Questions — Edexcel M3 (510 questions)

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Edexcel M3 2015 June Q5
6 marks Challenging +1.2
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00388805-5d60-4327-a10e-c0df74a0cb75-09_403_790_210_577} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a uniform solid spindle which is made by joining together the circular faces of two right circular cones. The common circular face has radius \(r\) and centre \(O\). The smaller cone has height \(h\) and the larger cone has height \(k h\). The point \(A\) lies on the circumference of the common circular face. The spindle is suspended from \(A\) and hangs freely in equilibrium with \(A O\) at an angle of \(30 ^ { \circ }\) to the vertical. Show that \(k = \frac { 4 r } { h \sqrt { 3 } } + 1\)
Edexcel M3 2015 June Q6
15 marks Challenging +1.2
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00388805-5d60-4327-a10e-c0df74a0cb75-11_186_1042_223_452} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Two points \(A\) and \(B\) are 6 m apart on a smooth horizontal floor. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic spring, of natural length 2.5 m and modulus of elasticity 20 N . The other end of the spring is attached to \(A\). A second light elastic spring, of natural length 1.5 m and modulus of elasticity 18 N , has one end attached to \(P\) and the other end attached to \(B\), as shown in Figure 3. Initially \(P\) rests in equilibrium at the point \(O\), where \(A O B\) is a straight line.
  1. Find the length of \(A O\). The particle \(P\) now receives an impulse of magnitude 6 N s acting in the direction \(O B\) and \(P\) starts to move towards \(B\).
  2. Show that \(P\) moves with simple harmonic motion about \(O\).
  3. Find the amplitude of the motion.
  4. Find the time taken by \(P\) to travel 1.2 m from \(O\).
Edexcel M3 2015 June Q7
15 marks Standard +0.8
  1. A solid smooth sphere, with centre \(O\) and radius \(r\), is fixed to a point \(A\) on a horizontal floor. A particle \(P\) is placed on the surface of the sphere at the point \(B\), where \(B\) is vertically above \(A\). The particle is projected horizontally from \(B\) with speed \(\frac { \sqrt { g r } } { 2 }\) and starts to move on the surface of the sphere. When \(O P\) makes an angle \(\theta\) with the upward vertical and \(P\) remains in contact with the sphere, the speed of \(P\) is \(v\).
    1. Show that \(v ^ { 2 } = \frac { g r } { 4 } ( 9 - 8 \cos \theta )\).
    The particle leaves the surface of the sphere when \(\theta = \alpha\).
  2. Find the value of \(\cos \alpha\). After leaving the surface of the sphere, \(P\) moves freely under gravity and hits the floor at the point \(C\). Given that \(r = 0.5 \mathrm {~m}\),
  3. find, to 2 significant figures, the distance \(A C\).
Edexcel M3 2016 June Q1
9 marks Standard +0.8
  1. A particle \(P\) of mass 0.5 kg is moving along the positive \(x\)-axis under the action of a resultant force. The force acts along the \(x\)-axis. At time \(t\) seconds, \(P\) is \(x\) metres from the origin \(O\) and is moving away from \(O\) in the positive \(x\) direction with speed \(\frac { 12 } { x + 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
    1. Find the magnitude of the force acting on \(P\) when \(x = 3\)
    Given that \(x = 4\) when \(t = 2\)
  2. find the value of \(t\) when \(x = 10\)
Edexcel M3 2016 June Q2
6 marks Standard +0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c1c51ff-6ae8-402d-b303-b656d26e4230-03_430_739_324_607} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a uniform triangular lamina \(A B C\) in which \(A B = 6 \mathrm {~cm} , B C = 9 \mathrm {~cm}\) and angle \(A B C = 90 ^ { \circ }\). The centre of mass of the lamina is \(G\). Use algebraic integration to find the distance of \(G\) from \(A B\).
