Questions — Edexcel (10514 questions)

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Edexcel M2 Q3
9 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9126ebb1-eaa7-4a40-953f-5dc819c9f479-4_698_1271_296_488} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A uniform plane lamina is in the shape of an isosceles triangle \(A B C\), where \(A B = A C\). The mid-point of \(B C\) is \(M , A M = 30 \mathrm {~cm}\) and \(B M = 40 \mathrm {~cm}\). The mid-points of \(A C\) and \(A B\) are \(D\) and \(E\) respectively. The triangular portion \(A D E\) is removed leaving a uniform plane lamina \(B C D E\) as shown in Fig. 2.
  1. Show that the centre of mass of the lamina \(B C D E\) is \(6 \frac { 2 } { 3 } \mathrm {~cm}\) from \(B C\).
    (6 marks)
    The lamina \(B C D E\) is freely suspended from \(D\) and hangs in equilibrium.
  2. Find, in degrees to one decimal place, the angle which \(D E\) makes with the vertical.
    (3 marks)
Edexcel M2 Q4
9 marks Standard +0.3
4. The resistance to the motion of a cyclist is modelled as \(k v ^ { 2 } \mathrm {~N}\), where \(k\) is a constant and \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the cyclist. The total mass of the cyclist and his bicycle is 100 kg . The cyclist freewheels down a slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\), at a constant speed of \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Show that \(k = 4\). The cyclist ascends a slope inclined at an angle \(\beta\) to the horizontal, where \(\sin \beta = \frac { 1 } { 40 }\), at a constant speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  2. Find the rate at which the cyclist is working.
    (6 marks)
Edexcel M2 Q5
9 marks Standard +0.3
5. A smooth sphere \(S\) of mass \(m\) is moving with speed \(u\) on a smooth horizontal plane. The sphere \(S\) collides with another smooth sphere \(T\), of equal radius to \(S\) but of mass \(k m\), moving in the same straight line and in the same direction with speed \(\lambda u , 0 < \lambda < \frac { 1 } { 2 }\). The coefficient of restitution between \(S\) and \(T\) is \(e\). Given that \(S\) is brought to rest by the impact,
  1. show that \(e = \frac { 1 + k \lambda } { k ( 1 - \lambda ) }\).
  2. Deduce that \(k > 1\).
Edexcel M2 Q6
9 marks Standard +0.3
6. At time \(t\) seconds the acceleration, a \(\mathrm { m } \mathrm { s } ^ { - 2 }\), of a particle \(P\) relative to a fixed origin \(O\), is given by \(\mathbf { a } = 2 \mathbf { i } + 6 t \mathbf { j }\). Initially the velocity of \(P\) is \(( 2 \mathbf { i } - 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
  1. Find the velocity of \(P\) at time \(t\) seconds. At time \(t = 2\) seconds the particle \(P\) is given an impulse ( \(3 \mathbf { i } - 1.5 \mathbf { j }\) ) Ns. Given that the particle \(P\) has mass 0.5 kg ,
  2. find the speed of \(P\) immediately after the impulse has been applied.
Edexcel M2 Q7
13 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9126ebb1-eaa7-4a40-953f-5dc819c9f479-6_675_1243_392_415} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} A shot is projected upwards from the top of a cliff with a velocity of \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) above the horizontal. It strikes the ground 52.5 m vertically below the level of the point of projection, as shown in Fig. 3. The motion of the shot is modelled as that of a particle moving freely under gravity. Find, to 3 significant figures,
  1. the horizontal distance from the point of projection at which the shot strikes the ground,
  2. the speed of the shot as it strikes the ground.
Edexcel M2 Q8
15 marks Standard +0.3
8. A particle \(P\) is projected up a line of greatest slope of a rough plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 2 }\). The particle is projected from the point \(O\) with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and comes to instantaneous rest at the point \(A\). By Using the Work-Energy principle, or otherwise,
  1. find, to 3 significant figures, the length \(O A\).
  2. Show that \(P\) will slide back down the plane.
  3. Find, to 3 significant figures, the speed of \(P\) when it returns to \(O\).
Edexcel M2 Specimen Q1
5 marks Moderate -0.3
  1. The vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane. A ball of mass 0.5 kg is moving with velocity \(- 20 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it is struck by a bat. The bat gives the ball an impulse of \(( 15 \mathbf { i } + 10 \mathbf { j } )\) Ns.
