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Edexcel M4 Q1
6 marks Standard +0.3
  1. A smooth sphere \(S\) is moving on a smooth horizontal plane with speed \(u\) when it collides with a smooth fixed vertical wall. At the instant of collision the direction of motion of \(S\) makes an angle of \(30 ^ { \circ }\) with the wall. The coefficient of restitution between \(S\) and the wall is \(\frac { 1 } { 3 }\).
Find the speed of \(S\) immediately after the collision.
Edexcel M4 Q2
8 marks Challenging +1.2
2. A car of mass 1000 kg , moving along a straight horizontal road, is driven by an engine which produces a constant power of 12 kW . The only resistance to the motion of the car is air resistance of magnitude \(10 v ^ { 2 } \mathrm {~N}\) where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car. Find the distance travelled by the car as its speed increases from \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
(8 marks)
Edexcel M4 Q3
10 marks Challenging +1.2
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{d57ea92a-4d6a-46bf-a6aa-bbd5083e8726-3_469_1163_1217_443}
\end{figure} A smooth uniform sphere \(A\), moving on a smooth horizontal table, collides with a second identical sphere \(B\) which is at rest on the table. When the spheres collide the line joining their centres makes an angle of \(30 ^ { \circ }\) with the direction of motion of \(A\), as shown in Fig. 1. The coefficient of restitution between the spheres is \(e\). The direction of motion of \(A\) is deflected through an angle \(\theta\) by the collision. Show that \(\tan \theta = \frac { ( 1 + e ) \sqrt { 3 } } { 5 - 3 e }\).
(10 marks)
Edexcel M4 Q4
10 marks Standard +0.8
4. A body falls vertically from rest and is subject to air resistance of a magnitude which is proportional to its speed. Given that its terminal speed is \(100 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the time it takes for the body to attain a speed of \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
(10 marks)
Edexcel M4 Q5
12 marks Standard +0.8
5. A particle \(P\) of mass \(m\) is fixed to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(2 m a n ^ { 2 }\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is released from rest at a point which is a distance \(2 a\) vertically below \(O\). The air resistance is modelled as having magnitude \(2 m n v\), where \(v\) is the speed of \(P\).
  1. Show that, when the extension of the string is \(x\), $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 n \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 n ^ { 2 } x = g$$
  2. Find \(x\) in terms of \(t\).
Edexcel M4 Q6
12 marks Challenging +1.8
6. Two particles \(P\) and \(Q\) have constant velocity vectors \(\mathbf { v } _ { P }\) and \(\mathbf { v } _ { Q }\) respectively. The magnitude of the velocity of \(P\) relative to \(Q\) is equal to the speed of \(P\). If the direction of motion of one of the particles is reversed, the magnitude of the velocity of \(P\) relative to \(Q\) is doubled. Find
  1. the ratio of the speeds of \(P\) and \(Q\),
  2. the cosine of the angle between the directions of motion of \(P\) and \(Q\).
Edexcel M4 Q7
17 marks Challenging +1.8
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{d57ea92a-4d6a-46bf-a6aa-bbd5083e8726-5_955_855_349_573}
\end{figure} A smooth wire \(A B\), in the shape of a circle of radius \(r\), is fixed in a vertical plane with \(A B\) vertical. A small smooth ring \(R\) of mass \(m\) is threaded on the wire and is connected by a light inextensible string to a particle \(P\) of mass \(m\). The length of the string is greater than the diameter of the circle. The string passes over a small smooth pulley which is fixed at the highest point \(A\) of the wire and angle \(R \hat { A } P = \theta\), as shown in Fig. 2.
  1. Show that the potential energy of the system is given by $$2 m g r \left( \cos \theta - \cos ^ { 2 } \theta \right) + \text { constant. }$$
  2. Hence determine the values of \(\theta , \theta \geq 0\), for which the system is in equilibrium. (6 marks)
  3. Determine the stability of each position of equilibrium. END
Edexcel M4 Specimen Q1
6 marks Moderate -0.3
  1. A particle \(P\) of mass 2 kg moves in a straight line along a smooth horizontal plane. The only horizontal force acting on \(P\) is a resistance of magnitude \(4 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is its speed. At time \(t = 0 \mathrm {~s} , P\) has a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find \(v\) in terms of \(t\).
    (6)
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{4737e682-1e1d-4c1a-91c2-7d051cb43aac-2_470_979_657_591}
\end{figure} A girl swims in still water at \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). She swims across a river which is 336 m wide and is flowing at \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). She sets off from a point \(A\) on one bank and lands at a point \(B\), which is directly opposite \(A\), on the other bank as shown in Fig. 1. Find
  1. the direction, relative to the earth, in which she swims,
  2. the time that she takes to cross the river.
