Questions M3 (796 questions)

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Edexcel M3 Q2
7 marks Moderate -0.3
2. A light elastic string \(A B\) has one end \(A\) attached to a fixed point on a ceiling. A particle \(P\) of mass 0.3 kg is attached to \(B\). When \(P\) hangs in equilibrium with \(A B\) vertical, \(A B = 100 \mathrm {~cm}\). The particle \(P\) is replaced by another particle \(Q\) of mass 0.5 kg . When \(Q\) hangs in equilibrium with \(A B\) vertical, \(A B = 110 \mathrm {~cm}\). Find
  1. the natural length of the string,
  2. the modulus of elasticity of the string.
Edexcel M3 Q3
10 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{45c51316-7d58-4c16-9b5f-1d7421060a88-3_485_855_1073_584} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(3 a\). The other end of the string is attached to a fixed point \(A\) which is a vertical distance \(a\) above a smooth horizontal table. The particle moves on the table in a circle whose centre \(O\) is vertically below \(A\), as shown in Fig. 1. The string is taut and the speed of \(P\) is \(2 \sqrt { } ( a g )\). Find
  1. the tension in the string,
  2. the normal reaction of the table on \(P\).
Edexcel M3 Q4
11 marks Challenging +1.2
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{45c51316-7d58-4c16-9b5f-1d7421060a88-4_332_1056_251_459} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} A small smooth bead \(B\) of mass 0.2 kg is threaded on a smooth horizontal wire. The point \(A\) is on the same horizontal level as the wire and at a perpendicular distance \(d\) from the wire. The point \(O\) is the point on the wire nearest to \(A\), as shown in Fig. 2. The bead experiences a force of magnitude \(5 ( A B )\) newtons in the direction \(B A\) towards \(A\). Initially \(B\) is at rest with \(O B = 2 \mathrm {~m}\).
  1. Prove that \(B\) moves with simple harmonic motion about \(O\), with period \(\frac { 2 \pi } { 5 } \mathrm {~s}\).
  2. Find the greatest speed of \(B\) in the motion.
  3. Find the time when \(B\) has first moved a distance 3 m from its initial position.
Edexcel M3 Q5
11 marks Standard +0.8
5. In a "test your strength" game at an amusement park, competitors hit one end of a small lever with a hammer, causing the other end of the lever to strike a ball which then moves in a vertical tube whose total height is adjustable. The ball is attached to one end of an elastic spring of natural length 3 m and modulus of elasticity 120 N . The mass of the ball is 2 kg . The other end of the spring is attached to the top of the tube. The ball is modelled as a particle, the spring as light and the tube is assumed to be smooth. The height of the tube is first set at 3 m . A competitor gives the ball an initial speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the height to which the ball rises before coming to rest. The tube is now adjusted by reducing its height to 2.5 m . The spring and the ball remain unchanged.
  2. Find the initial speed which the ball must now have if it is to rise by the same distance as in part (a).
    (5 marks)
Edexcel M3 Q6
14 marks Standard +0.8
6. (a) Show, by integration, that the centre of mass of a uniform right cone, of radius \(a\) and height \(h\), is a distance \(\frac { 3 } { 4 } h\) from the vertex of the cone. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{45c51316-7d58-4c16-9b5f-1d7421060a88-5_789_914_406_486} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} A uniform right cone \(C\), of radius \(a\) and height \(h\), has vertex \(A\). A solid \(S\) is formed by removing from \(C\) another cone, of radius \(\frac { 2 } { 3 } a\) and height \(\frac { 1 } { 2 } h\), with the same axis as \(C\). The plane faces of the two cones coincide, as shown in Fig. 3.
(b) Find the distance of the centre of mass of \(S\) from \(A\).
Edexcel M3 Q7
17 marks Challenging +1.2
7. A smooth solid hemisphere is fixed with its plane face on a horizontal table and its curved surface uppermost. The plane face of the hemisphere has centre \(O\) and radius \(a\). The point \(A\) is the highest point on the hemisphere. A particle \(P\) is placed on the hemisphere at \(A\). It is then given an initial horizontal speed \(u\), where \(u ^ { 2 } = \frac { 1 } { 2 } ( a g )\). When \(O P\) makes an angle \(\theta\) with \(O A\), and while \(P\) remains on the hemisphere, the speed of \(P\) is \(v\).
