Questions M3 (745 questions)

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Edexcel M3 2014 January Q1
  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
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
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
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
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
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
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
  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
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
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
  1. 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 .
Edexcel M3 2015 January Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3706a02d-95c6-4e7a-bf38-88b338d77892-09_270_919_267_557} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a uniform solid \(S\) formed by joining the plane faces of two solid right circular cones, of base radius \(r\), so that the centres of their bases coincide at \(O\). One cone, with vertex \(V\), has height \(4 r\) and the other cone has height \(k r\), where \(k > 4\)
  1. Find the distance of the centre of mass of \(S\) from \(O\).
    (4) The point \(A\) lies on the circumference of the common base of the cones. The solid is placed on a horizontal surface with VA in contact with the surface. Given that \(S\) rests in equilibrium,
  2. find the greatest possible value of \(k\). When \(S\) is suspended from \(A\) and hangs freely in equilibrium, \(O A\) makes an angle of \(12 ^ { \circ }\) with the downward vertical.
  3. Find the value of \(k\).
Edexcel M3 2015 January Q6
6. A smooth sphere, with centre \(O\) and radius \(a\), is fixed with its lowest point \(A\) on a horizontal floor. A particle \(P\) is placed on the surface of the sphere at the point \(B\), where \(B\) is vertically above \(A\). The particle is projected horizontally from \(B\) with speed \(\sqrt { \frac { a g } { 5 } }\) and moves along the surface of the sphere. When \(O P\) makes an angle \(\theta\) with the upward vertical, and \(P\) is still in contact with the sphere, the speed of \(P\) is \(v\).
  1. Show that \(v ^ { 2 } = \frac { a g } { 5 } ( 11 - 10 \cos \theta )\). The particle leaves the surface of the sphere at the point \(C\).
    Find
  2. the speed of \(P\) at \(C\) in terms of \(a\) and \(g\),
  3. the size of the angle between the floor and the direction of motion of \(P\) at the instant immediately before \(P\) hits the floor.
Edexcel M3 2015 January Q7
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 \(\lambda\). The other end of the string is attached to a fixed point \(A\) on a smooth plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The string lies along a line of greatest slope of the plane. The particle rests in equilibrium at the point \(B\), where \(B\) is lower than \(A\) and \(A B = \frac { 6 } { 5 } a\).
  1. Show that \(\lambda = \frac { 5 } { 2 } m g\). The particle is now pulled down a line of greatest slope to the point \(C\), where \(B C = \frac { 1 } { 5 } a\), and released from rest.
  2. Show that \(P\) moves with simple harmonic motion of period \(2 \pi \sqrt { \frac { 2 a } { 5 g } }\)
  3. Find, in terms of \(g\), the greatest magnitude of the acceleration of \(P\) while the string is taut. The midpoint of \(B C\) is \(D\) and the string becomes slack for the first time at the point \(E\).
  4. Find, in terms of \(a\) and \(g\), the time taken by \(P\) to travel directly from \(D\) to \(E\).
Edexcel M3 2016 January Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ffe0bc72-3136-48d9-9d5b-4a364d134070-02_503_524_121_712} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A hemispherical bowl of internal radius \(2 r\) is fixed with its circular rim horizontal. A particle \(P\) is moving in a horizontal circle of radius \(r\) on the smooth inner surface of the bowl, as shown in Figure 1. Particle \(P\) is moving with constant angular speed \(\omega\). Show that \(\omega = \sqrt { \frac { g \sqrt { 3 } } { 3 r } }\)
Edexcel M3 2016 January Q2
2. A particle \(P\) is moving in a straight line. At time \(t\) seconds, the distance of \(P\) from a fixed point \(O\) on the line is \(x\) metres and the acceleration of \(P\) is \(( 6 - 2 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\) in the direction of \(x\) increasing. When \(t = 0 , P\) is moving towards \(O\) with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  1. Find the velocity of \(P\) in terms of \(t\).
  2. Find the total distance travelled by \(P\) in the first 4 seconds.
Edexcel M3 2016 January Q3
3. A car of mass 800 kg is driven at constant speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) round a bend in a race track. Around the bend, the track is banked at \(20 ^ { \circ }\) to the horizontal and the path followed by the car can be modelled as a horizontal circle of radius 20 m . The car is modelled as a particle. The coefficient of friction between the car tyres and the track is 0.5 Given that the tyres do not slip sideways on the track, find the maximum value of \(v\).
Edexcel M3 2016 January Q4
4. Fixed points \(A\) and \(B\) are on a horizontal ceiling, where \(A B = 4 a\). A light elastic string has natural length \(3 a\) and modulus of elasticity \(\lambda\). One end of the string is attached to \(A\) and the other end is attached to \(B\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The particle hangs freely in equilibrium at the point \(C\), where \(C\) is at a distance \(\frac { 3 } { 2 } a\) vertically below the ceiling.
  1. Show that \(\lambda = \frac { 5 m g } { 4 }\)
    (5) The point \(D\) is the midpoint of \(A B\). The particle is now raised vertically upwards to \(D\), and released from rest.
  2. Find the speed of \(P\) as it passes through \(C\).
    \includegraphics[max width=\textwidth, alt={}]{ffe0bc72-3136-48d9-9d5b-4a364d134070-05_542_51_2026_1982}VIIV SIHI NI JIIIM IONOOVI4V SIHI NI JIIYM ION OO
Edexcel M3 2016 January Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ffe0bc72-3136-48d9-9d5b-4a364d134070-07_371_800_262_573} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(l\) and modulus of elasticity \(\lambda\). The other end of the string is attached to a fixed point \(A\) on a smooth plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\). The particle rests in equilibrium on the plane at the point \(B\) with the string lying along a line of greatest slope of the plane, as shown in Figure 2. Given that \(A B = \frac { 6 } { 5 } l\)
  1. show that \(\lambda = 3 \mathrm { mg }\) The particle is pulled down the line of greatest slope to the point \(C\), where \(B C = \frac { 1 } { 2 } l\), and released from rest.
