Questions M3 (745 questions)

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Edexcel M3 2008 January Q4
  1. A particle \(P\) of mass \(m\) lies on a smooth plane inclined at an angle \(30 ^ { \circ }\) to the horizontal. The particle is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(2 m g\). The other end of the string is attached to a fixed point \(O\) on the plane. The particle \(P\) is in equilibrium at the point \(A\) on the plane and the extension of the string is \(\frac { 1 } { 4 } a\). The particle \(P\) is now projected from \(A\) down a line of greatest slope of the plane with speed \(V\). It comes to instantaneous rest after moving a distance \(\frac { 1 } { 2 } a\).
By using the principle of conservation of energy,
  1. find \(V\) in terms of \(a\) and \(g\),
  2. find, in terms of \(a\) and \(g\), the speed of \(P\) when the string first becomes slack.
Edexcel M3 2008 January Q5
5. A car of mass \(m\) moves in a circular path of radius 75 m round a bend in a road. The maximum speed at which it can move without slipping sideways on the road is \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that this section of the road is horizontal,
  1. show that the coefficient of friction between the car and the road is 0.6 . The car comes to another bend in the road. The car's path now forms an arc of a horizontal circle of radius 44 m . The road is banked at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The coefficient of friction between the car and the road is again 0.6. The car moves at its maximum speed without slipping sideways.
  2. Find, as a multiple of \(m g\), the normal reaction between the car and road as the car moves round this bend.
  3. Find the speed of the car as it goes round this bend.
Edexcel M3 2008 January Q6
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{39c2d25a-a39b-4eb9-a17b-6e741ab5ae98-09_357_606_315_717}
\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 attached to a fixed point \(O\). At time \(t = 0 , P\) is projected vertically downwards with speed \(\sqrt { } \left( \frac { 5 } { 2 } g a \right)\) from a point \(A\) which is at the same level as \(O\) and a distance \(a\) from \(O\). When the string has turned through an angle \(\theta\) and the string is still taut, the speed of \(P\) is \(v\) and the tension in the string is \(T\), as shown in Figure 2.
  1. Show that \(v ^ { 2 } = \frac { g a } { 2 } ( 5 + 4 \sin \theta )\).
  2. Find \(T\) in terms of \(m , g\) and \(\theta\). The string becomes slack when \(\theta = \alpha\).
  3. Find the value of \(\alpha\). The particle is projected again from \(A\) with the same velocity as before. When \(P\) is at the same level as \(O\) for the first time after leaving \(A\), the string meets a small smooth peg \(B\) which has been fixed at a distance \(\frac { 1 } { 2 } a\) from \(O\). The particle now moves on an arc of a circle centre \(B\). Given that the particle reaches the point \(C\), which is \(\frac { 1 } { 2 } a\) vertically above the point \(B\), without the string going slack,
  4. find the tension in the string when \(P\) is at the point \(C\).
Edexcel M3 2008 January Q7
7. A particle \(P\) of mass 2 kg is attached to one end of a light elastic string, of natural length 1 m and modulus of elasticity 98 N . The other end of the string is attached to a fixed point \(A\). When \(P\) hangs freely below \(A\) in equilibrium, \(P\) is at the point \(E , 1.2 \mathrm {~m}\) below \(A\). The particle is now pulled down to a point \(B\) which is 0.4 m vertically below \(E\) and released from rest.
  1. Prove that, while the string is taut, \(P\) moves with simple harmonic motion about \(E\) with period \(\frac { 2 \pi } { 7 } \mathrm {~s}\).
  2. Find the greatest magnitude of the acceleration of \(P\) while the string is taut.
  3. Find the speed of \(P\) when the string first becomes slack.
  4. Find, to 3 significant figures, the time taken, from release, for \(P\) to return to \(B\) for the first time.
Edexcel M3 2009 January Q1
  1. A particle \(P\) of mass 3 kg is moving in a straight line. At time \(t\) seconds, \(0 \leqslant t \leqslant 4\), the only force acting on \(P\) is a resistance to motion of magnitude \(\left( 9 + \frac { 15 } { ( t + 1 ) ^ { 2 } } \right) \mathrm { N }\). At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(t = 4 , v = 0\).
Find the value of \(v\) when \(t = 0\).
Edexcel M3 2009 January Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8374fa0f-cb28-497f-8696-877d7d0762f1-03_467_622_242_635} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(3 m g\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held in equilibrium by a horizontal force of magnitude \(\frac { 4 } { 3 } m g\) applied to \(P\). This force acts in the vertical plane containing the string, as shown in Figure 1. Find (a) the tension in the string,
(b) the elastic energy stored in the string.
