and \(P B\) make angles \(\alpha\) with \(A B\), where \(\tan \alpha = \frac { 4 } { 3 }\), as shown in Fig. 1. Find \(\lambda\) in terms of \(m\) and \(g\).
Edexcel M3 2004 June Q3
10 marks Challenging +1.2
3. A particle \(P\) of mass \(m \mathrm {~kg}\) slides from rest down a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. When \(P\) has moved a distance \(x\) metres down the plane, the resistance to the motion of \(P\) from non-gravitational forces has magnitude \(m x ^ { 2 }\) newtons. Find
  1. the speed of \(P\) when \(x = 2\),
  2. the distance \(P\) has moved when it comes to rest for the first time.
Edexcel M3 2004 June Q4
11 marks Standard +0.3
4. A rough disc rotates in a horizontal plane with constant angular velocity \(\omega\) about a fixed vertical axis. A particle \(P\) of mass \(m\) lies on the disc at a distance \(\frac { 4 } { 3 } a\) from the axis. The coefficient of friction between \(P\) and the disc is \(\frac { 3 } { 5 }\). Given that \(P\) remains at rest relative to the disc,
  1. prove that \(\omega ^ { 2 } \leqslant \frac { 9 g } { 20 a }\). The particle is now connected to the axis by a horizontal light elastic string of natural length \(a\) and modulus of elasticity 2 mg . The disc again rotates with constant angular velocity \(\omega\) about the axis and \(P\) remains at rest relative to the disc at a distance \(\frac { 4 } { 3 } a\) from the axis.
  2. Find the greatest and least possible values of \(\omega ^ { 2 }\).
Edexcel M3 2004 June Q5
13 marks Standard +0.3
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{b9e9b91c-7e6d-4b84-9f0e-180b626887c2-3_522_659_1043_701}
\end{figure} A toy is formed by joining a uniform solid right circular cone, of base radius \(3 r\) and height \(4 r\), to a uniform solid cylinder, also of radius \(3 r\) and height \(4 r\). The cone and the cylinder are made from the same material, and the plane face of the cone coincides with a plane face of the cylinder, as shown in Fig. 2. The centre of this plane face is \(O\).
  1. Find the distance of the centre of mass of the toy from \(O\). The point \(A\) lies on the edge of the plane face of the cylinder which forms the base of the toy. The toy is suspended from \(A\) and hangs in equilibrium.
  2. Find, in degrees to one decimal place, the angle between the axis of symmetry of the toy and the vertical. The toy is placed with the curved surface of the cone on horizontal ground.
  3. Determine whether the toy will topple.
    (4)
Edexcel M3 2004 June Q6
15 marks Standard +0.8
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{b9e9b91c-7e6d-4b84-9f0e-180b626887c2-4_460_799_301_657}
\end{figure} Figure 3 represents the path of a skier of mass 70 kg moving on a ski-slope \(A B C D\). The path lies in a vertical plane. From \(A\) to \(B\), the path is modelled as a straight line inclined at \(60 ^ { \circ }\) to the horizontal. From \(B\) to \(D\), the path is modelled as an arc of a vertical circle of radius 50 m . The lowest point of the \(\operatorname { arc } B D\) is \(C\). At \(B\), the skier is moving downwards with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At \(D\), the path is inclined at \(30 ^ { \circ }\) to the horizontal and the skier is moving upwards. By modelling the slope as smooth and the skier as a particle, find
  1. the speed of the skier at \(C\),
  2. the normal reaction of the slope on the skier at \(C\),
  3. the speed of the skier at \(D\),
  4. the change in the normal reaction of the slope on the skier as she passes \(B\). The model is refined to allow for the influence of friction on the motion of the skier.
  5. State briefly, with a reason, how the answer to part (b) would be affected by using such a model. (No further calculations are expected.)
Edexcel M3 2004 June Q7
16 marks Challenging +1.3
7. A particle \(P\) of mass 0.3 kg is attached to one end of a light elastic spring. The other end of the spring is attached to a fixed point \(O\) on a smooth horizontal table. The spring has natural length 2 m and modulus of elasticity 21.6 N . The particle \(P\) is placed on the table at the point \(A\), where \(O A = 2 \mathrm {~m}\). The particle \(P\) is now pulled away from \(O\) to the point \(B\), where \(O A B\) is a straight line with \(O B = 3.5 \mathrm {~m}\). It is then released from rest.
