Questions M3 (796 questions)

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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 2008 June Q1
9 marks Standard +0.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-02_259_659_283_642} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A light elastic spring, of natural length \(L\) and modulus of elasticity \(\lambda\), has a particle \(P\) of mass \(m\) attached to one end. The other end of the spring is fixed to a point \(O\) on the closed end of a fixed smooth hollow tube of length \(L\). The tube is placed horizontally and \(P\) is held inside the tube with \(O P = \frac { 1 } { 2 } L\), as shown
in Figure 1. The particle \(P\) is released and passes through the open end of the tube with speed \(\sqrt { } ( 2 g L )\).
  1. Show that \(\lambda = 8 \mathrm { mg }\). The tube is now fixed vertically and \(P\) is held inside the tube with \(O P = \frac { 1 } { 2 } L\) and \(P\) above \(O\). The particle \(P\) is released and passes through the open top of the tube with speed \(u\).
  2. Find \(u\).
Edexcel M3 2008 June Q2
11 marks Standard +0.3
2. A particle \(P\) moves with simple harmonic motion and comes to rest at two points \(A\) and \(B\) which are 0.24 m apart on a horizontal line. The time for \(P\) to travel from \(A\) to \(B\) is 1.5 s . The midpoint of \(A B\) is \(O\). At time \(t = 0 , P\) is moving through \(O\), towards \(A\), with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the value of \(u\).
  2. Find the distance of \(P\) from \(B\) when \(t = 2 \mathrm {~s}\).
  3. Find the speed of \(P\) when \(t = 2 \mathrm {~s}\).
Edexcel M3 2008 June Q3
13 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-05_495_972_239_484} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a particle \(B\), of mass \(m\), attached to one end of a light elastic string. The other end of the string is attached to a fixed point \(A\), at a distance \(h\) vertically above a smooth horizontal table. The particle moves on the table in a horizontal circle with centre \(O\), where \(O\) is vertically below \(A\). The string makes a constant angle \(\theta\) with the downward vertical and \(B\) moves with constant angular speed \(\omega\) about \(O A\).
  1. Show that \(\omega ^ { 2 } \leqslant \frac { g } { h }\). The elastic string has natural length \(h\) and modulus of elasticity \(2 m g\).
    Given that \(\tan \theta = \frac { 3 } { 4 }\),
  2. find \(\omega\) in terms of \(g\) and \(h\).
Edexcel M3 2008 June Q4
13 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-07_454_614_239_662} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A uniform solid hemisphere, of radius \(6 a\) and centre \(O\), has a solid hemisphere of radius \(2 a\), and centre \(O\), removed to form a bowl \(B\) as shown in Figure 3.
  1. Show that the centre of mass of \(B\) is \(\frac { 30 } { 13 } a\) from \(O\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-07_735_614_1126_662} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} The bowl \(B\) is fixed to a plane face of a uniform solid cylinder made from the same material as \(B\). The cylinder has radius \(2 a\) and height \(6 a\) and the combined solid \(S\) has an axis of symmetry which passes through \(O\), as shown in Figure 4.
  2. Show that the centre of mass of \(S\) is \(\frac { 201 } { 61 } a\) from \(O\). The plane surface of the cylindrical base of \(S\) is placed on a rough plane inclined at \(12 ^ { \circ }\) to the horizontal. The plane is sufficiently rough to prevent slipping.
  3. Determine whether or not \(S\) will topple. \section*{
    \includegraphics[max width=\textwidth, alt={}]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-08_56_366_251_178}
    }
Edexcel M3 2008 June Q5
15 marks Standard +0.8
  1. 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 released from rest with the string taut and \(O P\) horizontal.
    1. Find the tension in the string when \(O P\) makes an angle of \(60 ^ { \circ }\) with the downward vertical.
    A particle \(Q\) of mass \(3 m\) is at rest at a distance \(a\) vertically below \(O\). When \(P\) strikes \(Q\) the particles join together and the combined particle of mass \(4 m\) starts to move in a vertical circle with initial speed \(u\).