(6)
Edexcel M3 2016 June Q3
9 marks Standard +0.3
3. One end of a light elastic string, of natural length 1.5 m and modulus of elasticity 14.7 N ,
3. One end of a light elastic string, of natural length 1.5 m and modulus of elasticity 14.7 N , is attached to a fixed point \(O\) on a ceiling. A particle \(P\) of mass 0.6 kg is attached to the free end of the string. The particle is held at \(O\) and released from rest. The particle comes to instantaneous rest for the first time at the point \(A\). Find
  1. the distance \(O A\),
  2. the magnitude of the instantaneous acceleration of \(P\) at \(A\).
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4c1c51ff-6ae8-402d-b303-b656d26e4230-05_620_956_118_500} \captionsetup{labelformat=empty} \caption{igure 2}
    \end{figure} A uniform solid \(S\) consists of two right circular cones of base radius \(r\). The smaller cone has height \(2 h\) and the centre of the plane face of this cone is \(O\). The larger cone has height \(k h\) where \(k > 2\). The two cones are joined so that their plane faces coincide, as shown in Figure 2.
    1. Show that the distance of the centre of mass of \(S\) from \(O\) is $$\frac { h } { 4 } ( k - 2 )$$ The point \(A\) lies on the circumference of the base of one of the cones. The solid is suspended by a string attached at \(A\) and hangs freely in equilibrium. Given that \(r = 3 h\) and \(k = 6\)
    2. find the size of the angle between \(A O\) and the vertical.
Edexcel M3 2016 June Q4
8 marks Standard +0.3
4. A uniform solid \(S\) consists of two right circular cones of base rate
has height \(2 h\) and the centre of the plane face of this cone is \(O\). T \(k h\) where \(k > 2\). The two cones are joined so that their plane fac
Figure 2 .
  1. Show that the distance of the centre of mass of \(S\) from \(O\) is \(\frac { h } { 4 } ( k - 2 )\)
Edexcel M3 2016 June Q5
13 marks Standard +0.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c1c51ff-6ae8-402d-b303-b656d26e4230-07_842_449_248_826} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A particle \(P\) of mass \(m\) is attached to the ends of two light inextensible strings. The other ends of the strings are attached to fixed points \(A\) and \(B\), where \(B\) is vertically below \(A\) and \(A B = l\). The particle is moving with constant angular speed \(\omega\) in a horizontal circle. Both strings are taut and inclined at \(30 ^ { \circ }\) to \(A B\), as shown in Figure 3.
    1. Show that the tension in \(A P\) is \(\frac { m \sqrt { 3 } } { 6 } \left( 2 g + l \omega ^ { 2 } \right)\)
    2. Find the tension in \(B P\).
  2. Show that the time taken by \(P\) to complete one revolution is less than \(\pi \sqrt { \frac { 2 l } { g } }\)
Edexcel M3 2016 June Q6
13 marks Challenging +1.2
6. One end of a light inextensible string of length \(l\) is attached to a particle \(P\) of mass \(2 m\). The other end of the string is attached to a fixed point \(A\). The particle is hanging freely at rest with the string vertical. The particle is then projected horizontally with speed \(\sqrt { \frac { 7 g l } { 2 } }\) (a) Find the speed of \(P\) at the instant when the string is horizontal.
(4) When the string is horizontal and \(P\) is moving upwards, the string comes into contact with a small smooth peg which is fixed at the point \(B\), where \(A B\) is horizontal and \(A B < l\). The particle then describes a complete semicircle with centre \(B\).
(b) Show that \(A B \geqslant \frac { 1 } { 2 } l\)
Edexcel M3 2016 June Q7
17 marks Challenging +1.2
7. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic spring, of natural length 1.2 m and modulus of elasticity 15 N . The other end of the spring is attached to a fixed point \(A\) on a smooth horizontal table. The particle is placed on the table at the point \(B\) where \(A B = 1.2 \mathrm {~m}\). The particle is pulled away from \(B\) to the point \(C\), where \(A B C\) is a straight line and \(B C = 0.8 \mathrm {~m}\), and is then released from rest.