Edexcel M3 2009 January Q3
3. A rough disc rotates about its centre in a horizontal plane with constant angular speed 80 revolutions per minute. A particle \(P\) lies on the disc at a distance 8 cm from the centre of the disc. The coefficient of friction between \(P\) and the disc is \(\mu\). Given that \(P\) remains at rest relative to the disc, find the least possible value of \(\mu\).
Edexcel M3 2009 January Q4
4. A small shellfish is attached to a wall in a harbour. The rise and fall of the water level is modelled as simple harmonic motion and the shellfish as a particle. On a particular day the minimum depth of water occurs at 1000 hours and the next time that this minimum depth occurs is at 2230 hours. The shellfish is fixed in a position 5 m above the level of the minimum depth of the water and 11 m below the level of the maximum depth of the water. Find
  1. the speed, in metres per hour, at which the water level is rising when it reaches the shellfish,
  2. the earliest time after 1000 hours on this day at which the water reaches the shellfish.
Edexcel M3 2009 January Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8374fa0f-cb28-497f-8696-877d7d0762f1-07_311_716_249_612} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} One end \(A\) of a light elastic string, of natural length \(a\) and modulus of elasticity \(6 m g\), is fixed at a point on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. A small ball \(B\) of mass \(m\) is attached to the other end of the string. Initially \(B\) is held at rest with the string lying along a line of greatest slope of the plane, with \(B\) below \(A\) and \(A B = a\). The ball is released and comes to instantaneous rest at a point \(C\) on the plane, as shown in Figure 2. Find
  1. the length \(A C\),
  2. the greatest speed attained by \(B\) as it moves from its initial position to \(C\).
Edexcel M3 2009 January Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8374fa0f-cb28-497f-8696-877d7d0762f1-09_433_376_242_781} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The region \(R\) is bounded by part of the curve with equation \(y = 4 - x ^ { 2 }\), the positive \(x\)-axis and the positive \(y\)-axis, as shown in Figure 3. The unit of length on both axes is one metre. A uniform solid \(S\) is formed by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis.
  1. Show that the centre of mass of \(S\) is \(\frac { 5 } { 8 } \mathrm {~m}\) from \(O\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{8374fa0f-cb28-497f-8696-877d7d0762f1-09_702_584_1138_676} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 shows a cross section of a uniform solid \(P\) consisting of two components, a solid cylinder \(C\) and the solid \(S\). The cylinder \(C\) has radius 4 m and length \(l\) metres. One end of \(C\) coincides with the plane circular face of \(S\). The point \(A\) is on the circumference of the circular face common to \(C\) and \(S\). When the solid \(P\) is freely suspended from \(A\), the solid \(P\) hangs with its axis of symmetry horizontal.
  2. Find the value of \(l\).
Edexcel M3 2009 January Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8374fa0f-cb28-497f-8696-877d7d0762f1-11_671_1077_276_429} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A particle is projected from the highest point \(A\) on the outer surface of a fixed smooth sphere of radius \(a\) and centre \(O\). The lowest point \(B\) of the sphere is fixed to a horizontal plane. The particle is projected horizontally from \(A\) with speed \(\frac { 1 } { 2 } \sqrt { } ( g a )\). The particle leaves the surface of the sphere at the point \(C\), where \(\angle A O C = \theta\), and strikes the plane at the point \(P\), as shown in Figure 5.
  1. Show that \(\cos \theta = \frac { 3 } { 4 }\).
  2. Find the angle that the velocity of the particle makes with the horizontal as it reaches \(P\).
Edexcel M3 2010 January Q1
  1. A particle \(P\) of mass 0.5 kg is moving along the positive \(x\)-axis. At time \(t\) seconds, \(P\) is moving under the action of a single force of magnitude \([ 4 + \cos ( \pi t ) ] \mathrm { N }\), directed away from the origin. When \(t = 1\), the particle \(P\) is moving away from the origin with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Find the speed of \(P\) when \(t = 1.5\), giving your answer to 3 significant figures.
Edexcel M3 2010 January Q2
2. A particle \(P\) moves in a straight line with simple harmonic motion of period 2.4 s about a fixed origin \(O\). At time \(t\) seconds the speed of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\). When \(t = 0 , P\) is at \(O\). When \(t = 0.4 , v = 4\). Find
  1. the greatest speed of \(P\),
  2. the magnitude of the greatest acceleration of \(P\).
Edexcel M3 2010 January Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d831556d-fdf3-4639-9a89-6d3b372d3446-05_556_576_224_687} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A bowl \(B\) consists of a uniform solid hemisphere, of radius \(r\) and centre \(O\), from which is removed a solid hemisphere, of radius \(\frac { 2 } { 3 } r\) and centre \(O\), as shown in Figure 1.