Find, to 3 significant figures, the speed of the ball immediately after it has been struck.
(5)
Edexcel M2 Specimen Q2
5 marks Moderate -0.3
2. A bullet of mass 6 grams passes horizontally through a fixed, vertical board. After the bullet has travelled 2 cm through the board its speed is reduced from \(400 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(250 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The board exerts a constant resistive force on the bullet. Find, to 3 significant figures, the magnitude of this resistive force.
(5)
Edexcel M2 Specimen Q3
7 marks Moderate -0.3
3. At time \(t\) seconds, a particle \(P\) has position vector \(\mathbf { r }\) metres relative to a fixed origin \(O\), where $$\mathbf { r } = \left( t ^ { 3 } - 3 t \right) \mathbf { i } + 4 t ^ { 2 } \mathbf { j } , t \geq 0$$ Find
  1. the velocity of \(P\) at time \(t\) seconds,
  2. the time when \(P\) is moving parallel to the vector \(\mathbf { i } + \mathbf { j }\).
    (5)
Edexcel M2 Specimen Q4
9 marks Standard +0.8
4. \section*{Figure 1}
\includegraphics[max width=\textwidth, alt={}]{0d3d35b1-e3c5-47ac-b05e-78cdf1eb3083-3_714_565_262_749}
A uniform ladder, of mass \(m\) and length \(2 a\), has one end on rough horizontal ground. The other end rests against a smooth vertical wall. A man of mass \(3 m\) stands at the top of the ladder and the ladder is in equilibrium. The coefficient of friction between the ladder and the ground is \(\frac { 1 } { 4 }\), and the ladder makes an angle \(\alpha\) with the vertical, as shown in Fig. 1. The ladder is in a vertical plane perpendicular to the wall. Show that \(\tan \alpha \leq \frac { 2 } { 7 }\).
Edexcel M2 Specimen Q5
11 marks Moderate -0.3
5. A straight road is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\). A lorry of mass 4800 kg moves up the road at a constant speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The non-gravitational resistance to the motion of the lorry is constant and has magnitude 2000 N .
  1. Find, in kW to 3 significant figures, the rate of working of the lorry's engine.
    (5) The road becomes horizontal. The lorry's engine continues to work at the same rate and the resistance to motion remains the same. Find
  2. the acceleration of the lorry immediately after the road becomes horizontal,
    (3)
  3. the maximum speed, in \(\mathrm { m } \mathrm { s } ^ { - 1 }\) to 3 significant figures, at which the lorry will go along the horizontal road.
    (3)
Edexcel M2 Specimen Q6
12 marks Moderate -0.3
6. A cricket ball is hit from a height of 0.8 m above horizontal ground with a speed of \(26 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\). The motion of the ball is modelled as that of a particle moving freely under gravity.
  1. Find, to 2 significant figures, the greatest height above the ground reached by the ball. When the ball has travelled a horizontal distance of 36 m , it hits a window.
  2. Find, to 2 significant figures, the height above the ground at which the ball hits the window.
  3. State one physical factor which could be taken into account in any refinement of the model which would make it more realistic. Figure 2
Edexcel M2 Specimen Q7
13 marks Standard +0.3
7. \includegraphics[max width=\textwidth, alt={}, center]{0d3d35b1-e3c5-47ac-b05e-78cdf1eb3083-4_360_472_1105_815} A uniform plane lamina \(A B C D E\) is formed by joining a uniform square \(A B D E\) with a uniform triangular lamina \(B C D\), of the same material, along the side \(B D\), as shown in Fig. 2. The lengths \(A B , B C\) and \(C D\) are \(18 \mathrm {~cm} , 15 \mathrm {~cm}\) and 15 cm respectively.
  1. Find the distance of the centre of mass of the lamina from \(A E\). The lamina is freely suspended from \(B\) and hangs in equilibrium.
  2. Find, in degrees to one decimal place, the angle which \(B D\) makes with the vertical.
Edexcel M2 Specimen Q8
13 marks Standard +0.3
8. A particle \(A\) of mass \(m\) is moving with speed \(3 u\) on a smooth horizontal table when it collides directly with a particle \(B\) of mass \(2 m\) which is moving in the opposite direction with speed \(u\). The direction of motion of \(A\) is reversed by the collision. The coefficient of restitution between \(A\) and \(B\) is \(e\).