    1. Show that \(P\) moves with simple harmonic motion with centre \(B\).
    2. Find the period of this motion.
  2. Find the speed of \(P\) when it reaches \(B\). The point \(D\) is the midpoint of \(A B\).
  3. Find the time taken for \(P\) to move directly from \(C\) to \(D\). When \(P\) first comes to instantaneous rest a particle \(Q\) of mass 0.3 kg is placed at \(B\). When \(P\) reaches \(B\) again, \(P\) strikes and adheres to \(Q\) to form a single particle \(R\).
  4. Show that \(R\) also moves with simple harmonic motion.
  5. Find the amplitude of this motion.
Edexcel M3 2017 June Q1
7 marks Standard +0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{698b44b5-801c-45ec-b9de-021e44487edb-02_672_732_226_589} \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 \(y = 4 - x ^ { 2 }\), the positive \(x\)-axis and the positive \(y\)-axis, as shown shaded in Figure 1. Use algebraic integration to find the \(x\) coordinate of the centre of mass of the lamina.
Edexcel M3 2017 June Q2
8 marks Moderate -0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{698b44b5-801c-45ec-b9de-021e44487edb-04_723_636_219_733} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(A\). The particle moves in a horizontal circle with constant angular speed \(\sqrt { 58.8 } \mathrm { rad } \mathrm { s } ^ { - 1 }\). The centre \(O\) of the circle is vertically below \(A\) and the string makes a constant angle \(\theta ^ { \circ }\) with the downward vertical, as shown in Figure 2. Given that the tension in the string is 1.2 mg , find
  1. the value of \(\theta\)
  2. the length of the string.
Edexcel M3 2017 June Q3
9 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 plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The particle is released from rest and slides down the plane against a force of magnitude \(\frac { 1 } { 2 } m x ^ { 2 }\) newtons acting towards \(O\), where \(x\) metres is the distance of \(P\) from \(O\).
    1. Find the speed of \(P\) when \(x = 3\)
    2. Find the distance \(P\) has moved when it first comes to instantaneous rest.
Edexcel M3 2017 June Q4
9 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{698b44b5-801c-45ec-b9de-021e44487edb-10_570_410_237_826} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A thin uniform right hollow cylinder, of radius \(a\) and height \(4 a\), has a base but no top. A thin uniform hemispherical shell, also of radius \(a\), is made of the same material as the cylinder. The hemispherical shell is attached to the open 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. Find the distance from \(O\) to the centre of mass of \(C\). The container is placed with its circular base on a plane which is inclined at \(\theta ^ { \circ }\) 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 \(\theta\).
Edexcel M3 2017 June Q5
12 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{698b44b5-801c-45ec-b9de-021e44487edb-14_565_696_219_721} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A hollow cylinder is fixed with its axis horizontal. A particle \(P\) moves in a vertical circle, with centre \(O\) and radius \(a\), on the smooth inner surface of the cylinder. The particle moves in a vertical plane which is perpendicular to the axis of the cylinder. The particle is projected vertically downwards with speed \(\sqrt { 7 a g }\) from the point \(A\), where \(O A\) is horizontal and \(O A = a\). When angle \(A O P = \theta\), the speed of \(P\) is \(v\), as shown in Figure 4.
  1. Show that \(v ^ { 2 } = a g ( 7 + 2 \sin \theta )\)
  2. Verify that \(P\) will move in a complete circle.
  3. Find the maximum value of \(v\).
Edexcel M3 2017 June Q6
13 marks Standard +0.8
  1. The ends of a light elastic string, of natural length 0.4 m and modulus of elasticity \(\lambda\) newtons, are attached to two fixed points \(A\) and \(B\) which are 0.6 m apart on a smooth horizontal table. The tension in the string is 8 N .