  1. Show that the distance of the centre of mass of \(B\) from \(O\) is \(\frac { 65 } { 152 } r\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d831556d-fdf3-4639-9a89-6d3b372d3446-05_526_1014_1292_478} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} The bowl \(B\) has mass \(M\). A particle of mass \(k M\) is attached to a point \(P\) on the outer rim of \(B\). The system is placed with a point \(C\) on its outer curved surface in contact with a horizontal plane. The system is in equilibrium with \(P , O\) and \(C\) in the same vertical plane. The line \(O P\) makes an angle \(\theta\) with the horizontal as shown in Figure 2. Given that \(\tan \theta = \frac { 4 } { 5 }\),
  2. find the exact value of \(k\). January 2010
Edexcel M3 2010 January Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d831556d-fdf3-4639-9a89-6d3b372d3446-08_388_521_279_710} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A particle \(P\) of weight 40 N is attached to one end of a light elastic string of natural length 0.5 m . The other end of the string is attached to a fixed point \(O\). A horizontal force of magnitude 30 N is applied to \(P\), as shown in Figure 3. The particle \(P\) is in equilibrium and the elastic energy stored in the string is 10 J . Calculate the length \(O P\).
Edexcel M3 2010 January Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d831556d-fdf3-4639-9a89-6d3b372d3446-10_590_858_242_575} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} One end \(A\) of a light inextensible string of length \(3 a\) is attached to a fixed point. A particle of mass \(m\) is attached to the other end \(B\) of the string. The particle is held in equilibrium at a distance \(2 a\) below the horizontal through \(A\), with the string taut. The particle is then projected with speed \(\sqrt { } ( 2 a g )\), in the direction perpendicular to \(A B\), in the vertical plane containing \(A\) and \(B\), as shown in Figure 4. In the subsequent motion the string remains taut. When \(A B\) is at an angle \(\theta\) below the horizontal, the speed of the particle is \(v\) and the tension in the string is \(T\).
  1. Show that \(v ^ { 2 } = 2 \operatorname { ag } ( 3 \sin \theta - 1 )\).
  2. Find the range of values of \(T\).
Edexcel M3 2010 January Q6
6. A bend of a race track is modelled as an arc of a horizontal circle of radius 120 m . The track is not banked at the bend. The maximum speed at which a motorcycle can be ridden round the bend without slipping sideways is \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The motorcycle and its rider are modelled as a particle and air resistance is assumed to be negligible.
  1. Show that the coefficient of friction between the motorcycle and the track is \(\frac { 2 } { 3 }\). The bend is now reconstructed so that the track is banked at an angle \(\alpha\) to the horizontal. The maximum speed at which the motorcycle can now be ridden round the bend without slipping sideways is \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The radius of the bend and the coefficient of friction between the motorcycle and the track are unchanged.
  2. Find the value of \(\tan \alpha\).
Edexcel M3 2010 January Q7
7. A light elastic string has natural length \(a\) and modulus of elasticity \(\frac { 3 } { 2 } m g\). A particle \(P\) of mass \(m\) is attached to one end of the string. The other end of the string is attached to a fixed point \(A\). The particle is released from rest at \(A\) and falls vertically. When \(P\) has fallen a distance \(a + x\), where \(x > 0\), the speed of \(P\) is \(v\).
  1. Show that \(v ^ { 2 } = 2 g ( a + x ) - \frac { 3 g x ^ { 2 } } { 2 a }\).
  2. Find the greatest speed attained by \(P\) as it falls. After release, \(P\) next comes to instantaneous rest at a point \(D\).
  3. Find the magnitude of the acceleration of \(P\) at \(D\).
Edexcel M3 2011 January Q1
  1. A particle \(P\) moves on the positive \(x\)-axis. When the distance of \(P\) from the origin \(O\) is \(x\) metres, the acceleration of \(P\) is \(( 7 - 2 x ) \mathrm { m } \mathrm { s } ^ { - 2 }\), measured in the positive \(x\)-direction. When \(t = 0 , P\) is at \(O\) and is moving in the positive \(x\)-direction with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance of \(P\) from \(O\) when \(P\) first comes to instantaneous rest.
    (6)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c150d518-2da6-4f08-90c4-a831a31020f9-03_433_485_260_733} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A toy is formed by joining a uniform solid hemisphere, of radius \(r\) and mass \(4 m\), to a uniform right circular solid cone of mass \(k m\). The cone has vertex \(A\), base radius \(r\) and height \(2 r\). The plane face of the cone coincides with the plane face of the hemisphere. The centre of the plane face of the hemisphere is \(O\) and \(O B\) is a radius of its plane face as shown in Figure 1. The centre of mass of the toy is at \(O\).