  1. Find an expression for \(v ^ { 2 }\).
  2. Show that, when \(\theta = \arccos 0.9 , P\) is still on the hemisphere.
  3. Find the value of \(\cos \theta\) when \(P\) leaves the hemisphere.
  4. Find the value of \(v\) when \(P\) leaves the hemisphere. After leaving the hemisphere \(P\) strikes the table at \(B\).
  5. Find the speed of \(P\) at \(B\).
  6. Find the angle at which \(P\) strikes the table. \section*{Alternative Question 2:}
    1. Two light elastic strings \(A B\) and \(B C\) are joined at \(B\). The string \(A B\) has natural length 1 m and modulus of elasticity 15 N . The string \(B C\) has natural length 1.2 m and modulus of elasticity 30 N . The ends \(A\) and \(C\) are attached to fixed points 3 m apart and the strings rest in equilibrium with \(A B C\) in a straight line.
    Find the tension in the combined string \(A C\).
Edexcel M3 Specimen Q1
7 marks Standard +0.3
1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{e256678d-89e8-48eb-aa8a-b8e027b62ef1-2_259_822_367_625}
\end{figure} A car moves round a bend in a road which is banked at an angle \(\alpha\) to the horizontal, as shown in Fig. 1. The car is modelled as a particle moving in a horizontal circle of radius 100 m . When the car moves at a constant speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), there is no sideways frictional force on the car. Find, in degrees to one decimal place, the value of \(\alpha\).
Edexcel M3 Specimen Q2
7 marks Standard +0.3
2. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{e256678d-89e8-48eb-aa8a-b8e027b62ef1-2_310_1122_1178_466}
\end{figure} Two elastic ropes each have natural length 30 cm and modulus of elasticity \(\lambda \mathrm { N }\). One end of each rope is attached to a lead weight \(P\) of mass 2 kg and the other ends are attached to two points \(A\) and \(B\) on a horizontal ceiling, where \(A B = 72 \mathrm {~cm}\). The weight hangs in equilibrium 15 cm below the ceiling, as shown in Fig. 2. By modelling \(P\) as a particle and the ropes as light elastic strings,
  1. find, to one decimal place, the value of \(\lambda\).
  2. State how you have used the fact that \(P\) is modelled as a particle.
Edexcel M3 Specimen Q3
8 marks Standard +0.8
3. A particle \(P\) of mass 0.5 kg moves away from the origin \(O\) along the positive \(x\)-axis under the action of a force directed towards \(O\) of magnitude \(\frac { 2 } { x ^ { 2 } } \mathrm {~N}\), where \(O P = x\) metres. When \(x = 1\), the speed of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance of \(P\) from \(O\) when its speed has been reduced to \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
(8)
Edexcel M3 Specimen Q4
10 marks Standard +0.8
4. A man of mass 75 kg is attached to one end of a light elastic rope of natural length 12 m . The other end of the rope is attached to a point on the edge of a horizontal ledge 19 m above the ground. The man steps off the ledge and falls vertically under gravity. The man is modelled as a particle falling from rest. He is brought to instantaneous rest by the rope when he is 1 m above the ground.
Find
  1. the modulus of elasticity of the rope,
    (5)
  2. the speed of the man when he is 2 m above the ground, giving your answer in \(\mathrm { m } \mathrm { s } ^ { - 1 }\) to 3 significant figures.
    (5)
Edexcel M3 Specimen Q5
13 marks Standard +0.3
5. \includegraphics[max width=\textwidth, alt={}, center]{e256678d-89e8-48eb-aa8a-b8e027b62ef1-3_423_357_918_847} A uniform solid, \(S\), is placed with its plane face on horizontal ground. The solid consists of a right circular cylinder, of radius \(r\) and height \(r\), joined to a right circular cone, of radius \(r\) and height \(h\). The plane face of the cone coincides with one of the plane faces of the cylinder, as shown in Fig. 3.
