Questions — Edexcel M3 (510 questions)

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Edexcel M3 2002 June Q1
6 marks Standard +0.3
A particle \(P\) moves in a straight line with simple harmonic motion about a fixed centre \(O\) with period 2 s. At time \(t\) seconds the speed of \(P\) is \(v\) m s\(^{-1}\). When \(t = 0\), \(v = 0\) and \(P\) is at a point \(A\) where \(OA = 0.25\) m. Find the smallest positive value of \(t\) for which \(AP = 0.375\) m. [6]
Edexcel M3 2002 June Q2
9 marks Standard +0.3
\includegraphics{figure_1} A metal ball \(B\) of mass \(m\) is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(A\). The ball \(B\) moves in a horizontal circle with centre \(O\) vertically below \(A\), as shown in Fig. 1. The string makes a constant angle \(\alpha°\) with the downward vertical and \(B\) moves with constant angular speed \(\sqrt{(2gk)}\), where \(k\) is a constant. The tension in the string is \(3mg\). By modelling \(B\) as a particle, find
  1. the value of \(\alpha\), [4]
  2. the length of the string. [5]
Edexcel M3 2002 June Q3
10 marks Standard +0.3
A particle \(P\) of mass 2.5 kg moves along the positive \(x\)-axis. It moves away from a fixed origin \(O\), under the action of a force directed away from \(O\). When \(OP = x\) metres the magnitude of the force is \(2e^{-0.1x}\) newtons and the speed of \(P\) is \(v\) m s\(^{-1}\). When \(x = 0\), \(v = 2\). Find
  1. \(v^2\) in terms of \(x\), [6]
  2. the value of \(x\) when \(v = 4\). [3]
  3. Give a reason why the speed of \(P\) does not exceed \(\sqrt{20}\) m s\(^{-1}\). [1]
Edexcel M3 2002 June Q4
10 marks Standard +0.3
A light elastic string \(AB\) of natural length 1.5 m has modulus of elasticity 20 N. The end \(A\) is fixed to a point on a smooth horizontal table. A small ball \(S\) of mass 0.2 kg is attached to the end \(B\). Initially \(S\) is at rest on the table with \(AB = 1.5\) m. The ball \(S\) is then projected horizontally directly away from \(A\) with a speed of 5 m s\(^{-1}\). By modelling \(S\) as a particle,
  1. find the speed of \(S\) when \(AS = 2\) m. [5]
When the speed of \(S\) is 1.5 m s\(^{-1}\), the string breaks.
  1. Find the tension in the string immediately before the string breaks. [5]
Edexcel M3 2002 June Q5
12 marks Standard +0.3
\includegraphics{figure_2} A model tree is made by joining a uniform solid cylinder to a uniform solid cone made of the same material. The centre \(O\) of the base of the cone is also the centre of one end of the cylinder, as shown in Fig. 2. The radius of the cylinder is \(r\) and the radius of the base of the cone is \(2r\). The height of the cone and the height of the cylinder are each \(h\). The centre of mass of the model is at the point \(G\).
  1. Show that \(OG = \frac{1}{14}h\). [8]
The model stands on a desk top with its plane face in contact with the desk top. The desk top is tilted until it makes an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \frac{r}{7h}\). The desk top is rough enough to prevent slipping and the model is about to topple.
  1. Find \(r\) in terms of \(h\). [4]
Edexcel M3 2002 June Q6
14 marks Standard +0.3
A light elastic string, of natural length \(4a\) and modulus of elasticity \(8mg\), has one end attached to a fixed point \(A\). A particle \(P\) of mass \(m\) is attached to the other end of the string and hangs in equilibrium at the point \(O\).
  1. Find the distance \(AO\). [2]
The particle is now pulled down to a point \(C\) vertically below \(O\), where \(OC = d\). It is released from rest. In the subsequent motion the string does not become slack.
  1. Show that \(P\) moves with simple harmonic motion of period \(\pi\sqrt{\frac{2a}{g}}\). [7]
The greatest speed of \(P\) during this motion is \(\frac{1}{2}\sqrt{(ga)}\).
