Questions M3 (745 questions)

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Edexcel M3 Q5
5. (a) Use integration to show that the centre of mass of a uniform solid right circular cone of height \(h\) is \(\frac { 3 } { 4 } h\) from the vertex of the cone.
(6 marks) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ad523c3f-9109-45a8-8399-80a4c2edeff7-4_419_424_372_721} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} A paperweight is made by removing material from the top half of a solid sphere of radius \(r\) so that the remaining solid consists of a hemisphere of radius \(r\) and a cone of height \(r\) and base radius \(r\) as shown in Figure 3.
(b) Find the distance of the centre of mass of the paperweight from its vertex.
(7 marks)
Edexcel M3 Q6
6. A car is travelling on a horizontal racetrack round a circular bend of radius 40 m . The coefficient of friction between the car and the road is \(\frac { 2 } { 5 }\).
  1. Find the maximum speed at which the car can travel round the bend without slipping, giving your answer correct to 3 significant figures.
    (5 marks)
    The owner of the track decides to bank the corner at an angle of \(25 ^ { \circ }\) in order to enable the cars to travel more quickly.
  2. Show that this increases the maximum speed at which the car can travel round the bend without slipping by 63\%, correct to the nearest whole number.
    (8 marks)
Edexcel M3 Q7
7. A particle is travelling along the \(x\)-axis. At time \(t = 0\), the particle is at \(O\) and it travels such that its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at a distance \(x\) metres from \(O\) is given by $$v = \frac { 2 } { x + 1 }$$ The acceleration of the particle is \(a \mathrm {~ms} ^ { - 2 }\).
  1. Show that \(a = \frac { - 4 } { ( x + 1 ) ^ { 3 } }\).
    (4 marks) The points \(A\) and \(B\) lie on the \(x\)-axis. Given that the particle travels \(d\) metres from \(O\) to \(A\) in \(T\) seconds and 4 metres from \(A\) to \(B\) in 9 seconds,
  2. show that \(d = 1.5\),
  3. find \(T\).
Edexcel M3 Q1
  1. A light elastic string has natural length \(a\) and modulus of elasticity 4 mg . One end of the string is attached to a fixed point \(A\) and a particle of mass \(m\) is attached to the other end.
The particle is released from rest at \(A\) and falls vertically until it comes to rest instantaneously at the point \(B\). Find the distance \(A B\) in terms of \(a\).
(7 marks)
Edexcel M3 Q2
2. A particle \(P\) of mass 0.25 kg is moving on a horizontal plane. At time \(t\) seconds the velocity, \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), of \(P\) relative to a fixed origin \(O\) is given by $$\mathbf { v } = \ln ( t + 1 ) \mathbf { i } - \mathrm { e } ^ { - 2 t } \mathbf { j } , t \leq 0 ,$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in the horizontal plane.
  1. Find the acceleration of \(P\) in terms of \(t\).
  2. Find, correct to 3 significant figures, the magnitude of the resultant force acting on \(P\) when \(t = 1\).
    (4 marks)
Edexcel M3 Q3
3. A coin of mass 5 grams is placed on a vinyl disc rotating on a record player. The distance between the centre of the coin and the centre of the disc is 0.1 m and the coefficient of friction between the coin and the disc is \(\mu\). The disc rotates at 45 revolutions per minute around a vertical axis at its centre and the coin moves with it and does not slide. By modelling the coin as a particle and giving your answers correct to an appropriate degree of accuracy, find
  1. the speed of the coin,
  2. the horizontal and vertical components of the force exerted on the coin by the disc. Given that the coin is on the point of moving,
  3. show that, correct to 2 significant figures, \(\mu = 0.23\).
Edexcel M3 Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cab238c9-f4e2-4637-a079-f74779548f49-3_447_506_205_657} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A stand used to reach high shelves in a storeroom is in the shape of a frustum of a cone. It is modelled as a uniform solid formed by removing a right circular cone of height \(2 h\) from a similar cone of height \(3 h\) and base radius \(3 r\) as shown in Figure 1.