  1. Show that the distance of the centre of mass of \(S\) from the ground is $$\frac { 6 r ^ { 2 } + 4 r h + h ^ { 2 } } { 4 ( 3 r + h ) }$$ (8) The solid is now placed with its plane face on a rough plane which is inclined at an angle \(\alpha\) to the horizontal. The plane is rough enough to prevent \(S\) from sliding. Given that \(h = 2 r\), and that \(S\) is on the point of toppling,
  2. find, to the nearest degree, the value of \(\alpha\).
    (5)
Edexcel M3 Specimen Q6
15 marks Challenging +1.2
6. 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 hanging in equilibrium below \(O\) when it receives a horizontal impulse giving it a speed \(u\), where \(u ^ { 2 } = 3 g a\). The string becomes slack when \(P\) is at the point \(B\). The line \(O B\) makes an angle \(\theta\) with the upward vertical.
  1. Show that \(\cos \theta = \frac { 1 } { 3 }\).
    (9)
  2. Show that the greatest height of \(P\) above \(B\) in the subsequent motion is \(\frac { 4 a } { 27 }\).
    (6)
Edexcel M3 Specimen Q7
15 marks Challenging +1.2
7. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity 6 mg . The other end of the string is attached to a fixed point \(O\). When the particle hangs in equilibrium with the string vertical, the extension of the string is \(e\).
  1. Find \(e\).
    (2) The particle is now pulled down a vertical distance \(\frac { 1 } { 3 } a\) below its equilibrium position and released from rest. At time \(t\) after being released, during the time when the string remains taut, the extension of the string is \(e + x\). By forming a differential equation for the motion of \(P\) while the string remains taut,
  2. show that during this time \(P\) moves with simple harmonic motion of period \(2 \pi \sqrt { \frac { a } { 6 g } }\).
    (6)
  3. Show that, while the string remains taut, the greatest speed of \(P\) is \(\frac { 1 } { 3 } \sqrt { } ( 6 g a )\).
  4. Find \(t\) when the string becomes slack for the first time. \section*{END}
Edexcel M3 Q5
12 marks Standard +0.8
5. A cyclist is travelling around a circular track which is banked at \(25 ^ { \circ }\) to the horizontal. The coefficient of friction between the cycle's tyres and the track is 0.6 . The cyclist moves with constant speed in a horizontal circle of radius 40 m , without the tyres slipping. Find the maximum speed of the cyclist.
AQA M3 Q5
Moderate -0.3
5 A football is kicked from a point \(O\) on a horizontal football ground with a velocity of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(30 ^ { \circ }\). During the motion, the horizontal and upward vertical displacements of the football from \(O\) are \(x\) metres and \(y\) metres respectively.
  1. Show that \(x\) and \(y\) satisfy the equation $$y = x \tan 30 ^ { \circ } - \frac { g x ^ { 2 } } { 800 \cos ^ { 2 } 30 ^ { \circ } }$$
  2. On its downward flight the ball hits the horizontal crossbar of the goal at a point which is 2.5 m above the ground. Using the equation given in part (a), find the horizontal distance from \(O\) to the goal.
    (4 marks) \includegraphics[max width=\textwidth, alt={}, center]{fc5bfc4b-68bb-4a23-874b-87e9558dc990-04_330_1411_1902_303}
  3. State two modelling assumptions that you have made.
AQA M3 Q6
Standard +0.8
6 Two smooth billiard balls \(A\) and \(B\), of identical size and equal mass, move towards each other on a horizontal surface and collide. Just before the collision, \(A\) has velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at \(30 ^ { \circ }\) to the line of centres of the balls, and \(B\) has velocity \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at \(60 ^ { \circ }\) to the line of centres, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fc5bfc4b-68bb-4a23-874b-87e9558dc990-05_508_1420_532_294} The coefficient of restitution between the balls is \(\frac { 1 } { 2 }\).
