Questions M3 (745 questions)

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Edexcel M3 2022 June Q3
  1. In this question you must show all stages of your working.
\section*{Solutions relying entirely on calculator technology are not acceptable.} A particle \(P\) is moving along a straight line. At time \(t\) seconds, \(P\) is a distance \(x\) metres from a fixed point \(O\) on the line and is moving away from \(O\) with speed \(\frac { 50 } { 2 x + 3 } \mathrm {~ms} ^ { - 1 }\)
  1. Find the deceleration of \(P\) when \(x = 12\) Given that \(x = 4\) when \(t = 1\)
  2. find the value of \(t\) when \(x = 12\)
Edexcel M3 2022 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2e837bb9-4ada-4f0f-8b21-2730611335f2-12_357_737_260_664} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} One end of a light elastic string, of natural length \(l\) and modulus of elasticity \(\lambda\), is fixed to a point \(A\) on a smooth plane. The plane is inclined at \(30 ^ { \circ }\) to the horizontal. A small ball \(B\) of mass \(m\) is attached to the other end of the elastic string. Initially, \(B\) is held at rest at the point \(C\) on the plane with the elastic string lying along a line of greatest slope of the plane. The point \(C\) is below \(A\) and \(A C = l\), as shown in Figure 2 . The ball is released and comes to instantaneous rest at a point \(D\) on the plane.
The points \(A , C\) and \(D\) all lie along a line of greatest slope of the plane and \(A D = \frac { 5 l } { 4 }\)
The ball is modelled as a particle and air resistance is modelled as being negligible.
Using the model,
  1. show that \(\lambda = 4 \mathrm { mg }\)
  2. find, in terms of \(g\) and \(l\), the greatest speed of \(B\) as it moves from \(C\) to \(D\)
Edexcel M3 2022 June Q5
  1. (a) 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 \(\frac { 4 } { 3 } \pi r ^ { 3 }\) ]
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2e837bb9-4ada-4f0f-8b21-2730611335f2-16_355_574_571_749} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A uniform solid hemisphere of radius \(r\) is joined to a uniform solid right circular cone made of the same material to form a toy. The cone has base radius \(r\) and height \(k r\). The centre of the base of the cone is \(O\). The plane face of the cone coincides with the plane face of the hemisphere, as shown in Figure 3. The toy can rest in equilibrium on a horizontal plane with any point of the curved surface of the hemisphere in contact with the plane.
(b) Find the exact value of \(k\)
Edexcel M3 2022 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2e837bb9-4ada-4f0f-8b21-2730611335f2-20_499_748_244_653} \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 \(a\). 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, as shown in Figure 4. The particle is projected vertically downwards with speed \(\sqrt { \frac { 9 a g } { 5 } }\)
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 \(S\).
  1. Show that \(S = \frac { 3 } { 5 } m g ( 5 \cos \theta + 3 )\) At the instant when the string becomes slack, the speed of \(P\) is \(v\)
  2. Show that \(v = \sqrt { \frac { 3 a g } { 5 } }\)
  3. Find the maximum height of \(P\) above the horizontal level of \(O\)
Edexcel M3 2022 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2e837bb9-4ada-4f0f-8b21-2730611335f2-24_165_1392_258_338} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows two fixed points, \(A\) and \(B\), which are 5 m apart on a smooth horizontal floor. A particle \(P\) of mass 1.25 kg is attached to one end of a light elastic string, of natural length 2 m and modulus of elasticity 20 N . The other end of the string is attached to \(A\) A second light elastic string, of natural length 1.2 m and modulus of elasticity \(\lambda\) newtons, has one end attached to \(P\) and the other end attached to \(B\) Initially \(P\) rests in equilibrium at the point \(O\), where \(A O = 3 \mathrm {~m}\)
  1. Show that \(\lambda = 15\) The particle is now projected along the floor towards \(B\)
    At time \(t\) seconds, \(P\) is a displacement \(x\) metres from \(O\) in the direction \(O B\)
  2. Show that, while both strings are taut, \(P\) moves with simple harmonic motion where \(\ddot { x } = - 18 x\) The initial speed of \(P\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  3. Find the speed of \(P\) at the instant when the string \(P B\) becomes slack. Both strings are taut for \(T\) seconds during one complete oscillation.