  1. Find \(d\) in terms of \(a\). [3]
Instead of being pulled down a distance \(d\), the particle is pulled down a distance \(a\). Without further calculation,
  1. describe briefly the subsequent motion of \(P\). [2]
Edexcel M3 2002 June Q7
14 marks Standard +0.3
\includegraphics{figure_3} A particle of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(O\). The particle is hanging at the point \(A\), which is vertically below \(O\). It is projected horizontally with speed \(u\). When the particle is at the point \(P\), \(\angle AOP = \theta\), as shown in Fig. 3. The string oscillates through an angle \(\alpha\) on either side of \(OA\) where \(\cos \alpha = \frac{2}{3}\).
  1. Find \(u\) in terms of \(g\) and \(l\). [4]
When \(\angle AOP = \theta\), the tension in the string is \(T\).
  1. Show that \(T = \frac{mg}{3}(9\cos\theta - 4)\). [6]
  2. Find the range of values of \(T\). [4]
Edexcel M3 2003 June Q1
6 marks Standard +0.8
A particle \(P\) of mass \(m\) is held at a point \(A\) on a rough horizontal plane. The coefficient of friction between \(P\) and the plane is \(\frac{2}{3}\). The particle is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(4mg\). The other end of the string is attached to a fixed point \(O\) on the plane, where \(OA = \frac{3}{4}a\). The particle \(P\) is released from rest and comes to rest at a point \(B\), where \(OB < a\). Using the work-energy principle, or otherwise, calculate the distance \(AB\). [6]
Edexcel M3 2003 June Q2
6 marks Standard +0.3
A car moves round a bend which is banked at a constant angle of \(10°\) to the horizontal. When the car is travelling at a constant speed of \(18 \text{ m s}^{-1}\), there is no sideways frictional force on the car. The car is modelled as a particle moving in a horizontal circle of radius \(r\) metres. Calculate the value of \(r\). [6]
Edexcel M3 2003 June Q3
9 marks Standard +0.3
A toy car of mass \(0.2\) kg is travelling in a straight line on a horizontal floor. The car is modelled as a particle. At time \(t = 0\) the car passes through a fixed point \(O\). After \(t\) seconds the speed of the car is \(v \text{ m s}^{-1}\) and the car is at a point \(P\) with \(OP = x\) metres. The resultant force on the car is modelled as \(\frac{1}{5}x(4 - 3x)\) N in the direction \(OP\). The car comes to instantaneous rest when \(x = 6\). Find
  1. an expression for \(v^2\) in terms of \(x\), [7]
  2. the initial speed of the car. [2]
Edexcel M3 2003 June Q4
11 marks Standard +0.3
\includegraphics{figure_1} A particle \(P\) of mass \(m\) is attached to the ends of two light inextensible strings \(AP\) and \(BP\) each of length \(l\). The ends \(A\) and \(B\) are attached to fixed points, with \(A\) vertically above \(B\) and \(AB = \frac{3}{4}l\), as shown in Fig. 1. The particle \(P\) moves in a horizontal circle with constant angular speed \(\omega\). The centre of the circle is the mid-point of \(AB\) and both strings remain taut.
  1. Show that the tension \(AP\) is \(\frac{1}{6}m(3l\omega^2 + 4g)\). [7]
  2. Find, in terms of \(m\), \(l\), \(\omega\) and \(g\), an expression for the tension in \(BP\). [2]
  3. Deduce that \(\omega^2 \geq \frac{4g}{3l}\). [2]
Edexcel M3 2003 June Q5
13 marks Standard +0.3
A particle \(P\) of mass \(0.8\) kg is attached to one end \(A\) of a light elastic spring \(OA\), of natural length \(60\) cm and modulus of elasticity \(12\) N. The spring is placed on a smooth horizontal table and the end \(O\) is fixed. The particle \(P\) is pulled away from \(O\) to a point \(B\), where \(OB = 85\) cm, and is released from rest.
  1. Prove that the motion of \(P\) is simple harmonic with period \(\frac{2\pi}{5}\) s. [5]
  2. Find the greatest magnitude of the acceleration of \(P\) during the motion. [2]
Two seconds after being released from rest, \(P\) passes through the point \(C\).