  1. Show that the speed of \(B\) immediately after the collision is \(\frac { 1 } { 3 } ( 1 + 4 e ) u\).
    (6)
  2. Show that \(e > \frac { 1 } { 8 }\).
    (3) Subsequently \(B\) hits a wall fixed at right angles to the line of motion of \(A\) and \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 2 }\). After \(B\) rebounds from the wall, there is a further collision between \(A\) and \(B\).
  3. Show that \(e < \frac { 1 } { 4 }\).
    (4) END
Edexcel M3 2014 January Q1
5 marks Standard +0.3
  1. A particle \(P\) of mass 0.5 kg moves along the positive \(x\)-axis under the action of a single force of magnitude \(F\) newtons. The force acts along the \(x\)-axis in the direction of \(x\) increasing. When \(P\) is \(x\) metres from the origin \(O\), it is moving away from \(O\) with speed \(\sqrt { \left( 8 x ^ { \frac { 3 } { 2 } } - 4 \right) } \mathrm { ms } ^ { - 1 }\).
Find \(F\) when \(P\) is 4 m from \(O\).
Edexcel M3 2014 January Q2
9 marks Standard +0.8
2. A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring, of natural length \(l\) and modulus of elasticity \(2 m g\). The other end of the spring is attached to a fixed point \(A\) on a rough horizontal plane. The particle is held at rest on the plane at a point \(B\), where \(A B = \frac { 1 } { 2 } l\), and released from rest. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\) Find the distance of \(P\) from \(B\) when \(P\) first comes to rest.
Edexcel M3 2014 January Q3
8 marks Challenging +1.2
3. A light rod \(A B\) of length \(2 a\) has a particle \(P\) of mass \(m\) attached to \(B\). The rod is rotating in a vertical plane about a fixed smooth horizontal axis through \(A\). Given that the greatest tension in the rod is \(\frac { 9 m g } { 8 }\), find, to the nearest degree, the angle between the rod and the downward vertical when the speed of \(P\) is \(\sqrt { \left( \frac { a g } { 20 } \right) }\).
Edexcel M3 2014 January Q4
10 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2c0bb9ea-31a6-42f1-9e2e-d792eee8fd10-05_568_620_269_653} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the region \(R\) bounded by the curve with equation \(y = \mathrm { e } ^ { - x }\), the line \(x = 1\), the \(x\)-axis and the \(y\)-axis. 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 \(\frac { \pi } { 2 } \left( 1 - \mathrm { e } ^ { - 2 } \right)\).
  2. Find, in terms of e, the distance of the centre of mass of \(S\) from \(O\).
Edexcel M3 2014 January Q5
12 marks Standard +0.8
5. A solid \(S\) consists of a uniform solid hemisphere of radius \(r\) and a uniform solid circular cylinder of radius \(r\) and height \(3 r\). The circular face of the hemisphere is joined to one of the circular faces of the cylinder, so that the centres of the two faces coincide. The other circular face of the cylinder has centre \(O\). The mass per unit volume of the hemisphere is \(3 k\) and the mass per unit volume of the cylinder is \(k\).
  1. Show that the distance of the centre of mass of \(S\) from \(O\) is \(\frac { 9 r } { 4 }\) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{2c0bb9ea-31a6-42f1-9e2e-d792eee8fd10-07_501_1082_653_422} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} The solid \(S\) is held in equilibrium by a horizontal force of magnitude \(P\). The circular face of \(S\) has one point in contact with a fixed rough horizontal plane and is inclined at an angle \(\alpha\) to the horizontal. The force acts through the highest point of the circular face of \(S\) and in the vertical plane through the axis of the cylinder, as shown in Figure 2. The coefficient of friction between \(S\) and the plane is \(\mu\). Given that \(S\) is on the point of slipping along the plane in the same direction as \(P\),
  2. show that \(\mu = \frac { 1 } { 8 } ( 9 - 4 \cot \alpha )\).