  4. Find the value of \(T\)
Edexcel M3 2023 June Q1
  1. In this question you must show all stages in your working.
Solutions relying on calculator technology are not acceptable. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dceb2432-117c-40fe-bf3d-782beeb42e41-02_579_1059_386_502} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The finite region \(R\), shown shaded in Figure 1, is bounded by the \(x\)-axis, the line with equation \(x = 3\), the curve with equation \(y = \sqrt { ( x + 1 ) }\) and the \(y\)-axis.
Find the \(\boldsymbol { y }\) coordinate of the centre of mass of a uniform lamina in the shape of \(R\).
Edexcel M3 2023 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dceb2432-117c-40fe-bf3d-782beeb42e41-04_424_510_246_767} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A light elastic string \(A B\) has modulus of elasticity \(2 m g\) and natural length \(k a\), where \(k\) is a constant.
The end \(A\) of the elastic string is attached to a fixed point. The other end \(B\) is attached to a particle of mass \(m\). The particle is held in equilibrium, with the elastic string taut, by a force that acts in a direction that is perpendicular to the string. The line of action of the force and the elastic string lie in the same vertical plane. The string makes an angle \(\theta\) with the downward vertical at \(A\), as shown in Figure 2. Given that the length \(A B = \frac { 21 } { 10 } a\) and \(\tan \theta = \frac { 3 } { 4 }\), find the value of \(k\).
Edexcel M3 2023 June Q3
  1. A uniform solid right circular cone \(C\) has base radius \(r\), height \(H\) and vertex \(V\). A uniform solid \(S\), shown in Figure 3, is formed by removing from \(C\) a uniform solid right circular cone of height \(h ( h < H )\) that has the same base and axis of symmetry as \(C\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dceb2432-117c-40fe-bf3d-782beeb42e41-08_725_1152_422_461} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure}
  1. Show that the distance of the centre of mass of \(S\) from \(V\) is $$\frac { 1 } { 4 } ( 3 H - h )$$ The solid \(S\) is suspended by two vertical light strings. The first string is attached to \(S\) at \(V\) and the second string is attached to \(S\) at a point on the circumference of the circular base of \(S\).
    The solid \(S\) hangs freely in equilibrium with its axis of symmetry horizontal.
    The tension in the first string is \(T _ { 1 }\) and the tension in the second string is \(T _ { 2 }\)
  2. Find \(\frac { T _ { 1 } } { T _ { 2 } }\), giving your answer in terms of \(H\) and \(h\), in its simplest form.
Edexcel M3 2023 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dceb2432-117c-40fe-bf3d-782beeb42e41-12_360_1004_246_534} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A car is travelling round a circular track. The track is banked at an angle \(\alpha\) to the horizontal, as shown in Figure 4. The car and driver are modelled as a particle.
The car moves round the track with constant speed in a horizontal circle of radius \(r\).
When the car is moving with speed \(\frac { 1 } { 2 } \sqrt { g r }\) round the circle, there is no sideways friction between the tyres of the car and the track.
  1. Show that \(\tan \alpha = \frac { 1 } { 4 }\) The sideways friction between the tyres of the car and the track has coefficient of friction \(\mu\), where \(\mu < 4\) The maximum speed at which the car can move round the circle without slipping sideways is \(V\).
  2. Find \(V\) in terms of \(\mu , r\) and \(g\).
Edexcel M3 2023 June Q5
  1. The centre of the Earth is the point \(O\) and the Earth is modelled as a fixed sphere of radius \(R\).
    At time \(t = 0\), a particle \(P\) is projected vertically upwards with speed \(U\) from a point \(A\) on the surface of the Earth.
At time \(t\) seconds, where \(t \geqslant 0\)
  • \(\quad P\) is a distance \(x\) from \(O\)
  • \(P\) is moving with speed \(v\)
  • \(P\) has acceleration of magnitude \(\frac { g R ^ { 2 } } { x ^ { 2 } }\) directed towards \(O\)
Air resistance is modelled as being negligible.
  1. Show that \(v ^ { 2 } = \frac { 2 g R ^ { 2 } } { x } + U ^ { 2 } - 2 g R\) Particle \(P\) is first moving with speed \(\frac { 1 } { 2 } \sqrt { g R }\) at the point \(B\).
  2. Given that \(U = \sqrt { g R }\) find, in terms of \(R\), the distance \(A B\).
  3. Find, in terms of \(g\) and \(R\), the smallest value of \(U\) that would ensure that \(P\) never comes to rest, explaining your reasoning.
