Questions M3 (796 questions)

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Edexcel M3 Q3
9 marks Challenging +1.2
Two particles \(A\) and \(B\), of masses \(M\) kg and \(m\) kg respectively, are connected by a light inextensible string passing over a smooth fixed pulley. \(B\) is placed on a smooth horizontal table and \(A\) hangs freely, as shown. \(B\) is attached to a spring of natural length \(l\) m and modulus of elasticity \(\lambda\) N, whose other end is fixed to a vertical wall. \includegraphics{figure_3} The system starts to move from rest when the string is taut and the spring neither extended nor compressed. \(A\) does not reach the ground, nor does \(B\) reach the pulley, during the motion.
  1. Show that the maximum extension of the spring is \(\frac{2Mgl}{\lambda}\) m. [3 marks]
  2. If \(M = 3\), \(m = 1.5\) and \(\lambda = 35l\), find the speed of \(A\) when the extension in the spring is \(0.5\) m. [6 marks]
Edexcel M3 Q4
11 marks Challenging +1.8
A particle \(P\) of mass \(m\) kg moves along a straight line under the action of a force of magnitude \(\frac{km}{x^2}\) N, where \(k\) is a constant, directed towards a fixed point \(O\) on the line, where \(OP = x\) m. \(P\) starts from rest at \(A\), at a distance \(a\) m from \(O\). When \(OP = x\) m, the speed of \(P\) is \(v\) ms\(^{-1}\).
  1. Show that \(v = \sqrt{\frac{2k(a-x)}{ax}}\). [6 marks]
\(B\) is the point half-way between \(O\) and \(A\). When \(k = \frac{1}{2}\) and \(a = 1\), the time taken by \(P\) to travel from \(A\) to \(B\) is \(T\) seconds Assuming the result that, for \(0 \leq x \leq 1\), \(\int \sqrt{\frac{x}{1-x}} dx = \arcsin(\sqrt{x}) - \sqrt{x(1-x^2)} + \text{constant}\),
  1. find the value of \(T\). [5 marks]
Edexcel M3 Q5
13 marks Standard +0.8
A car moves round a circular racing track of radius 100 m, which is banked at an angle of 4° to the horizontal.
  1. Show that when its speed is 8.28 ms\(^{-1}\), there is no sideways force acting on the car. [4 marks]
  2. When the speed of the car is 12.5 ms\(^{-1}\), find the smallest value of the coefficient of friction between the car and the track which will prevent side-slip. [9 marks]
Edexcel M3 Q6
13 marks Standard +0.8
The diagram shows a particle \(P\) of mass \(m\) kg moving on the inner surface of a smooth fixed hemispherical bowl of radius \(r\) m which is fixed with its axis vertical. \(P\) moves at a constant speed in a horizontal circle, at a depth \(h\) m below the top of the bowl. \includegraphics{figure_6}
  1. Show that the force \(R\) exerted on \(P\) by the bowl has magnitude \(\frac{mgr}{h}\) N. [4 marks]
  2. Find, in terms of \(g\), \(h\) and \(r\), the constant speed of \(P\). [4 marks]
The bowl is now inverted and \(P\) moves on the smooth outer surface at a height \(h\) above the plane face under the action of a force of magnitude \(mg\) applied tangentially as shown. The reaction of the surface of the sphere on \(P\) now has magnitude \(S\) N. \includegraphics{figure_6b}
  1. Given that \(r = 2h\), prove that \(S < \frac{1}{6}R\). [5 marks]
Edexcel M3 Q7
15 marks Challenging +1.8
A particle \(P\) of mass \(m\) kg is fixed to one end of a light elastic string of modulus \(mg\) N and natural length \(l\) m. The other end of the string is attached to a fixed point \(O\) on a rough horizontal table. Initially \(P\) is at rest in limiting equilibrium on the table at the point \(X\) where \(OX = \frac{5l}{4}\) m.
  1. Find the coefficient of friction between \(P\) and the table. [2 marks]
\(P\) is now given a small displacement \(x\) m horizontally along \(OX\), away from \(O\). While \(P\) is in motion, the frictional resistance remains constant at its limiting value.
  1. Show that as long as the string remains taut, \(P\) performs simple harmonic motion with \(X\) as the centre. [4 marks]
If \(P\) is held at the point where the extension in the string is \(l\) m and then released,
  1. show that the string becomes slack after a time \(\left(\frac{\pi}{2} + \arcsin\left(\frac{1}{3}\right)\right)\sqrt{\frac{l}{g}}\) s. [5 marks]
  2. Determine the speed of \(P\) when it reaches \(O\). [4 marks]
Edexcel M3 Q1
6 marks Moderate -0.3
A cyclist travels on a banked track inclined at \(8°\) to the horizontal. He moves in a horizontal circle of radius 10 m at a constant speed of \(v\) ms\(^{-1}\). If there is no sideways frictional force on the cycle, calculate the value of \(v\). [6 marks]
Edexcel M3 Q2
9 marks Standard +0.3
The figure shows a particle \(P\), of mass 0·8 kg, attached to the ends of two light elastic strings. \(AP\) has natural length 20 cm and modulus of elasticity \(\lambda\) N. \(BP\) has natural length 20 cm and modulus of elasticity \(\mu\) N. \(A\) and \(B\) are fixed to points on the same horizontal level so that \(AB = 50\) cm. When \(P\) is suspended in equilibrium, \(AP = 30\) cm and \(BP = 40\) cm. Calculate the values of \(\lambda\) and \(\mu\). \includegraphics{figure_2} [9 marks]
Edexcel M3 Q3
10 marks Challenging +1.8
Suraiya, whose mass is \(m\) kg, takes a running jump into a swimming pool so that she begins to swim in a straight line with speed 0·2 ms\(^{-1}\). She continues to move in the same straight line, the only force acting on her being a resistance of magnitude \(mv^2 \sin \left(\frac{t}{100}\right)\) N, where \(v\) ms\(^{-1}\) is her speed at time \(t\) seconds after entering the pool and \(0 \leq t \leq 50\pi\).