  2. Show that, while the string remains taut, \(P\) moves with simple harmonic motion about centre \(B\).
  3. Find the greatest magnitude of the acceleration of \(P\) while the string remains taut. The point \(D\) is the midpoint of \(B C\). The time taken by \(P\) to move directly from \(D\) to the point where the string becomes slack for the first time is \(k \sqrt { \frac { l } { g } }\), where \(k\) is a constant.
  4. Find, to 2 significant figures, the value of \(k\).
Edexcel M3 2016 January Q6
6. (a) Use algebraic integration to show that the centre of mass of a uniform solid hemisphere of radius \(r\) is at a distance \(\frac { 3 } { 8 } r\) from the centre of its plane face.
[0pt] [You may assume that the volume of a sphere of radius \(r\) is \(\frac { 4 } { 3 } \pi r ^ { 3 }\) ]
(5) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ffe0bc72-3136-48d9-9d5b-4a364d134070-09_351_597_598_678} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A uniform solid hemisphere of mass \(m\) and radius \(r\) is joined to a uniform solid right circular cone to form a solid \(S\). The cone has mass \(M\), base radius \(r\) and height \(4 r\). The vertex of the cone is \(O\). The plane face of the cone coincides with the plane face of the hemisphere, as shown in Figure 3.
(b) Find the distance of the centre of mass of \(S\) from \(O\). The point \(A\) lies on the circumference of the base of the cone. The solid is placed on a horizontal table with \(O A\) in contact with the table. The solid remains in equilibrium in this position.
(c) Show that \(M \geqslant \frac { 1 } { 10 } m\)
Edexcel M3 2016 January Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ffe0bc72-3136-48d9-9d5b-4a364d134070-11_581_641_262_678} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A particle 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 a horizontal axis through \(O\). The particle is projected with speed \(u\) from a point \(A\), where \(O A\) makes an angle \(\alpha\) with the upward vertical through \(O\), as shown in Figure 4. The particle moves in complete vertical circles. Given that \(\cos \alpha = \frac { 4 } { 5 }\)
  1. show that \(u > \sqrt { \frac { 2 g l } { 5 } }\) As the rod rotates, the least tension in the rod is \(T\) and the greatest tension is \(4 T\).
  2. Show that \(u = \sqrt { \frac { 17 } { 5 } g l }\)
    \includegraphics[max width=\textwidth, alt={}]{ffe0bc72-3136-48d9-9d5b-4a364d134070-12_2639_1830_121_121}
Edexcel M3 2017 January Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{85d8fc7d-8863-419e-8eef-8751a6fb6315-02_483_702_255_612} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The shaded region \(R\) is bounded by the curve with equation \(y ^ { 2 } = 9 ( 4 - x )\), the positive \(x\)-axis and the positive \(y\)-axis, as shown in Figure 1. A uniform solid \(S\) is formed by rotating \(R\) through \(360 ^ { \circ }\) about the \(x\)-axis.
Use algebraic integration to find the \(x\) coordinate of the centre of mass of \(S\).
Edexcel M3 2017 January Q2
2. A particle \(P\) of mass 0.6 kg is moving along the positive \(x\)-axis in the positive direction. The only force acting on \(P\) acts in the direction of \(x\) increasing and has magnitude \(\left( 3 t + \frac { 1 } { 2 } \right) \mathrm { N }\), where \(t\) seconds is the time after \(P\) leaves the origin \(O\). When \(t = 0 , P\) is at rest at \(O\).
  1. Find an expression, in terms of \(t\), for the velocity of \(P\) at time \(t\) seconds. The particle passes through the point \(A\) with speed \(\frac { 10 } { 3 } \mathrm {~ms} ^ { - 1 }\).
  2. Find the distance \(O A\).
Edexcel M3 2017 January Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{85d8fc7d-8863-419e-8eef-8751a6fb6315-04_647_684_260_635} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A uniform right circular solid cylinder has radius \(4 a\) and height \(6 a\). A solid hemisphere of radius \(3 a\) is removed from the cylinder forming a solid \(S\). The upper plane face of the cylinder coincides with the plane face of the hemisphere. The centre of the upper plane face of the cylinder is \(O\) and this is also the centre of the plane face of the hemisphere, as shown in Figure 2. Find the distance from \(O\) to the centre of mass of \(S\).
(6)
Edexcel M3 2017 January Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{85d8fc7d-8863-419e-8eef-8751a6fb6315-05_654_515_267_712} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A light inextensible string has its ends attached to two fixed points \(A\) and \(B\). The point \(A\) is vertically above \(B\) and \(A B = 7 a\). A particle \(P\) of mass \(m\) is fixed to the string and moves with constant angular speed \(\omega\) in a horizontal circle of radius \(4 a\). The centre of the circle is \(C\), where \(C\) lies on \(A B\) and \(A C = 3 a\), as shown in Figure 3. Both parts of the string are taut.
  1. Show that the tension in \(A P\) is \(\frac { 5 } { 7 } m \left( 4 a \omega ^ { 2 } + g \right)\).
  2. Find the tension in \(B P\).
  3. Deduce that \(\omega \geqslant \sqrt { \frac { g } { k a } }\), stating the value of \(k\).