Edexcel M3 2023 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{dceb2432-117c-40fe-bf3d-782beeb42e41-20_134_653_243_707} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held at rest with the string taut and horizontal and is then projected vertically downwards with speed \(u\), as shown in Figure 5. Air resistance is modelled as being negligible.
At the instant when the string has turned through an angle \(\theta\) and the string is taut, the tension in the string is \(T\).
  1. Show that \(T = \frac { m u ^ { 2 } } { a } + 3 m g \sin \theta\) Given that \(u = 2 \sqrt { \frac { 3 a g } { 5 } }\)
  2. find, in terms of \(a\) and \(g\), the speed of \(P\) at the instant when the string goes slack.
  3. Hence find, in terms of \(a\), the maximum height of \(P\) above \(O\) in the subsequent motion.
Edexcel M3 2023 June Q7
  1. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(l\). The other end of the string is attached to a fixed point on a ceiling. The particle \(P\) hangs in equilibrium at a distance \(D\) below the ceiling.
The particle \(P\) is now pulled vertically downwards until it is a distance \(3 l\) below the ceiling and released from rest. Given that \(P\) comes to instantaneous rest just before it reaches the ceiling,
  1. show that \(D = \frac { 5 l } { 3 }\)
  2. Show that, while the elastic string is stretched, \(P\) moves with simple harmonic motion, with period \(2 \pi \sqrt { \frac { 2 l } { 3 g } }\)
  3. Find, in terms of \(g\) and \(l\), the exact time from the instant when \(P\) is released to the instant when the elastic string first goes slack.
Edexcel M3 2024 June Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{176ae8f8-7de9-4993-825a-6067614526ae-02_314_677_296_696} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A light elastic string \(A B\) has natural length \(4 a\) and modulus of elasticity \(\lambda\). The end \(A\) is attached to a fixed point and the end \(B\) is attached to a particle of mass \(m\). The particle is held in equilibrium, with the string stretched, by a horizontal force of magnitude \(k m g\).
The line of action of the horizontal force lies in the vertical plane containing the elastic string.
The string \(A B\) makes an angle \(\alpha\) with the vertical, where \(\tan \alpha = \frac { 4 } { 3 }\)
With the particle in this position, \(A B = 5 a\), as shown in Figure 1.
  1. Show that \(\lambda = \frac { 20 m g } { 3 }\)
  2. Find the value of \(k\).
Edexcel M3 2024 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{176ae8f8-7de9-4993-825a-6067614526ae-04_351_563_296_751} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A thin hemispherical shell, with centre \(O\) and radius \(a\), is fixed with its open end uppermost and horizontal. A particle \(P\) of mass \(m\) moves in a horizontal circle on the smooth inner surface of the shell. The vertical distance of \(P\) below the level of \(O\) is \(d\), as shown in Figure 2.
  1. Find, in terms of \(m , g , d\) and \(a\), the magnitude of the force exerted on \(P\) by the inner surface of the hemisphere. The particle moves with constant speed \(v\).
  2. Find \(v\) in terms of \(g , a\) and \(d\).
Edexcel M3 2024 June Q3
  1. A particle \(P\) is moving along the \(x\)-axis.
At time \(t\) seconds, where \(t \geqslant 0\), the displacement of \(P\) from the origin \(O\) is \(x\) metres and \(P\) is moving with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\) direction. The acceleration of \(P\) is \(\frac { 3 \sqrt { x + 1 } } { 4 } \mathrm {~ms} ^ { - 2 }\) in the positive \(x\) direction.
When \(t = 0 , x = 15\) and \(v = 8\)
  1. Show that \(v = ( x + 1 ) ^ { \frac { 3 } { 4 } }\)
  2. Find \(t\) in terms of \(v\).
Edexcel M3 2024 June Q4
  1. In a harbour, the water level rises and falls with the tides with simple harmonic motion.
On a particular day, the depths of water in the harbour at low and high tide are 4 m and 10 m respectively. Low tide occurs at 12:00 and high tide occurs at 18:20
  1. Find, in \(\mathrm { mh } ^ { - 1 }\), the speed at which the water level is rising on this particular day at 13:35 A ship can only safely enter the harbour when the depth of water is at least 8.5 m .