Edexcel M4 Specimen Q3
10 marks Challenging +1.2
3. A ball of mass \(m\) is thrown vertically upwards from the ground. When its speed is \(v\) the magnitude of the air resistance is modelled as being \(m k v ^ { 2 }\), where \(k\) is a positive constant. The ball is projected with speed \(\sqrt { \frac { g } { k } }\). By modelling the ball as a particle,
  1. find the greatest height reached by the ball.
  2. State one physical factor which is ignored in this model.
Edexcel M4 Specimen Q4
11 marks Challenging +1.2
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{4737e682-1e1d-4c1a-91c2-7d051cb43aac-3_417_986_303_534}
\end{figure} Two smooth uniform spheres \(A\) and \(B\), of equal radius, are moving on a smooth horizontal plane. Sphere \(A\) has mass 3 kg and velocity \(( 2 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), and sphere \(B\) has mass 5 kg and velocity \(( - \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). When the spheres collide the line joining their centres is parallel to \(\mathbf { i }\), as shown in Fig. 2. Given that the direction of \(A\) is deflected through a right angle by the collision, find
  1. the velocity of \(A\) after the collision,
  2. the coefficient of restitution between the spheres.
Edexcel M4 Specimen Q5
12 marks Challenging +1.2
5. An elastic string spring of modulus \(2 m g\) and natural length \(l\) is fixed at one end. To the other end is attached a mass \(m\) which is allowed to hang in equilibrium. The mass is then pulled vertically downwards through a distance \(l\) and released from rest. The air resistance is modelled as having magnitude \(2 m \omega v\), where \(v\) is the speed of the particle and \(\omega = \sqrt { \frac { g } { l } }\). The particle is at distance \(x\) from its equilibrium position at time \(t\).
  1. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \omega \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 \omega ^ { 2 } x = 0\).
  2. Find the general solution of this differential equation.
  3. Hence find the period of the damped harmonic motion.
Edexcel M4 Specimen Q6
14 marks Standard +0.3
6. Two horizontal roads cross at right angles. One is directed from south to north, and the other from east to west. A tractor travels north on the first road at a constant speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at noon is 200 m south of the junction. A car heads west on the second road at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at noon is 960 m east of the junction.
  1. Find the magnitude and direction of the velocity of the car relative to the tractor.
  2. Find the shortest distance between the car and the tractor.
Edexcel M4 Specimen Q7
16 marks Challenging +1.2
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{4737e682-1e1d-4c1a-91c2-7d051cb43aac-4_558_1180_845_440}
\end{figure} A uniform rod \(A B\) has mass \(m\) and length \(2 a\). The end \(A\) is smoothly hinged at a fixed point on a fixed straight horizontal wire. A smooth light ring \(R\) is threaded on the wire. The ring \(R\) is attached by a light elastic string, of natural length \(a\) and modulus of elasticity \(m g\), to the end \(B\) of the rod. The end \(B\) is always vertically below \(R\) and angle \(\angle R A B = \theta\), as shown in Fig. 3.
  1. Show that the potential energy of the system is $$m g a \left( 2 \sin ^ { 2 } \theta - 3 \sin \theta \right) + \text { constant }$$ (6)
  2. Hence determine the value of \(\theta , \theta < \frac { \pi } { 2 }\), for which the system is in equilibrium.
  3. Determine whether this position of equilibrium is stable or unstable. END
OCR M4 2002 January Q1
4 marks Moderate -0.8
1 A wheel rotating about a fixed axis is slowing down with constant angular deceleration. Initially the angular speed is \(24 \mathrm { rad } \mathrm { s } ^ { - 1 }\). In the first 5 seconds the wheel turns through 96 radians.
  1. Find the angular deceleration.
  2. Find the total angle the wheel turns through before coming to rest.
OCR M4 2002 January Q2
5 marks Challenging +1.2
2 A uniform solid of revolution is formed by rotating the region bounded by the \(x\)-axis, the line \(x = 1\) and the curve \(y = x ^ { 2 }\) for \(0 \leqslant x \leqslant 1\), about the \(x\)-axis. The units are metres, and the density of the solid is \(5400 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\). Find the moment of inertia of this solid about the \(x\)-axis.
OCR M4 2002 January Q3
6 marks Challenging +1.2
3 A uniform rectangular lamina \(A B C D\) of mass 0.6 kg has sides \(A B = 0.4 \mathrm {~m}\) and \(A D = 0.3 \mathrm {~m}\). The lamina is free to rotate about a fixed horizontal axis which passes through \(A\) and is perpendicular to the lamina.
  1. Find the moment of inertia of the lamina about the axis.
  2. Find the approximate period of small oscillations in a vertical plane.
OCR M4 2002 January Q4
8 marks Challenging +1.2
4 A uniform circular disc has mass \(m\), radius \(a\) and centre \(C\). The disc is free to rotate in a vertical plane about a fixed horizontal axis passing through a point \(A\) on the disc, where \(C A = \frac { 1 } { 3 } a\).