  1. Prove that \(P\) moves with simple harmonic motion of period \(\frac { \pi } { 3 } \mathrm {~s}\).
  2. Find the speed of \(P\) when it reaches \(A\). The point \(C\) is the mid-point of \(A B\).
  3. Find, in terms of \(\pi\), the time taken for \(P\) to reach \(C\) for the first time. Later in the motion, \(P\) collides with a particle \(Q\) of mass 0.2 kg which is at rest at \(A\).
    After the impact, \(P\) and \(Q\) coalesce to form a single particle \(R\).
  4. Show that \(R\) also moves with simple harmonic motion and find the amplitude of this motion. END
Edexcel M3 2005 June Q1
6 marks Standard +0.3
1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{fecee25b-e5d9-4669-89a1-6ae445090126-2_336_624_306_683}
\end{figure} A particle of mass 0.8 kg is attached to one end of a light elastic spring, of natural length 2 m and modulus of elasticity 20 N . The other end of the spring is attached to a fixed point \(O\) on a smooth plane which is inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 3 } { 4 }\). The particle is held on the plane at a point which is 1.6 m down a line of greatest slope of the plane from \(O\), as shown in Figure 1. The particle is then released from rest. Find the initial acceleration of the particle.
(Total 6 marks)
Edexcel M3 2005 June Q2
9 marks Standard +0.3
2. A closed container \(C\) consists of a thin uniform hollow hemispherical bowl of radius \(a\), together with a lid. The lid is a thin uniform circular disc, also of radius \(a\). The centre \(O\) of the disc coincides with the centre of the hemispherical bowl. The bowl and its lid are made of the same material.
  1. Show that the centre of mass of \(C\) is at a distance \(\frac { 1 } { 3 } a\) from \(O\). The container \(C\) has mass \(M\). A particle of mass \(\frac { 1 } { 2 } M\) is attached to the container at a point \(P\) on the circumference of the lid. The container is then placed with a point of its curved surface in contact with a horizontal plane. The container rests in equilibrium with \(P , O\) and the point of contact in the same vertical plane.
  2. Find, to the nearest degree, the angle made by the line \(P O\) with the horizontal.
Edexcel M3 2005 June Q3
9 marks Standard +0.3
3. A light elastic string has natural length \(2 l\) and modulus of elasticity \(4 m g\). One end of the string is attached to a fixed point \(A\) and the other end to a fixed point \(B\), where \(A\) and \(B\) lie on a smooth horizontal table, with \(A B = 4 l\). A particle \(P\) of mass \(m\) is attached to the mid-point of the string. The particle is released from rest at the point of the line \(A B\) which is \(\frac { 5 l } { 3 }\) from \(B\). The speed of \(P\) at the mid-point of \(A B\) is \(V\).
  1. Find \(V\) in terms of \(g\) and \(L\).
  2. Explain why \(V\) is the maximum speed of \(P\).
    (Total 9 marks)
Edexcel M3 2005 June Q4
10 marks Standard +0.3
4. A particle \(P\) of mass \(m\) moves on the smooth inner surface of a spherical bowl of internal radius \(r\). The particle moves with constant angular speed in a horizontal circle, which is at a depth \(\frac { 1 } { 2 } r\) below the centre of the bowl.
  1. Find the normal reaction of the bowl on \(P\).
  2. Find the time for \(P\) to complete one revolution of its circular path.
    (6)
    (Total 10 marks)
Edexcel M3 2005 June Q5
13 marks Standard +0.3
5. A smooth solid sphere, with centre \(O\) and radius \(a\), is fixed to the upper surface of a horizontal table. A particle \(P\) is placed on the surface of the sphere at a point \(A\), where \(O A\) makes an angle \(\alpha\) with the upward vertical, and \(0 < \alpha < \frac { \pi } { 2 }\). The particle is released from rest. When \(O P\) makes an angle \(\theta\) with the upward vertical, and \(P\) is still on the surface of the sphere, the speed of \(P\) is \(v\).
  1. Show that \(v ^ { 2 } = 2 g a ( \cos \alpha - \cos \theta )\). Given that \(\cos \alpha = \frac { 3 } { 4 }\), find
  2. the value of \(\theta\) when \(P\) loses contact with the sphere,
  3. the speed of \(P\) as it hits the table.