Edexcel M3 2014 January Q6
15 marks Standard +0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2c0bb9ea-31a6-42f1-9e2e-d792eee8fd10-09_1089_1072_278_466} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A light inextensible string of length \(14 a\) has its ends attached to two fixed points \(A\) and \(B\), where \(A\) is vertically above \(B\) and \(A B = 10 a\). A particle of mass \(m\) is attached to the string at the point \(P\), where \(A P = 8 a\). The particle moves in a horizontal circle with constant angular speed \(\omega\) and with both parts of the string taut, as shown in Figure 3.
  1. Show that angle \(A P B = 90 ^ { \circ }\).
  2. Show that the time for the particle to make one complete revolution is less than $$2 \pi \sqrt { \left( \frac { 32 a } { 5 g } \right) } .$$
Edexcel M3 2014 January Q7
16 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2c0bb9ea-31a6-42f1-9e2e-d792eee8fd10-11_517_254_278_845} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A smooth hollow narrow tube of length \(l\) has one open end and one closed end. The tube is fixed in a vertical position with the closed end at the bottom. A light elastic spring has natural length \(l\) and modulus of elasticity \(8 m g\). The spring is inside the tube and has one end attached to a fixed point \(O\) on the closed end of the tube. The other end of the spring is attached to a particle \(P\) of mass \(m\). The particle rests in equilibrium at a distance \(e\) below the top of the tube, as shown in Figure 4.
  1. Find \(e\) in terms of \(l\). The particle \(P\) is now held inside the tube at a distance \(\frac { 1 } { 2 } l\) below the top of the tube and released from rest at time \(t = 0\)
  2. Prove that \(P\) moves with simple harmonic motion of period \(2 \pi \sqrt { \left( \frac { l } { 8 g } \right) }\). The particle \(P\) passes through the open top of the tube with speed \(u\).
  3. Find \(u\) in terms of \(g\) and \(l\).
  4. Find the time taken for \(P\) to first attain a speed of \(\sqrt { \left( \frac { 9 g l } { 32 } \right) }\).
Edexcel M3 2015 January Q1
6 marks Standard +0.8
  1. A particle \(P\) of mass 3 kg is moving along the horizontal \(x\)-axis. At time \(t = 0 , P\) passes through the origin \(O\) moving in the positive \(x\) direction. At time \(t\) seconds, \(O P = x\) metres and the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t\) seconds, the resultant force acting on \(P\) is \(\frac { 9 } { 2 } ( 26 - x ) \mathrm { N }\), measured in the positive \(x\) direction. For \(t > 0\) the maximum speed of \(P\) is \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Find \(v ^ { 2 }\) in terms of \(x\).
Edexcel M3 2015 January Q2
9 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3706a02d-95c6-4e7a-bf38-88b338d77892-03_547_671_260_648} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A uniform lamina is in the shape of the region \(R\) which is bounded by the curve with equation \(y = \frac { 3 } { x ^ { 2 } }\), the lines \(x = 1\) and \(x = 3\), and the \(x\)-axis, as shown in Figure 1. The centre of mass of the lamina has coordinates \(( \bar { x } , \bar { y } )\).
Use algebraic integration to find
  1. the value of \(\bar { x }\),
  2. the value of \(\bar { y }\).
Edexcel M3 2015 January Q3
8 marks Standard +0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3706a02d-95c6-4e7a-bf38-88b338d77892-05_828_624_264_676} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A light inextensible string has one end attached to a fixed point \(A\) and the other end attached to a particle \(P\) of mass \(m\). An identical string has one end attached to the fixed point \(B\), where \(B\) is vertically below \(A\) and \(A B = 4 a\), and the other end attached to \(P\), as shown in Figure 2. The particle is moving in a horizontal circle with constant angular speed \(\omega\), with both strings taut and inclined at \(30 ^ { \circ }\) to the vertical. The tension in the upper string is twice the tension in the lower string. Find \(\omega\) in terms of \(a\) and \(g\).
Edexcel M3 2015 January Q4
11 marks Standard +0.8
A light elastic string has natural length 5 m and modulus of elasticity 20 N . The ends of the string are attached to two fixed points \(A\) and \(B\), which are 6 m apart on a horizontal ceiling. A particle \(P\) is attached to the midpoint of the string and hangs in equilibrium at a point which is 4 m below \(A B\).
  1. Calculate the weight of \(P\). The particle is now raised to the midpoint of \(A B\) and released from rest.
  2. Calculate the speed of \(P\) when it has fallen 4 m .