  1. Find the speed of \(B\) immediately after the collision.
  2. Find the angle between the velocity of \(B\) and the line of centres of the balls immediately after the collision.
AQA M3 Q7
Challenging +1.8
7 A projectile is fired from a point \(O\) on the slope of a hill which is inclined at an angle \(\alpha\) to the horizontal. The projectile is fired up the hill with velocity \(U\) at an angle \(\theta\) above the hill and first strikes it at a point \(A\). The projectile is modelled as a particle and the hill is modelled as a plane with \(O A\) as a line of greatest slope.
    1. Find, in terms of \(U , g , \alpha\) and \(\theta\), the time taken by the projectile to travel from \(O\) to \(A\).
    2. Hence, or otherwise, show that the magnitude of the component of the velocity of the projectile perpendicular to the hill, when it strikes the hill at the point \(A\), is the same as it was initially at \(O\).
  2. The projectile rebounds and strikes the hill again at a point \(B\). The hill is smooth and the coefficient of restitution between the projectile and the hill is \(e\). \includegraphics[max width=\textwidth, alt={}, center]{fc5bfc4b-68bb-4a23-874b-87e9558dc990-06_428_1332_1023_338} Find the ratio of the time of flight from \(O\) to \(A\) to the time of flight from \(A\) to \(B\). Give your answer in its simplest form.
AQA M3 2006 June Q1
7 marks Moderate -0.5
1 The time \(T\) taken for a simple pendulum to make a single small oscillation is thought to depend only on its length \(l\), its mass \(m\) and the acceleration due to gravity \(g\). By using dimensional analysis:
  1. show that \(T\) does not depend on \(m\);
  2. express \(T\) in terms of \(l , g\) and \(k\), where \(k\) is a dimensionless constant.
AQA M3 2006 June Q2
12 marks Standard +0.3
2 Three smooth spheres \(A , B\) and \(C\) of equal radii and masses \(m , m\) and \(2 m\) respectively lie at rest on a smooth horizontal table. The centres of the spheres lie in a straight line with \(B\) between \(A\) and \(C\). The coefficient of restitution between any two spheres is \(e\). The sphere \(A\) is projected directly towards \(B\) with speed \(u\) and collides with \(B\).
  1. Find, in terms of \(u\) and \(e\), the speed of \(B\) immediately after the impact between \(A\) and \(B\).
  2. The sphere \(B\) subsequently collides with \(C\). The speed of \(C\) immediately after this collision is \(\frac { 3 } { 8 } u\). Find the value of \(e\).
AQA M3 2006 June Q3
9 marks Standard +0.3
3 A ball of mass 0.45 kg is travelling horizontally with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it strikes a fixed vertical bat directly and rebounds from it. The ball stays in contact with the bat for 0.1 seconds. At time \(t\) seconds after first coming into contact with the bat, the force exerted on the ball by the bat is \(1.4 \times 10 ^ { 5 } \left( t ^ { 2 } - 10 t ^ { 3 } \right)\) newtons, where \(0 \leqslant t \leqslant 0.1\). In this simple model, ignore the weight of the ball and model the ball as a particle.
  1. Show that the magnitude of the impulse exerted by the bat on the ball is 11.7 Ns , correct to three significant figures.
  2. Find, to two significant figures, the speed of the ball immediately after the impact.
  3. Give a reason why the speed of the ball immediately after the impact is different from the speed of the ball immediately before the impact.
AQA M3 2006 June Q4
10 marks Moderate -0.3
4 The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due east and due north respectively.
Two cyclists, Aazar and Ben, are cycling on straight horizontal roads with constant velocities of \(( 6 \mathbf { i } + 12 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and \(( 12 \mathbf { i } - 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) respectively. Initially, Aazar and Ben have position vectors \(( 5 \mathbf { i } - \mathbf { j } ) \mathrm { km }\) and \(( 18 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }\) respectively, relative to a fixed origin.