  2. Find the earliest time after 12:00 on this particular day at which it is safe for the ship to enter the harbour, giving your answer to the nearest minute.
Edexcel M3 2024 June Q5
  1. A uniform right solid circular cone \(C\) has radius \(r\) and height \(4 r\).
    1. Show, using algebraic integration, that the distance of the centre of mass of \(C\) from its vertex is \(3 r\).
      [0pt] [You may assume that the volume of \(C\) is \(\frac { 4 } { 3 } \pi r ^ { 3 }\) ]
    A uniform solid \(S\), shown below in Figure 3, is formed by removing from \(C\) a uniform solid right circular cylinder of height \(r\) and radius \(\frac { 1 } { 2 } r\), where the centre of one end of the cylinder coincides with the centre of the plane face of \(C\) and the axis of the cylinder coincides with the axis of \(C\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{176ae8f8-7de9-4993-825a-6067614526ae-12_661_1194_861_440} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure}
  2. Show that the distance of the centre of mass of \(S\) from the vertex of \(C\) is \(\frac { 75 } { 26 } r\) A rough plane is inclined at an angle \(\alpha\) to the horizontal.
    The solid \(S\) rests in equilibrium with its plane face in contact with the inclined plane.
    Given that \(S\) is on the point of toppling,
  3. find the exact value of \(\tan \alpha\)
Edexcel M3 2024 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{176ae8f8-7de9-4993-825a-6067614526ae-16_739_921_299_699} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A fixed solid sphere has centre \(O\) and radius \(r\).
A particle \(P\) of mass \(m\) is held at rest on the smooth surface of the sphere at \(A\), the highest point of the sphere.
The particle \(P\) is then projected horizontally from \(A\) with speed \(u\) and moves on the surface of the sphere.
At the instant when \(P\) reaches the point \(B\) on the sphere, where angle \(A O B = \theta , P\) is moving with speed \(v\), as shown in Figure 4. At this instant, \(P\) loses contact with the surface of the sphere.
  1. Show that $$\cos \theta = \frac { 2 g r + u ^ { 2 } } { 3 g r }$$ In the subsequent motion, the particle \(P\) crosses the horizontal through \(O\) at the point \(C\), also shown in Figure 4. At the instant \(P\) passes through \(C , P\) is moving at an angle \(\alpha\) to the horizontal.
    Given that \(u ^ { 2 } = \frac { 2 g r } { 5 }\)
  2. find the exact value of \(\tan \alpha\).
Edexcel M3 2024 June Q7
  1. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(l\) and modulus of elasticity 2 mg . The other end of the string is attached to a fixed point \(A\) on a smooth horizontal table. The particle \(P\) is at rest at the point \(B\) on the table, where \(A B = l\).
At time \(t = 0 , P\) is projected along the table with speed \(U\) in the direction \(A B\).
At time \(t\)
  • the elastic string has not gone slack
  • \(B P = x\)
  • the speed of \(P\) is \(v\)
    1. Show that
$$v ^ { 2 } = U ^ { 2 } - \frac { 2 g x ^ { 2 } } { l }$$
  • By differentiating this equation with respect to \(x\), prove that, before the elastic string goes slack, \(P\) moves with simple harmonic motion with period \(\pi \sqrt { \frac { 2 l } { g } }\) Given that \(U = \sqrt { \frac { g l } { 2 } }\)
  • find, in terms of \(l\) and \(g\), the exact total time, from the instant it is projected from \(B\), that it takes \(P\) to travel a total distance of \(\frac { 3 } { 4 } l\) along the table.
    •
    Edexcel M3 2021 October Q1
    1. A particle \(P\) is moving in a straight line with simple harmonic motion of period 4 s . The centre of the motion is the point \(O\)
    At time \(t = 0 , P\) passes through \(O\) At time \(t = 0.5 \mathrm {~s} , P\) is moving with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
    1. Show that the amplitude of the motion is \(\frac { 4 \sqrt { 2 } } { \pi } \mathrm {~m}\)
    2. Find the maximum speed of \(P\)
    Edexcel M3 2021 October Q2
    2. In this question solutions relying on calculator technology are not acceptable. A particle \(P\) of mass 2 kg is moving along the positive \(x\)-axis.