  1. Show that the centre of mass of the stand is a distance of \(\frac { 33 } { 76 } h\) from its larger plane face.
    (7 marks)
    The stand is stored hanging in equilibrium from a point on the circumference of the larger plane face. Given that \(h = 2 r\),
  2. find, correct to the nearest degree, the acute angle which the plane faces of the stand make with the vertical.
    (4 marks)
Edexcel M3 Q5
5. A particle of mass 0.8 kg is moving along the positive \(x\)-axis at a speed of \(5 \mathrm {~ms} ^ { - 1 }\) away from the origin \(O\). When the particle is 2 metres from \(O\) it becomes subject to a single force directed towards \(O\). The magnitude of the force is \(\frac { k } { x ^ { 2 } } \mathrm {~N}\) when the particle is \(x\) metres from \(O\). Given that when the particle is 4 m from \(O\) its speed has been reduced to \(3 \mathrm {~ms} ^ { - 1 }\),
  1. show that \(k = \frac { 128 } { 5 }\),
  2. find the distance of the particle from \(O\) when it comes to instantaneous rest. (4 marks)
Edexcel M3 Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cab238c9-f4e2-4637-a079-f74779548f49-4_206_977_201_470} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows a particle \(P\) of mass \(m\) which lies on a smooth horizontal table. It is attached to a point \(A\) on the table by a light elastic spring of natural length \(3 a\) and modulus of elasticity \(\lambda\), and to a point \(B\) on the table by a light elastic spring of natural length \(2 a\) and modulus of elasticity \(2 \lambda\). The distance between the points \(A\) and \(B\) is \(7 a\).
  1. Show that in equilibrium \(A P = \frac { 9 } { 2 } a\). The particle is released from rest at a point \(Q\) where \(Q\) lies on the line \(A B\) and \(A Q = 5 a\).
  2. Prove that the subsequent motion of the particle is simple harmonic with a period of \(\pi \sqrt { \frac { 3 m a } { \lambda } }\).
    (9 marks)
Edexcel M3 Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cab238c9-f4e2-4637-a079-f74779548f49-4_300_952_1201_497} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 shows a vertical cross-section through part of a ski slope consisting of a horizontal section \(A B\) followed by a downhill section \(B C\). The point \(O\) is on the same horizontal level as \(C\) and \(B C\) is a circular arc of radius 30 m and centre \(O\), such that \(\angle B O C = 90 ^ { \circ }\). A skier of mass 60 kg is skiing at \(12 \mathrm {~ms} ^ { - 1 }\) along \(A B\).
  1. Assuming that friction and air resistance may be neglected, find the magnitude of the loss in reaction between the skier and the surface at \(B\).
    (4 marks)
    The skier subsequently leaves the slope at the point \(P\).
  2. Find, correct to 3 significant figures, the speed at which the skier leaves the slope.
  3. Find, correct to 3 significant figures, the speed of the skier immediately before hitting the ground again at the point \(D\) which is on the same horizontal level as \(C\).
Edexcel M3 Q1
  1. The mechanism for releasing the ball on a pinball machine contains a light elastic spring of natural length 15 cm and modulus of elasticity \(\lambda\).
The spring is held compressed to a length of 9 cm by a force of 4.5 N .
  1. Find \(\lambda\).
  2. Find the work done in compressing the spring from a length of 9 cm to a length of 5 cm .
    (4 marks)
Edexcel M3 Q2
2. A small bead \(P\) is threaded onto a smooth circular wire of radius 0.8 m and centre \(O\) which is fixed in a vertical plane. The bead is projected from the point vertically below \(O\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves in complete circles about \(O\).
  1. Suggest a suitable model for the bead.
  2. Given that the minimum speed of \(P\) is \(60 \%\) of its maximum speed, use the principle of conservation of energy to show that \(u = 7\).
    (6 marks)
Edexcel M3 Q3
3. At time \(t\) seconds the acceleration, \(a \mathrm {~ms} ^ { - 2 }\), of a particle is given by $$a = \frac { 4 } { ( 1 + t ) ^ { 3 } }$$ When \(t = 0\), the particle has velocity \(1 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and displacement 3 m from a fixed origin \(O\).