  1. Find, as a vector in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of Ben relative to Aazar.
  2. The position vector of Ben relative to Aazar at time \(t\) hours after they start is \(\mathbf { r } \mathrm { km }\). Show that $$\mathbf { r } = ( 13 + 6 t ) \mathbf { i } + ( 6 - 20 t ) \mathbf { j }$$
  3. Find the value of \(t\) when Aazar and Ben are closest together.
  4. Find the closest distance between Aazar and Ben.
AQA M3 2006 June Q5
13 marks Moderate -0.3
5 A football is kicked from a point \(O\) on a horizontal football ground with a velocity of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation of \(30 ^ { \circ }\). During the motion, the horizontal and upward vertical displacements of the football from \(O\) are \(x\) metres and \(y\) metres respectively.
  1. Show that \(x\) and \(y\) satisfy the equation $$y = x \tan 30 ^ { \circ } - \frac { g x ^ { 2 } } { 800 \cos ^ { 2 } 30 ^ { \circ } }$$
  2. On its downward flight the ball hits the horizontal crossbar of the goal at a point which is 2.5 m above the ground. Using the equation given in part (a), find the horizontal distance from \(O\) to the goal.
    (4 marks) \includegraphics[max width=\textwidth, alt={}, center]{f8c04360-f54b-4d08-aee9-fe28612918d0-3_330_1411_1902_303}
  3. State two modelling assumptions that you have made.
AQA M3 2006 June Q6
11 marks Standard +0.3
6 Two smooth billiard balls \(A\) and \(B\), of identical size and equal mass, move towards each other on a horizontal surface and collide. Just before the collision, \(A\) has velocity \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at \(30 ^ { \circ }\) to the line of centres of the balls, and \(B\) has velocity \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction inclined at \(60 ^ { \circ }\) to the line of centres, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{f8c04360-f54b-4d08-aee9-fe28612918d0-4_508_1420_532_294} The coefficient of restitution between the balls is \(\frac { 1 } { 2 }\).
  1. Find the speed of \(B\) immediately after the collision.
  2. Find the angle between the velocity of \(B\) and the line of centres of the balls immediately after the collision.
AQA M3 2006 June Q7
13 marks Challenging +1.2
7 A projectile is fired from a point \(O\) on the slope of a hill which is inclined at an angle \(\alpha\) to the horizontal. The projectile is fired up the hill with velocity \(U\) at an angle \(\theta\) above the hill and first strikes it at a point \(A\). The projectile is modelled as a particle and the hill is modelled as a plane with \(O A\) as a line of greatest slope.
    1. Find, in terms of \(U , g , \alpha\) and \(\theta\), the time taken by the projectile to travel from \(O\) to \(A\).
    2. Hence, or otherwise, show that the magnitude of the component of the velocity of the projectile perpendicular to the hill, when it strikes the hill at the point \(A\), is the same as it was initially at \(O\).
  2. The projectile rebounds and strikes the hill again at a point \(B\). The hill is smooth and the coefficient of restitution between the projectile and the hill is \(e\). \includegraphics[max width=\textwidth, alt={}, center]{f8c04360-f54b-4d08-aee9-fe28612918d0-5_428_1332_1023_338} Find the ratio of the time of flight from \(O\) to \(A\) to the time of flight from \(A\) to \(B\). Give your answer in its simplest form.
AQA M3 2007 June Q1
8 marks Moderate -0.5
1 The magnitude of the gravitational force, \(F\), between two planets of masses \(m _ { 1 }\) and \(m _ { 2 }\) with centres at a distance \(x\) apart is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { x ^ { 2 } }$$ where \(G\) is a constant.
  1. By using dimensional analysis, find the dimensions of \(G\).
  2. The lifetime, \(t\), of a planet is thought to depend on its mass, \(m\), its initial radius, \(R\), the constant \(G\) and a dimensionless constant, \(k\), so that $$t = k m ^ { \alpha } R ^ { \beta } G ^ { \gamma }$$ where \(\alpha , \beta\) and \(\gamma\) are constants.
    Find the values of \(\alpha , \beta\) and \(\gamma\).