  2. Show that \(u = \sqrt { } \left( \frac { g a } { 8 } \right)\). The combined particle comes to instantaneous rest at \(A\).
  3. Find
    1. the angle that the string makes with the downward vertical when the combined particle is at \(A\),
    2. the tension in the string when the combined particle is at \(A\).
      \section*{LU \(\_\_\_\_\)}
Edexcel M3 2008 June Q6
14 marks Standard +0.8
  1. A particle \(P\) of mass 0.5 kg moves along the positive \(x\)-axis. It moves away from the origin \(O\) under the action of a single force directed away from \(O\). When \(O P = x\) metres, the magnitude of the force is \(\frac { 3 } { ( x + 1 ) ^ { 3 } } \mathrm {~N}\) and the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    Initially \(P\) is at rest at \(O\).
    1. Show that \(v ^ { 2 } = 6 \left( 1 - \frac { 1 } { ( x + 1 ) ^ { 2 } } \right)\).
    2. Show that the speed of \(P\) never reaches \(\sqrt { } 6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
    3. Find \(x\) when \(P\) has been moving for 2 seconds.
    \section*{LL \(\_\_\_\_\)}
Edexcel M3 2010 June Q1
7 marks Moderate -0.8
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{049ea68c-d15f-41f8-860e-0816d36a2748-02_458_516_281_712} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A garden game is played with a small ball \(B\) of mass \(m\) attached to one end of a light inextensible string of length 13l. The other end of the string is fixed to a point \(A\) on a vertical pole as shown in Figure 1. The ball is hit and moves with constant speed in a horizontal circle of radius \(5 l\) and centre \(C\), where \(C\) is vertically below \(A\). Modelling the ball as a particle, find
  1. the tension in the string,
  2. the speed of the ball.
Edexcel M3 2010 June Q2
10 marks Standard +0.8
2. A particle \(P\) of mass \(m\) is above the surface of the Earth at distance \(x\) from the centre of the Earth. The Earth exerts a gravitational force on \(P\). The magnitude of this force is inversely proportional to \(x ^ { 2 }\). At the surface of the Earth the acceleration due to gravity is \(g\). The Earth is modelled as a sphere of radius \(R\).
  1. Prove that the magnitude of the gravitational force on \(P\) is \(\frac { m g R ^ { 2 } } { x ^ { 2 } }\). A particle is fired vertically upwards from the surface of the Earth with initial speed \(3 U\). At a height \(R\) above the surface of the Earth the speed of the particle is \(U\).
  2. Find \(U\) in terms of \(g\) and \(R\).
Edexcel M3 2010 June Q3
9 marks Challenging +1.2
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{049ea68c-d15f-41f8-860e-0816d36a2748-05_342_718_255_610} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle of mass 0.5 kg is attached to one end of a light elastic spring of natural length 0.9 m and modulus of elasticity \(\lambda\) newtons. The other end of the spring is attached to a fixed point \(O\) on a rough plane which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\). The coefficient of friction between the particle and the plane is 0.15 . The particle is held on the plane at a point which is 1.5 m down the line of greatest slope from \(O\), as shown in Figure 2. The particle is released from rest and first comes to rest again after moving 0.7 m up the plane. Find the value of \(\lambda\).
Edexcel M3 2010 June Q4
10 marks Standard +0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{049ea68c-d15f-41f8-860e-0816d36a2748-07_431_604_260_667} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A container is formed by removing a right circular solid cone of height \(4 l\) from a uniform solid right circular cylinder of height \(6 l\). The centre \(O\) of the plane face of the cone coincides with the centre of a plane face of the cylinder and the axis of the cone coincides with the axis of the cylinder, as shown in Figure 3. The cylinder has radius \(2 l\) and the base of the cone has radius \(l\).
  1. Find the distance of the centre of mass of the container from \(O\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{049ea68c-d15f-41f8-860e-0816d36a2748-07_460_588_1254_676} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} The container is placed on a plane which is inclined at an angle \(\theta ^ { \circ }\) to the horizontal. The open face is uppermost, as shown in Figure 4. The plane is sufficiently rough to prevent the container from sliding. The container is on the point of toppling.