  1. Find, to 2 significant figures, the speed of \(P\) as it passes through \(C\). [5]
  2. State the direction in which \(P\) is moving 2 s after being released. [1]
Edexcel M3 2003 June Q6
14 marks Challenging +1.2
\includegraphics{figure_2} A particle is at 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 \(u\), where \(u < \sqrt{ag}\). The particle leaves the sphere at the point \(C\), where \(OC\) makes an angle \(\theta\) with the upward vertical, as shown in Fig. 2.
  1. Find an expression for \(\cos \theta\) in terms of \(u\), \(g\) and \(a\). [7]
The particle strikes the plane with speed \(\sqrt{\frac{9ag}{2}}\).
  1. Find, to the nearest degree, the value of \(\theta\). [7]
Edexcel M3 2003 June Q7
16 marks Standard +0.3
\includegraphics{figure_3} The shaded region \(R\) is bounded by part of the curve with equation \(y = \frac{1}{4}(x - 2)^2\), the \(x\)-axis and the \(y\)-axis, as shown in Fig. 3. The unit of length on both axes is 1 cm. A uniform solid \(S\) is made by rotating \(R\) through \(360°\) about the \(x\)-axis. Using integration,
  1. calculate the volume of the solid \(S\), leaving your answer in terms of \(\pi\), [4]
  2. show that the centre of mass of \(S\) is \(\frac{4}{5}\) cm from its plane face. [7]
\includegraphics{figure_4} A tool is modelled as having two components, a solid uniform cylinder \(C\) and the solid \(S\). The diameter of \(C\) is 4 cm and the length of \(C\) is 8 cm. One end of \(C\) coincides with the plane face of \(S\). The components are made of different materials. The weight of \(C\) is \(10W\) newtons and the weight of \(S\) is \(2W\) newtons. The tool lies in equilibrium with its axis of symmetry horizontal on two smooth supports \(A\) and \(B\), which are at the ends of the cylinder, as shown in Fig. 4.
  1. Find the magnitude of the force of the support \(A\) on the tool. [5]
Edexcel M3 2006 June Q1
5 marks Standard +0.3
A uniform solid is formed by rotating the region enclosed between the curve with equation \(y = \sqrt{x}\), the \(x\)-axis and the line \(x = 4\), through one complete revolution about the \(x\)-axis. Find the distance of the centre of mass of the solid from the origin \(O\). [5]
Edexcel M3 2006 June Q2
10 marks Standard +0.3
A bowl consists of a uniform solid metal hemisphere, of radius \(a\) and centre \(O\), from which is removed the solid hemisphere of radius \(\frac{1}{4}a\) with the same centre \(O\).
  1. Show that the distance of the centre of mass of the bowl from \(O\) is \(\frac{45}{112}a\). [5]
The bowl is fixed with its plane face uppermost and horizontal. It is now filled with liquid. The mass of the bowl is \(M\) and the mass of the liquid is \(kM\), where \(k\) is a constant. Given that the distance of the centre of mass of the bowl and liquid together from \(O\) is \(\frac{17}{48}a\),
  1. Find the value of \(k\). [5]
Edexcel M3 2006 June Q3
11 marks Standard +0.3
A particle \(P\) of mass \(0.2\) kg oscillates with simple harmonic motion between the points \(A\) and \(B\), coming to rest at both points. The distance \(AB\) is \(0.2\) m, and \(P\) completes \(5\) oscillations every second.
  1. Find, to \(3\) significant figures, the maximum resultant force exerted on \(P\). [6]
When the particle is at \(A\), it is struck a blow in the direction \(BA\). The particle now oscillates with simple harmonic motion with the same frequency as previously but twice the amplitude.
  1. Find, to \(3\) significant figures, the speed of the particle immediately after it has been struck. [5]
Edexcel M3 2006 June Q4
11 marks Standard +0.3
\includegraphics{figure_1} A hollow cone, of base radius \(3a\) and height \(4a\), is fixed with its axis vertical and vertex \(V\) downwards, as shown in Figure 1. A particle moves in a horizontal circle with centre \(C\), on the smooth inner surface of the cone with constant angular speed \(\sqrt{\frac{8g}{9a}}\). Find the height of \(C\) above \(V\). [11]
Edexcel M3 2006 June Q5
12 marks Challenging +1.2
Two light elastic strings each have natural length \(0.75\) m and modulus of elasticity \(49\) N. A particle \(P\) of mass \(2\) kg is attached to one end of each string. The other ends of the strings are attached to fixed points \(A\) and \(B\), where \(AB\) is horizontal and \(AB = 1.5\) m. \includegraphics{figure_2} The particle is held at the mid-point of \(AB\). The particle is released from rest, as shown in Figure 2.