    1. Show that \(\lambda = 16\)
    A particle \(P\) is attached to the midpoint of the string. The particle \(P\) is now pulled horizontally in a direction perpendicular to \(A B\) to a point 0.4 m from the midpoint of \(A B\). The particle is held at rest by a horizontal force of magnitude \(F\) newtons acting in a direction perpendicular to \(A B\), as shown in Figure 5 below. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{698b44b5-801c-45ec-b9de-021e44487edb-18_623_796_792_573} \captionsetup{labelformat=empty} \caption{Figure 5}
    \end{figure}
  2. Find the value of \(F\). The particle is released from rest. Given that the mass of \(P\) is 0.3 kg ,
  3. find the speed of \(P\) as it crosses the line \(A B\).
Edexcel M3 2017 June Q7
17 marks Challenging +1.2
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{698b44b5-801c-45ec-b9de-021e44487edb-24_173_968_223_488} \captionsetup{labelformat=empty} \caption{Figure 6}
\end{figure} The fixed points \(A\) and \(B\) are 4 m apart on a smooth horizontal floor. One end of a light elastic string, of natural length 1.8 m and modulus of elasticity 45 N , is attached to a particle \(P\) and the other end is attached to \(A\). One end of another light elastic string, of natural length 1.2 m and modulus of elasticity 20 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 6.
  1. Show that \(A O = 2.2 \mathrm {~m}\). The point \(C\) lies on the straight line \(A O B\) with \(A C = 2.7 \mathrm {~m}\). The mass of \(P\) is 0.6 kg . The particle \(P\) is held at \(C\) and then released from rest.
  2. Show that, while both strings are taut, \(P\) moves with simple harmonic motion with centre \(O\). The point \(D\) lies on the straight line \(A O B\) with \(A D = 1.8 \mathrm {~m}\). When \(P\) reaches \(D\) the string \(P B\) breaks.
  3. Find the time taken by \(P\) to move directly from \(C\) to \(A\).
Edexcel M3 2018 June Q1
5 marks Standard +0.3
  1. A rough disc is rotating in a horizontal plane with constant angular speed \(\omega\) about a vertical axis through the centre of the disc. A particle \(P\) is placed on the disc at a distance \(r\) from the axis. The coefficient of friction between \(P\) and the disc is \(\mu\).
Given that \(P\) does not slip on the disc, show that $$\omega \leqslant \sqrt { \frac { \mu g } { r } }$$
Edexcel M3 2018 June Q2
11 marks Standard +0.3
2. A light elastic string has natural length 1.2 m and modulus of elasticity \(\lambda\) newtons. One end of the string is attached to a fixed point \(O\). A particle of mass 0.5 kg is attached to the other end of the string. The particle is moving with constant angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle with the string stretched. The circle has radius 0.9 m and its centre is vertically below \(O\). The string is inclined at \(60 ^ { \circ }\) to the horizontal. Find
  1. the value of \(\lambda\),
  2. the value of \(\omega\).
Edexcel M3 2018 June Q3
7 marks Challenging +1.2
3. A particle \(P\) of mass \(m\) moves in a straight line away from the centre of the Earth. The Earth is modelled as a sphere of radius \(R\). When \(P\) is at a distance \(x , x \geqslant R\), from the centre of the Earth, the force exerted by the Earth on \(P\) is directed towards the centre of the Earth and has magnitude \(\frac { m g R ^ { 2 } } { x ^ { 2 } }\). When \(P\) is at a distance \(2 R\) from the surface of the Earth, the speed of \(P\) is \(\sqrt { \frac { g R } { 3 } }\). Assuming that air resistance can be ignored, find the distance of \(P\) from the surface of the Earth when the speed of \(P\) is \(2 \sqrt { \frac { g R } { 3 } }\).
Edexcel M3 2018 June Q4
7 marks Challenging +1.2
4. One end of a light elastic string, of modulus of elasticity \(2 m g\) and natural length \(l\), is fixed to a point \(O\) on a rough plane. The plane is inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\). The other end of the string is attached to a particle \(P\) of mass \(m\) which is held at rest on the plane at the point \(O\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\). The particle is released from rest and slides down the plane, coming to instantaneous rest at the point \(A\), where \(O A = k l\). Given that \(k > 1\), find, to 3 significant figures, the value of \(k\).