  1. Find an expression for the velocity of the particle in terms of \(t\).
  2. Show that when \(t = 3\) the particle is 10.5 m from \(O\).
Edexcel M3 Q4
4. A particle of mass 0.5 kg is moving on a straight line with simple harmonic motion. At time \(t = 0\) the particle is instantaneously at rest at the point \(A\). It next comes instantaneously to rest 3 seconds later at the point \(B\) where \(A B = 4 \mathrm {~m}\).
  1. For the motion of the particle write down
    1. the period,
    2. the amplitude.
  2. Find the maximum kinetic energy of the particle in terms of \(\pi\). The point \(C\) lies on \(A B\) at a distance of 1.2 m from \(B\).
  3. Find the time it takes the particle to travel directly from \(A\) to \(C\), giving your answer in seconds correct to 2 decimal places.
    (4 marks)
Edexcel M3 Q5
5. When a particle of mass \(M\) is at a distance of \(x\) metres from the centre of the moon, the gravitational force, \(F\) N, acting on it and directed towards the centre of the moon is given by $$F = \frac { \left( 4.90 \times 10 ^ { 12 } \right) M } { x ^ { 2 } }$$ A rocket is projected vertically into space from a point on the surface of the moon with initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that the radius of the moon is \(\left( 1.74 \times 10 ^ { 6 } \right) \mathrm { m }\),
  1. show that the speed of the rocket, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when it is \(x\) metres from the centre of the moon is given by $$v ^ { 2 } = u ^ { 2 } + \frac { a } { x } - b$$ where \(a\) and \(b\) are constants which should be found correct to 3 significant figures.
  2. Find, correct to 2 significant figures, the minimum value of \(u\) needed for the rocket to escape the moon's gravitational attraction.
Edexcel M3 Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8b7133ed-3748-46cb-99d2-570ee33c7393-4_526_620_196_598} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a bowl formed by removing from a solid hemisphere of radius \(\frac { 3 } { 2 } r\) a smaller hemisphere of radius \(r\) having the same axis of symmetry and the same plane face.
  1. Show that the centre of mass of the bowl is a distance of \(\frac { 195 } { 304 } r\) from its plane face.
    (7 marks)
    The bowl has mass \(M\) and is placed with its curved surface on a smooth horizontal plane. A stud of mass \(\frac { 1 } { 2 } M\) is attached to the outer rim of the bowl. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8b7133ed-3748-46cb-99d2-570ee33c7393-4_517_729_1318_539} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} When the bowl is in equilibrium its plane surface is inclined at an angle \(\alpha\) to the horizontal as shown in Figure 2.
  2. Find tan \(\alpha\).
    (6 marks)
Edexcel M3 Q7
7. A cyclist is travelling round a circular bend of radius 25 m on a track which is banked at an angle of \(35 ^ { \circ }\) to the horizontal. In a model of the situation, the cyclist and her bicycle are represented by a particle of mass 60 kg and air resistance and friction are ignored. Using this model and assuming that the cyclist is not slipping,
  1. find, correct to 3 significant figures, the speed at which she is travelling. In tests it is found that the cyclist must travel at a minimum speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to prevent the bicycle from slipping down the slope. A more refined model is now used with a coefficient of friction between the bicycle and the track of \(\mu\). Using this model,
  2. show that \(\mu = 0.227\), correct to 3 significant figures,
  3. find, correct to 2 significant figures, the maximum speed at which the cyclist can travel without slipping up the slope. END
Edexcel M3 Q1
  1. The velocity, \(\mathbf { v ~ c m ~ s } { } ^ { - 1 }\), at time \(t\) seconds, of a radio-controlled toy is modelled by the formula
$$\mathbf { v } = \mathrm { e } ^ { 2 t } \mathbf { i } + 2 t \mathbf { j } ,$$ where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors.
  1. Find the acceleration of the toy in terms of \(t\).
  2. Find, correct to 2 significant figures, the time at which the acceleration of the toy is parallel to the vector \(( 4 \mathbf { i } + \mathbf { j } )\).