  1. Find the speed of \(P\) when it has fallen a distance of \(1\) m. [6]
Given instead that \(P\) hangs in equilibrium vertically below the mid-point of \(AB\), with \(\angle APB = 2\alpha\),
  1. show that \(\tan \alpha + 5 \sin \alpha = 5\). [6]
Edexcel M3 2006 June Q6
13 marks Standard +0.3
A particle moving in a straight line starts from rest at the point \(O\) at time \(t = 0\). At time \(t\) seconds, the velocity \(v\) m s\(^{-1}\) of the particle is given by $$v = 3t(t - 4), \quad 0 \leq t \leq 5,$$ $$v = 75t^{-1}, \quad 5 \leq t \leq 10.$$
  1. Sketch a velocity-time graph for the particle for \(0 \leq t \leq 10\). [3]
  2. Find the set of values of \(t\) for which the acceleration of the particle is positive. [2]
  3. Show that the total distance travelled by the particle in the interval \(0 \leq t \leq 5\) is \(39\) m. [3]
  4. Find, to \(3\) significant figures, the value of \(t\) at which the particle returns to \(O\). [5]
Edexcel M3 2006 June Q7
13 marks Challenging +1.2
One end of a light inextensible string of length \(l\) is attached to a particle \(P\) of mass \(m\). The other end is attached to a fixed point \(A\). The particle is hanging freely at rest with the string vertical when it is projected horizontally with speed \(\sqrt{\frac{5gl}{2}}\).
  1. Find the speed of \(P\) when the string is horizontal. [4]
When the string is horizontal it comes into contact with a small smooth fixed peg which is at the point \(B\), where \(AB\) is horizontal, and \(AB < l\). Given that the particle then describes a complete semicircle with centre \(B\),
  1. Find the least possible value of the length \(AB\). [9]
Edexcel M3 2007 June Q1
9 marks Standard +0.3
The rudder on a ship is modelled as a uniform plane lamina having the same shape as the region \(R\) which is enclosed between the curve with equation \(y = 2x - x^2\) and the \(x\)-axis.
  1. Show that the area of \(R\) is \(\frac{4}{3}\). [4]
  2. Find the coordinates of the centre of mass of the lamina. [5]
Edexcel M3 2007 June Q2
10 marks Standard +0.3
An open container \(C\) is modelled as a thin uniform hollow cylinder of radius \(h\) and height \(h\) with a base but no lid. The centre of the base is \(O\).
  1. Show that the distance of the centre of mass of \(C\) from \(O\) is \(\frac{1}{4}h\). [5]
The container is filled with uniform liquid. Given that the mass of the container is \(M\) and the mass of the liquid is \(M\),
  1. find the distance of the centre of mass of the filled container from \(O\). [5]
Edexcel M3 2007 June Q3
9 marks Standard +0.8
A spacecraft \(S\) of mass \(m\) moves in a straight line towards the centre of the earth. The earth is modelled as a fixed sphere of radius \(R\). When \(S\) is at a distance \(x\) from the centre of the earth, the force exerted by the earth on \(S\) is directed towards the centre of the earth and has magnitude \(\frac{k}{x^2}\), where \(k\) is a constant.
  1. Show that \(k = mgR^2\). [2]
Given that \(S\) starts from rest when its distance from the centre of the earth is \(2R\), and that air resistance can be ignored,
  1. find the speed of \(S\) as it crashes into the surface of the earth. [7]
Edexcel M3 2007 June Q4
9 marks Standard +0.3
A light inextensible string of length \(l\) has one end attached to a fixed point \(A\). The other end is attached to a particle \(P\) of mass \(m\). The particle moves with constant speed \(v\) in a horizontal circle with the string taut. The centre of the circle is vertically below \(A\) and the radius of the circle is \(r\). Show that $$gr^2 = v^2\sqrt{l^2 - r^2}.$$ [9]