  1. Find the moment of inertia of the disc about this axis. The disc is released from rest with \(C A\) horizontal.
  2. Find the initial angular acceleration of the disc.
  3. State the direction of the force acting on the disc at \(A\) immediately after release, and find its magnitude.
OCR M4 2002 January Q5
8 marks Challenging +1.2
5 The region bounded by the \(x\)-axis, the \(y\)-axis, the line \(x = \ln 5\) and the curve \(y = \mathrm { e } ^ { x }\) for \(0 \leqslant x \leqslant \ln 5\), is occupied by a uniform lamina.
  1. Show that the centre of mass of this lamina has \(x\)-coordinate $$\frac { 5 } { 4 } \ln 5 - 1$$
  2. Find the \(y\)-coordinate of the centre of mass.
OCR M4 2002 January Q6
8 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{98647526-b52a-4316-9a09-48d756b8f599-3_117_913_251_630} An arm on a fairground ride is modelled as a uniform rod \(A B\), of mass 75 kg and length 7.2 m , with a particle of mass 124 kg attached at \(B\). The arm can rotate about a fixed horizontal axis perpendicular to the rod and passing through the point \(P\) on the rod, where \(A P = 1.2 \mathrm {~m}\).
  1. Show that the moment of inertia of the arm about the axis is \(5220 \mathrm {~kg} \mathrm {~m} ^ { 2 }\).
  2. The arm is released from rest with \(A B\) horizontal, and a frictional couple of constant moment 850 N m opposes the motion. Find the angular speed of the arm when \(B\) is first vertically below \(P\).
OCR M4 2002 January Q7
9 marks Standard +0.3
7 At midnight, ship \(A\) is 70 km due north of ship \(B\). Ship \(A\) travels with constant velocity \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in the direction with bearing \(140 ^ { \circ }\). Ship \(B\) travels with constant velocity \(15 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in the direction with bearing \(025 ^ { \circ }\).
  1. Find the magnitude and direction of the velocity of \(A\) relative to \(B\).
  2. Find the distance between the ships when they are at their closest, and find the time when this occurs.
OCR M4 2002 January Q8
12 marks Challenging +1.2
8 \includegraphics[max width=\textwidth, alt={}, center]{98647526-b52a-4316-9a09-48d756b8f599-3_493_748_1393_708} The diagram shows a uniform rod \(A B\), of mass \(m\) and length \(2 a\), free to rotate in a vertical plane about a fixed horizontal axis through \(A\). A light elastic string has natural length \(a\) and modulus of elasticity \(\frac { 1 } { 2 } m g\). The string joins \(B\) to a light ring \(R\) which slides along a smooth horizontal wire fixed at a height \(a\) above \(A\) and in the same vertical plane as \(A B\). The string \(B R\) remains vertical. The angle between \(A B\) and the horizontal is denoted by \(\theta\), where \(0 < \theta < \pi\).
  1. Taking the reference level for gravitational potential energy to be the horizontal through \(A\), show that the total potential energy of the system is $$m g a \left( \sin ^ { 2 } \theta - \sin \theta \right) .$$
  2. Find the three values of \(\theta\) for which the system is in equilibrium.
  3. For each position of equilibrium, determine whether it is stable or unstable.
OCR M4 2004 January Q1
5 marks Moderate -0.8
1 A wheel is rotating about a fixed axis, and is slowing down with constant angular deceleration \(0.3 \mathrm { rad } \mathrm { s } ^ { - 2 }\).
  1. Find the angle the wheel turns through as its angular speed changes from \(8 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
  2. Find the time taken for the wheel to make its final complete revolution before coming to rest.
OCR M4 2004 January Q2
5 marks Standard +0.3
2 A rod \(A B\) of variable density has length 2 m . At a distance \(x\) metres from \(A\), the rod has mass per unit length ( \(0.7 - 0.3 x ) \mathrm { kg } \mathrm { m } ^ { - 1 }\). Find the distance of the centre of mass of the rod from \(A\).
OCR M4 2004 January Q3
7 marks Challenging +1.2
3 From a speedboat, a ship is sighted on a bearing of \(045 ^ { \circ }\). The ship has constant velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction with bearing \(120 ^ { \circ }\). The speedboat travels in a straight line with constant speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and intercepts the ship.
  1. Find the bearing of the course of the speedboat.
  2. Find the magnitude of the velocity of the ship relative to the speedboat.
OCR M4 2004 January Q4
7 marks Challenging +1.2
4 The region between the curve \(y = \frac { x ^ { 2 } } { a }\) and the \(x\)-axis for \(0 \leqslant x \leqslant a\) is occupied by a uniform lamina with mass \(m\). Show that the moment of inertia of this lamina about the \(x\)-axis is \(\frac { 1 } { 7 } m a ^ { 2 }\).