    (Total 13 marks)
Edexcel M3 2005 June Q6
14 marks Standard +0.3
6. The rise and fall of the water level in a harbour is modelled as simple harmonic motion. On a particular day the maximum and minimum depths of water in the harbour are 10 m and 4 m and these occur at 1100 hours and 1700 hours respectively.
  1. Find the speed, in \(\mathrm { m } \mathrm { h } ^ { - 1 }\), at which the water level in the harbour is falling at 1600 hours on this particular day.
  2. Find the total time, between 1100 hours and 2300 hours on this particular day, for which the depth of water in the harbour is less than 5.5 m .
    (Total 14 marks)
Edexcel M3 2005 June Q7
14 marks Standard +0.8
7. A particle \(P\) of mass \(\frac { 1 } { 3 } \mathrm {~kg}\) moves along the positive \(x\)-axis under the action of a single force. The force is directed towards the origin \(O\) and has magnitude \(\frac { k } { ( x + 1 ) ^ { 2 } } \mathrm {~N}\), where \(O P = x\) metres and \(k\) is a constant. Initially \(P\) is moving away from \(O\). At \(x = 1\) the speed of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and at \(x = 8\) the speed of \(P\) is \(\sqrt { } 2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the value of \(k\).
  2. Find the distance of \(P\) from \(O\) when \(P\) first comes to instantaneous rest.
    (Total 14 marks)
Edexcel M3 2006 June Q1
5 marks Standard +0.8
  1. A uniform solid is formed by rotating the region enclosed between the curve with equation \(y = \sqrt { } x\), the \(x\)-axis and the line \(x = 4\), through one complete revolution about the \(x\)-axis. Find the distance of the centre of mass of the solid from the origin \(O\).
    (5)
  2. A bowl consists of a uniform solid metal hemisphere, of radius \(a\) and centre \(O\), from which is removed the solid hemisphere of radius \(\frac { 1 } { 2 } a\) with the same centre \(O\).
    1. Show that the distance of the centre of mass of the bowl from \(O\) is \(\frac { 45 } { 112 } a\).
    The bowl is fixed with its plane face uppermost and horizontal. It is now filled with liquid. The mass of the bowl is \(M\) and the mass of the liquid is \(k M\), where \(k\) is a constant. Given that the distance of the centre of mass of the bowl and liquid together from \(O\) is \(\frac { 17 } { 48 } a\),
  3. find the value of \(k\).
    (5)
Edexcel M3 2006 June Q3
11 marks Standard +0.8
3. A particle \(P\) of mass 0.2 kg oscillates with simple harmonic motion between the points \(A\) and \(B\), coming to rest at both points. The distance \(A B\) is 0.2 m , and \(P\) completes 5 oscillations every second.
  1. Find, to 3 significant figures, the maximum resultant force exerted on \(P\).
    (6) When the particle is at \(A\), it is struck a blow in the direction \(B A\). The particle now oscillates with simple harmonic motion with the same frequency as previously but twice the amplitude.
  2. Find, to 3 significant figures, the speed of the particle immediately after it has been struck.
    (5)
Edexcel M3 2006 June Q4
11 marks Standard +0.8
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{e5998843-0bec-4b64-bba4-0e2212e4929b-3_700_743_354_740}
\end{figure} A hollow cone, of base radius \(3 a\) and height \(4 a\), is fixed with its axis vertical and vertex \(V\) downwards, as shown in Figure 1. A particle moves in a horizontal circle with centre \(C\), on the smooth inner surface of the cone with constant angular speed \(\sqrt { \frac { 8 g } { 9 a } }\). Find the height of \(C\) above \(V\).
Edexcel M3 2006 June Q5
12 marks Challenging +1.2
5. Two light elastic strings each have natural length 0.75 m and modulus of elasticity 49 N . A particle \(P\) of mass 2 kg is attached to one end of each string. The other ends of the strings are attached to fixed points \(A\) and \(B\), where \(A B\) is horizontal and \(A B = 1.5 \mathrm {~m}\). \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{e5998843-0bec-4b64-bba4-0e2212e4929b-3_205_668_1813_717}
\end{figure} The particle is held at the mid-point of \(A B\). The particle is released from rest, as shown in Figure 2.