  2. Find the value of \(\theta\).
Edexcel M3 2010 June Q5
12 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{049ea68c-d15f-41f8-860e-0816d36a2748-10_474_465_269_735} \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 fixed at the point \(O\). The particle is initially held with \(O P\) horizontal and the string taut. It is then projected vertically upwards with speed \(u\), where \(u ^ { 2 } = 5 a g\). When \(O P\) has turned through an angle \(\theta\) the speed of \(P\) is \(v\) and the tension in the string is \(T\), as shown in Figure 5.
  1. Find, in terms of \(a , g\) and \(\theta\), an expression for \(v ^ { 2 }\).
  2. Find, in terms of \(m , g\) and \(\theta\), an expression for \(T\).
  3. Prove that \(P\) moves in a complete circle.
  4. Find the maximum speed of \(P\).
Edexcel M3 2010 June Q6
12 marks Standard +0.3
  1. At time \(t = 0\), a particle \(P\) is at the origin \(O\) moving with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the \(x\)-axis in the positive \(x\)-direction. At time \(t\) seconds \(( t > 0 )\), the acceleration of \(P\) has magnitude \(\frac { 3 } { ( t + 1 ) ^ { 2 } } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and is directed towards \(O\).
    1. Show that at time \(t\) seconds the velocity of \(P\) is \(\left( \frac { 3 } { t + 1 } - 1 \right) \mathrm { m } \mathrm { s } ^ { - 1 }\).
    2. Find, to 3 significant figures, the distance of \(P\) from \(O\) when \(P\) is instantaneously at rest.
Edexcel M3 2010 June Q7
15 marks Challenging +1.2
  1. A light elastic string, of natural length \(3 a\) and modulus of elasticity \(6 m g\), has one end attached to a fixed point \(A\). A particle \(P\) of mass \(2 m\) is attached to the other end of the string and hangs in equilibrium at the point \(O\), vertically below \(A\).
    1. Find the distance \(A O\).
    The particle is now raised to point \(C\) vertically below \(A\), where \(A C > 3 a\), and is released from rest.
  2. Show that \(P\) moves with simple harmonic motion of period \(2 \pi \sqrt { } \left( \frac { a } { g } \right)\). It is given that \(O C = \frac { 1 } { 4 } a\).
  3. Find the greatest speed of \(P\) during the motion. The point \(D\) is vertically above \(O\) and \(O D = \frac { 1 } { 8 } a\). The string is cut as \(P\) passes through \(D\), moving upwards.
  4. Find the greatest height of \(P\) above \(O\) in the subsequent motion.
Edexcel M3 2011 June Q1
6 marks Standard +0.8
  1. A particle \(P\) of mass 0.5 kg moves on the positive \(x\)-axis under the action of a single force directed towards the origin \(O\). At time \(t\) seconds the distance of \(P\) from \(O\) is \(x\) metres, the magnitude of the force is \(0.375 x ^ { 2 } \mathrm {~N}\) and the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
When \(t = 0 , O P = 8 \mathrm {~m}\) and \(P\) is moving towards \(O\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(v ^ { 2 } = 260 - \frac { 1 } { 2 } \chi ^ { 3 }\).
  2. Find the distance of \(P\) from \(O\) at the instant when \(v = 5\).
Edexcel M3 2011 June Q2
9 marks Standard +0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{826ad8ff-6e5c-4224-88ba-e78b79d1bc21-03_438_661_223_644} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The shaded region \(R\) is bounded by the curve with equation \(y = 9 - x ^ { 2 }\), 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. Find the \(x\)-coordinate of the centre of mass of \(S\).
Edexcel M3 2011 June Q3
9 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{826ad8ff-6e5c-4224-88ba-e78b79d1bc21-04_542_469_219_735} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A solid consists of a uniform solid right cylinder of height \(5 l\) and radius \(3 l\) joined to a uniform solid hemisphere of radius \(3 l\). The plane face of the hemisphere coincides with a circular end of the cylinder and has centre \(O\), as shown in Figure 2. The density of the hemisphere is twice the density of the cylinder.