    At time \(t\) seconds, where \(t \geqslant 0 , P\) is \(x\) metres from the origin \(O\) and is moving away from \(O\) with speed \(v \mathrm {~ms} ^ { - 1 }\) where \(v = \frac { 1 } { \sqrt { ( 2 x + 1 ) } }\)
    1. Find the magnitude of the resultant force acting on \(P\) when its speed is \(\frac { 1 } { 3 } \mathrm {~ms} ^ { - 1 }\) When \(t = 0 , P\) is at \(O\)
    2. Find the value of \(t\) when \(P\) is 7.5 m from \(O\)
    Edexcel M3 2021 October Q3
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{9777abb8-a564-40d5-8d96-d5649913737b-08_307_437_244_756} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length \(l\) and modulus of elasticity \(k m g\), where \(k\) is a constant. The other end of the spring is fixed to horizontal ground. The particle \(P\) rests in equilibrium, with the spring vertical, at the point \(E\).
    The point \(E\) is at a height \(\frac { 3 } { 5 } l\) above the ground, as shown in Figure 1.
    1. Show that \(k = \frac { 5 } { 2 }\) The particle \(P\) is now moved a distance \(\frac { 1 } { 4 } l\) vertically downwards from \(E\) and released from rest. Air resistance is modelled as being negligible.
    2. Show that \(P\) moves with simple harmonic motion.
    3. Find the speed of \(P\) as it passes through \(E\).
    4. Find the time from the instant \(P\) is released to the first instant it passes through \(E\).
    Edexcel M3 2021 October Q4
    1. A light elastic string has natural length \(2 a\) and modulus of elasticity \(2 m g\).
    One end of the elastic string is attached to a fixed point \(O\). A particle \(P\) of mass \(\frac { 1 } { 2 } m\) is attached to the other end of the elastic string. The point \(A\) is vertically below \(O\) with \(O A = 4 a\). Particle \(P\) is held at \(A\) and released from rest. The speed of \(P\) at the instant when it has moved a distance \(a\) upwards is \(\sqrt { 3 a g }\) Air resistance to the motion of \(P\) is modelled as having magnitude \(k m g\), where \(k\) is a constant. Using the model and the work-energy principle,
    1. show that \(k = \frac { 1 } { 4 }\) Particle \(P\) is now held at \(O\) and released from rest. As \(P\) moves downwards, it reaches its maximum speed as it passes through the point \(B\).
    2. Find the distance \(O B\).
    Edexcel M3 2021 October Q5
    5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{9777abb8-a564-40d5-8d96-d5649913737b-16_730_634_246_657} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} A small smooth ring \(R\) of mass \(m\) is threaded on to a thin smooth fixed vertical pole. One end of a light inextensible string of length \(2 l\) is attached to a point \(A\) on the pole. The other end of the string is attached to \(R\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The particle \(P\) moves with constant angular speed in a horizontal circle, with both halves of the string taut, and \(A R = \frac { 6 l } { 5 }\), as shown in Figure 2. It may be assumed that in this motion the string does not wrap itself around the pole and that at any instant, the triangle \(A P R\) lies in a vertical plane.
    1. Show that the tension in the lower half of the string is \(\frac { 5 m g } { 3 }\)
    2. Find, in terms of \(l\) and \(g\), the time for \(P\) to complete one revolution.
    Edexcel M3 2021 October Q6
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{9777abb8-a564-40d5-8d96-d5649913737b-20_534_551_248_699} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} A light rod of length \(a\) is free to rotate in a vertical plane about a horizontal axis through one end \(O\). A particle \(P\) of mass \(m\) is attached to the other end of the rod. The particle \(P\) is held at rest with the rod making an angle \(\alpha\) with the upward vertical through \(O\), where \(\tan \alpha = \frac { 3 } { 4 }\) The particle \(P\) is then projected with speed \(u\) in a direction which is perpendicular to the rod. At the instant when the rod makes an angle \(\theta\) with the upward vertical through \(O\), the speed of \(P\) is \(v\), as shown in Figure 3. Air resistance is assumed to be negligible.
    1. Show that \(v ^ { 2 } = u ^ { 2 } + \frac { 2 a g } { 5 } ( 4 - 5 \cos \theta )\) It is given that \(u ^ { 2 } = \frac { 6 a g } { 5 }\) and \(P\) moves in complete vertical circles. When \(\theta = \beta\), the force exerted on \(P\) by the rod is zero.
    2. Find the value of \(\cos \beta\)