  1. Find the speed of \(P\) when it has fallen a distance of 1 m . Given instead that \(P\) hangs in equilibrium vertically below the mid-point of \(A B\), with \(\angle A P B = 2 \alpha\),
  2. show that \(\tan \alpha + 5 \sin \alpha = 5\).
    (6)
Edexcel M3 2006 June Q6
13 marks Standard +0.3
6. A particle moving in a straight line starts from rest at the point \(O\) at time \(t = 0\). At time \(t\) seconds, the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the particle is given by $$\begin{aligned} & v = 3 t ( t - 4 ) , \quad 0 \leq t \leq 5 \\ & v = 75 t ^ { - 1 } , \quad 5 \leq t \leq 10 \end{aligned}$$
  1. Sketch a velocity-time graph for the particle for \(0 \leq t \leq 10\).
  2. Find the set of values of \(t\) for which the acceleration of the particle is positive.
  3. Show that the total distance travelled by the particle in the interval \(0 \leq t \leq 5\) is 39 m .
  4. Find, to 3 significant figures, the value of \(t\) at which the particle returns to \(O\).
Edexcel M3 2006 June Q7
13 marks Challenging +1.2
7. One end of a light inextensible string of length \(l\) is attached to a particle \(P\) of mass \(m\). The other end is attached to a fixed point \(A\). The particle is hanging freely at rest with the string vertical when it is projected horizontally with speed \(\sqrt { \frac { 5 g l } { 2 } }\).
  1. Find the speed of \(P\) when the string is horizontal. When the string is horizontal it comes into contact with a small smooth fixed peg which is at the point \(B\), where \(A B\) is horizontal, and \(A B < l\). Given that the particle then describes a complete semicircle with centre \(B\),
  2. find the least possible value of the length \(A B\).
Edexcel M3 2007 June Q1
9 marks Standard +0.8
  1. The rudder on a ship is modelled as a uniform plane lamina having the same shape as the region \(R\) which is enclosed between the curve with equation \(y = 2 x - x ^ { 2 }\) and the \(x\)-axis.
    1. Show that the area of \(R\) is \(\frac { 4 } { 3 }\).
    2. Find the coordinates of the centre of mass of the lamina.
    3. An open container \(C\) is modelled as a thin uniform hollow cylinder of radius \(h\) and height \(h\) with a base but no lid. The centre of the base is \(O\).
    4. Show that the distance of the centre of mass of \(C\) from \(O\) is \(\frac { 1 } { 3 } h\).
    The container is filled with uniform liquid. Given that the mass of the container is \(M\) and the mass of the liquid is \(M\),
  2. find the distance of the centre of mass of the filled container from \(O\).
Edexcel M3 2007 June Q3
9 marks Standard +0.3
3. A spacecraft \(S\) of mass \(m\) moves in a straight line towards the centre of the earth. The earth is modelled as a fixed sphere of radius \(R\). When \(S\) is at a distance \(x\) from the centre of the earth, the force exerted by the earth on \(S\) is directed towards the centre of the earth and has magnitude \(\frac { k } { x ^ { 2 } }\), where \(k\) is a constant.
  1. Show that \(k = m g R ^ { 2 }\). Given that \(S\) starts from rest when its distance from the centre of the earth is \(2 R\), and that air resistance can be ignored,
  2. find the speed of \(S\) as it crashes into the surface of the earth.
Edexcel M3 2007 June Q4
9 marks Moderate -0.8
4. A light inextensible string of length \(l\) has one end attached to a fixed point \(A\). The other end is attached to a particle \(P\) of mass \(m\). The particle moves with constant speed \(v\) in a horizontal circle with the string taut. The centre of the circle is vertically below \(A\) and the radius of the circle is \(r\). Show that $$g r ^ { 2 } = v ^ { 2 } \sqrt { } \left( l ^ { 2 } - r ^ { 2 } \right) .$$
Edexcel M3 2007 June Q5
11 marks Standard +0.3
  1. A particle \(P\) moves on the \(x\)-axis with simple harmonic motion about the origin \(O\) as centre. When \(P\) is a distance 0.04 m from \(O\), its speed is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the magnitude of its acceleration is \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    1. Find the period of the motion.
    The amplitude of the motion is \(a\) metres. Find
  2. the value of \(a\),
  3. the total time, within one complete oscillation, for which the distance \(O P\) is greater than \(\frac { 1 } { 2 } a\) metres.