  1. Find the distance of the centre of mass of the solid from \(O\).
    (5) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{826ad8ff-6e5c-4224-88ba-e78b79d1bc21-04_618_807_1327_571} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The solid is now placed with its circular face on a plane inclined at an angle \(\theta ^ { \circ }\) to the horizontal, as shown in Figure 3. The plane is sufficiently rough to prevent the solid slipping. The solid is on the point of toppling.
  2. Find the value of \(\theta\).
Edexcel M3 2011 June Q4
12 marks Standard +0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{826ad8ff-6e5c-4224-88ba-e78b79d1bc21-07_805_460_214_740} \captionsetup{labelformat=empty} \caption{Figure 4}
\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 in a horizontal circle of radius \(3 a\) with angular speed \(\omega\). The centre of the circle is \(C\) where \(C\) lies on \(A B\) and \(A C = 4 a\), as shown in Figure 4. Both parts of the string are taut.
  1. Show that the tension in \(A P\) is \(\frac { 5 } { 7 } m \left( 3 a \omega ^ { 2 } + g \right)\).
  2. Find the tension in \(B P\).
  3. Deduce that \(\omega \geqslant \frac { 1 } { 2 } \sqrt { } \left( \frac { g } { a } \right)\).
Edexcel M3 2011 June Q5
12 marks Standard +0.8
  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 \(3 m g\). 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 \(O A = \frac { 7 } { 6 } l\). The coefficient of friction between \(P\) and the table is \(\mu\).
    1. Show that \(\mu \geqslant \frac { 1 } { 2 }\).
    The particle is now moved along the table to the point \(B\), where \(O B = \frac { 3 } { 2 } l\), and released from rest. Given that \(\mu = \frac { 1 } { 2 }\), find
  2. the speed of \(P\) at the instant when the string becomes slack,
  3. the total distance moved by \(P\) before it comes to rest again.
Edexcel M3 2011 June Q6
12 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{826ad8ff-6e5c-4224-88ba-e78b79d1bc21-11_574_540_226_701} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A particle \(P\) 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 point \(B\) is vertically above \(O\) and the point \(C\) is vertically below \(O\), with \(O B = O C = a\), as shown in Figure 5. The particle is projected vertically upwards with speed \(3 \sqrt { } ( a g )\).
  1. Show that \(P\) will pass through \(B\).
  2. Find the speed of \(P\) as it reaches \(C\). As \(P\) passes through \(C\) it receives an impulse. Immediately after this, the speed of \(P\) is \(\frac { 5 } { 12 } \sqrt { } ( 11 a g )\) and the direction of motion of \(P\) is unchanged.
  3. Find the angle between the string and the downward vertical when \(P\) comes to instantaneous rest.
Edexcel M3 2011 June Q7
15 marks Challenging +1.2
  1. A particle \(P\) of mass 0.5 kg is attached to the mid-point of a light elastic string of natural length 1.4 m and modulus of elasticity 2 N . The ends of the string are attached to the points \(A\) and \(B\) on a smooth horizontal table, where \(A B = 2 \mathrm {~m}\). The mid-point of \(A B\) is \(O\) and the point \(C\) is on the table between \(O\) and \(B\) where \(O C = 0.2 \mathrm {~m}\). At time \(t = 0\) the particle is released from rest at \(C\). At time \(t\) seconds the length of the string \(A P\) is \(( 1 + x ) \mathrm { m }\).
    1. Show that the tension in \(B P\) is \(\frac { 2 } { 7 } ( 3 - 10 x ) \mathrm { N }\).
    2. Find, in terms of \(x\), the tension in \(A P\).
    3. Show that \(P\) performs simple harmonic motion with period \(2 \pi \sqrt { } \left( \frac { 7 } { 80 } \right)\) s.
    4. Find the greatest speed of \(P\) during the motion.
    The point \(D\) lies between \(O\) and \(A\), where \(O D = 0.1 \mathrm {~m}\).
  2. Find the time taken by \(P\) to move directly from \(C\) to \(D\).