  3. Explain why this model is unlikely to be realistic for large values of \(t\).
Edexcel M3 Q2
2. A particle \(P\) of mass 0.4 kg is moving in a straight line through a fixed point \(O\). At time \(t\) seconds after it passes through \(O\), the distance \(O P\) is \(x\) metres and the resultant force acting on \(P\) is of magnitude ( \(5 + 4 \mathrm { e } ^ { - x }\) ) N in the direction \(O P\). When \(x = 1 , P\) is at the point \(A\).
  1. Find, correct to 3 significant figures, the work done in moving \(P\) from \(O\) to \(A\). Given that \(P\) passes through \(O\) with speed \(2 \mathrm {~ms} ^ { - 1 }\),
  2. find, correct to 3 significant figures, the speed of \(P\) as it passes through \(A\).
Edexcel M3 Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00776cc0-0214-4029-8ef1-c1cba89f4b87-2_382_796_1640_479} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A popular racket game involves a tennis ball of mass 0.1 kg which is attached to one end of a light inextensible string. The other end of the string is attached to the top of a fixed rigid pole. A boy strikes the ball such that it moves in a horizontal circle with angular speed \(4 \mathrm { rad } \mathrm { s } ^ { - 1 }\) and the string makes an angle of \(60 ^ { \circ }\) with the downward vertical as shown in Figure 1.
  1. Find the tension in the string.
  2. Find the length of the string.
Edexcel M3 Q4
4. A particle moves with simple harmonic motion along a straight line. When the particle is 3 cm from its centre of motion it has a speed of \(8 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\) and an acceleration of magnitude \(12 \mathrm {~cm} \mathrm {~s} ^ { - 2 }\).
  1. Show that the period of the motion is \(\pi\) seconds.
  2. Find the amplitude of the motion.
  3. Hence, find the greatest speed of the particle.
Edexcel M3 Q5
5. A physics student is set the task of finding the mass of an object without using a set of scales. She decides to use a light elastic string of natural length 2 m and modulus of elasticity 280 N attached to two points \(A\) and \(B\) which are on the same horizontal level and 2.4 m apart. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00776cc0-0214-4029-8ef1-c1cba89f4b87-3_307_1072_993_438} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} She attaches the object to the midpoint of the string so that it hangs in equilibrium 0.35 m below \(A B\) as shown in Figure 2.
  1. Explain why it is reasonable to assume that the tensions in each half of the string are equal.
  2. Find the mass of the object.
  3. Find the elastic potential energy of the string when the object is suspended from it.
Edexcel M3 Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{00776cc0-0214-4029-8ef1-c1cba89f4b87-4_455_540_201_660} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} Figure 3 shows part of the curve \(y = x ^ { 2 } + 1\). The shaded region enclosed by the curve, the coordinate axes and the line \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
  1. Find the coordinates of the centre of mass of the solid obtained. The solid is suspended from a point on its larger circular rim and hangs in equilibrium.
  2. Find, correct to the nearest degree, the acute angle which the plane surfaces of the solid make with the vertical.
    (3 marks)
Edexcel M3 Q7
7. A particle of mass 0.5 kg is hanging vertically at one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a fixed point. The particle is given an initial horizontal speed of \(u \mathrm {~ms} ^ { - 1 }\).
  1. Show that the particle will perform complete circles if \(u \geq \sqrt { 3 g }\). Given that \(u = 5\),
  2. find, correct to the nearest degree, the angle through which the string turns before it becomes slack,
  3. find, correct to the nearest centimetre, the greatest height the particle reaches above its position when the string becomes slack.
Edexcel M3 Q1
  1. A particle \(P\) of mass 1.5 kg moves from rest at the origin such that at time \(t\) seconds it is subject to a single force of magnitude \(( 4 t + 3 ) \mathrm { N }\) in the direction of the positive \(x\)-axis.
    1. Find the magnitude of the impulse exerted by the force during the interval \(1 \leq t \leq 4\).
    Given that at time \(T\) seconds, \(P\) has a speed of \(22 \mathrm {~ms} ^ { - 1 }\),
  2. find the value of \(T\) correct to 3 significant figures.