\includegraphics[max width=\textwidth, alt={}, center]{2cf74ba3-857a-4ce9-ab5b-e6203b279161-13_152_72_118_127} \includegraphics[max width=\textwidth, alt={}]{2cf74ba3-857a-4ce9-ab5b-e6203b279161-13_90_1620_123_203} □ ⟶ \(\_\_\_\_\) T
Edexcel M3 2018 June Q5
13 marks Challenging +1.2
5. A uniform solid hemisphere has radius \(r\). The centre of the plane face of the hemisphere is \(O\).
  1. Use algebraic integration to show that the distance from \(O\) to the centre of mass of the hemisphere is \(\frac { 3 } { 8 } r\).
    [0pt] [You may assume that the volume of a sphere of radius \(r\) is \(\frac { 4 } { 3 } \pi r ^ { 3 }\) ] \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2cf74ba3-857a-4ce9-ab5b-e6203b279161-14_378_602_740_671} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A solid \(S\) is formed by joining a uniform solid hemisphere of radius \(a\) to a uniform solid hemisphere of radius \(\frac { 1 } { 2 } a\). The plane faces of the hemispheres are joined together so that their centres coincide at \(O\), as shown in Figure 1. The mass per unit volume of the smaller hemisphere is \(k\) times the mass per unit volume of the larger hemisphere.
  2. Find the distance from \(O\) to the centre of mass of \(S\). When \(S\) is placed on a horizontal plane with any point on the curved surface of the larger hemisphere in contact with the plane, \(S\) remains in equilibrium.
  3. Find the value of \(k\).
Edexcel M3 2018 June Q6
15 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2cf74ba3-857a-4ce9-ab5b-e6203b279161-18_481_606_246_667} \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 \(a\). The other end of the string is attached to a fixed point \(O\). The particle is held at the point \(A\), where \(O A = a\) and \(O A\) is horizontal. The particle is projected vertically upwards with speed \(u\), as shown in Figure 2. When the string makes an angle \(\theta\) with the horizontal through \(O\) and the string is still taut, the tension in the string is \(T\).
  1. Show that \(T = \frac { m } { a } \left( u ^ { 2 } - 3 a g \sin \theta \right)\) The particle moves in complete circles.
  2. Find, in terms of \(a\) and \(g\), the minimum value of \(u\). Given that the least tension in the string is \(S\) and the greatest tension in the string is \(4 S\),
  3. find, in terms of \(a\) and \(g\), an expression for \(u\).
Edexcel M3 2018 June Q7
17 marks Challenging +1.2
7. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic string. The string has natural length \(l\) metres and modulus of elasticity 29.4 N . The other end of the string is attached to a fixed point \(A\). The particle hangs freely in equilibrium at the point \(B\), where \(B\) is vertically below \(A\) and \(A B = 1.4 \mathrm {~m}\).
  1. Show that \(l = 1.2\) The point \(C\) is vertically below \(A\) and \(A C = 1.8 \mathrm {~m}\). The particle is pulled down to \(C\) and released from rest.
  2. Show that, while the string is taut, \(P\) moves with simple harmonic motion.
  3. Calculate the speed of \(P\) at the instant when the string first becomes slack. The particle first comes to instantaneous rest at the point \(D\).
  4. Find the time taken by \(P\) to return directly from \(D\) to \(C\).
Edexcel M3 Q1
6 marks Standard +0.3
  1. A particle \(P\) moves on the positive \(x\)-axis. When the displacement of \(P\) from \(O\) is \(x\) metres, its acceleration is \(( 6 - 4 x ) \mathrm { m } \mathrm { s } ^ { - 2 }\), measured in the direction of \(x\) increasing. Initially \(P\) is at \(O\) and the velocity of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction \(O x\).
Find the distance of \(P\) from \(O\) when \(P\) is instantaneously at rest.