  1. Find the value of \(k\). A metal stud of mass \(\lambda m\) is attached to the toy at \(A\). The toy is now suspended by a light string attached to \(B\) and hangs freely at rest. The angle between \(O B\) and the vertical is \(30 ^ { \circ }\).
  2. Find the value of \(\lambda\).
Edexcel M3 2011 January Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c150d518-2da6-4f08-90c4-a831a31020f9-05_613_793_278_571} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The region \(R\) is bounded by the curve with equation \(y = \mathrm { e } ^ { x }\), the line \(x = 1\), the line \(x = 2\) and the \(x\)-axis as shown in Figure 2. A uniform solid \(S\) is formed by rotating \(R\) through \(2 \pi\) about the \(x\)-axis.
  1. Show that the volume of \(S\) is \(\frac { 1 } { 2 } \pi \left( \mathrm { e } ^ { 4 } - \mathrm { e } ^ { 2 } \right)\).
  2. Find, to 3 significant figures, the \(x\)-coordinate of the centre of mass of \(S\).
Edexcel M3 2011 January Q4
  1. A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds its displacement, \(x\) metres, from the origin \(O\) is given by \(x = 5 \sin \left( \frac { 1 } { 3 } \pi t \right)\).
    1. Prove that \(P\) is moving with simple harmonic motion.
    2. Find the period and the amplitude of the motion.
    3. Find the maximum speed of \(P\).
    The points \(A\) and \(B\) on the positive \(x\)-axis are such that \(O A = 2 \mathrm {~m}\) and \(O B = 3 \mathrm {~m}\).
  2. Find the time taken by \(P\) to travel directly from \(A\) to \(B\).
Edexcel M3 2011 January Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c150d518-2da6-4f08-90c4-a831a31020f9-09_728_732_157_598} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A small ball \(P\) of mass \(m\) is attached to the ends of two light inextensible strings of length \(l\). The other ends of the strings are attached to fixed points \(A\) and \(B\), where \(A\) is vertically above \(B\). Both strings are taut and \(A P\) is perpendicular to \(B P\) as shown in Figure 3. The system rotates about the line \(A B\) with constant angular speed \(\omega\). The ball moves in a horizontal circle.
  1. Find, in terms of \(m , g , l\) and \(\omega\), the tension in \(A P\) and the tension in \(B P\).
  2. Show that \(\omega ^ { 2 } > \frac { g \sqrt { } 2 } { l }\).
Edexcel M3 2011 January Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c150d518-2da6-4f08-90c4-a831a31020f9-11_485_711_244_589} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A small ball of mass \(3 m\) is attached to the ends of two light elastic strings \(A P\) and \(B P\), each of natural length \(l\) and modulus of elasticity \(k m g\). The ends \(A\) and \(B\) of the strings are attached to fixed points on the same horizontal level, with \(A B = 2 l\). The mid-point of \(A B\) is \(C\). The ball hangs in equilibrium at a distance \(\frac { 3 } { 4 } l\) vertically below \(C\) as shown in Figure 4.
  1. Show that \(k = 10\) The ball is now pulled vertically downwards until it is at a distance \(\frac { 12 } { 5 } l\) below \(C\). The ball is released from rest.
  2. Find the speed of the ball as it reaches \(C\).
Edexcel M3 2011 January Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c150d518-2da6-4f08-90c4-a831a31020f9-13_414_522_233_712} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light rod of length \(l\). The other end of the rod is attached to a fixed point \(O\). The rod can turn freely in a vertical plane about \(O\). The particle is projected with speed \(u\) from a point \(A\), where \(O A\) makes an angle \(\alpha\) with the upward vertical through \(O\) and \(0 < \alpha < \frac { \pi } { 2 }\). When \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\) the speed of \(P\) is \(v\) as shown in Figure 5.
  1. Show that \(v ^ { 2 } = u ^ { 2 } + 2 g l ( \cos \alpha - \cos \theta )\). It is given that \(\cos \alpha = \frac { 3 } { 5 }\) and that \(P\) moves in a complete vertical circle.
  2. Show that \(u > 2 \sqrt { } \left( \frac { g l } { 5 } \right)\). As the rod rotates the least tension in the rod is \(T\) and the greatest tension is \(5 T\).
  3. Show that \(u ^ { 2 } = \frac { 33 } { 10 } \mathrm { gl }\).
Edexcel M3 2012 January Q1
  1. A particle of mass 0.8 kg is attached to one end of a light elastic string of natural length 0.6 m . The other end of the string is attached to a fixed point \(A\). The particle is released from rest at \(A\) and comes to instantaneous rest 1.1 m below \(A\).
Find the modulus of elasticity of the string.