  1. Find an expression for \(v\) in terms of \(t\). [7 marks]
  2. Calculate her greatest and least speeds during her motion. [3 marks]
Edexcel M3 Q4
12 marks Challenging +1.2
A uniform lamina is in the shape of the region enclosed by the coordinate axes and the curve with equation \(y = 1 + \cos x\), as shown. \includegraphics{figure_4}
  1. Show by integration that the centre of mass of the lamina is at a distance \(\frac{\pi^2 - 4}{2\pi}\) from the \(y\)-axis. [9 marks]
Given that the centre of mass is at a distance 0·75 units from the \(x\)-axis, and that \(P\) is the point \((0, 2)\) and \(O\) is the origin \((0, 0)\),
  1. find, to the nearest degree, the angle between the line \(OP\) and the vertical when the lamina is freely suspended from \(P\). [3 marks]
Edexcel M3 Q5
12 marks Standard +0.8
A particle \(P\), of mass 0·5 kg, rests on the surface of a rough horizontal table. The coefficient of friction between \(P\) and the table is 0·5. \(P\) is connected to a particle \(Q\), of mass 0·2 kg, by a light inextensible string passing through a small smooth hole at a point \(O\) on the table, such that the distance \(OQ\) is 0·4 m. \(Q\) moves in a horizontal circle while \(P\) remains in limiting equilibrium. \includegraphics{figure_5}
  1. Calculate the angle \(\theta\) which \(OQ\) makes with the vertical. [4 marks]
  2. Show that the speed of \(Q\) is 1·33 ms\(^{-1}\). [3 marks]
The motion is altered so that \(Q\) hangs at rest below \(O\) and \(P\) moves in a horizontal circle on the table with speed 0·84 ms\(^{-1}\), at a constant distance \(r\) m from \(O\) but tending to slip away from \(O\).
  1. Find the value of \(r\). [5 marks]
Edexcel M3 Q6
12 marks Standard +0.3
The figure shows a swing consisting of two identical vertical light springs attached symmetrically to a light horizontal cross-bar and supported from a strong fixed horizontal beam. When a mass of 24 kg is placed at the mid-point of the cross-bar, both springs extend by 30 cm to the position \(A\), as shown. \includegraphics{figure_6} Each spring has natural length \(l\) m and modulus of elasticity \(\lambda\) N.
  1. Show that \(\lambda = 392l\). [2 marks]
The 24 kg mass is left on the bar and the bar is then displaced downwards by a further 20 cm.
  1. Prove that the system comprising the bar and the mass now performs simple harmonic motion with the centre of oscillation at the level \(A\). [5 marks]
  2. Calculate the number of oscillations made per second in this motion. [3 marks]
  3. Find the maximum acceleration which the mass experiences during the motion. [2 marks]
OCR M3 2009 June Q1
6 marks Moderate -0.3
A smooth sphere of mass 0.3 kg bounces on a fixed horizontal surface. Immediately before the sphere bounces the components of its velocity horizontally and vertically downwards are \(4 \text{ m s}^{-1}\) and \(6 \text{ m s}^{-1}\) respectively. The speed of the sphere immediately after it bounces is \(5 \text{ m s}^{-1}\).
  1. Show that the vertical component of the velocity of the sphere immediately after impact is \(3 \text{ m s}^{-1}\), and hence find the coefficient of restitution between the surface and the sphere. [3]
  2. State the direction of the impulse on the sphere and find its magnitude. [3]
OCR M3 2009 June Q2
8 marks Standard +0.3
\includegraphics{figure_2} Two uniform rods, \(AB\) and \(BC\), are freely jointed to each other at \(B\), and \(C\) is freely jointed to a fixed point. The rods are in equilibrium in a vertical plane with \(A\) resting on a rough horizontal surface. This surface is \(1.5\) m below the level of \(B\) and the horizontal distance between \(A\) and \(B\) is \(3\) m (see diagram). The weight of \(AB\) is \(80\) N and the frictional force acting on \(AB\) at \(A\) is \(14\) N.
  1. Write down the horizontal component of the force acting on \(AB\) at \(B\) and show that the vertical component of this force is \(33\) N upwards. [4]
  2. Given that the force acting on \(BC\) at \(C\) has magnitude \(50\) N, find the weight of \(BC\). [4]
OCR M3 2009 June Q3
10 marks Standard +0.8
\includegraphics{figure_3} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses \(4\) kg and \(2\) kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision both spheres have speed \(3 \text{ m s}^{-1}\). The spheres are moving in opposite directions, each at \(60°\) to the line of centres (see diagram). After the collision \(A\) moves in a direction perpendicular to the line of centres.
  1. Show that the speed of \(B\) is unchanged as a result of the collision, and find the angle that the new direction of motion of \(B\) makes with the line of centres. [8]
  2. Find the coefficient of restitution between the spheres. [2]
OCR M3 2009 June Q4
11 marks Challenging +1.2
A motor-cycle, whose mass including the rider is \(120\) kg, is decelerating on a horizontal straight road. The motor-cycle passes a point \(A\) with speed \(40 \text{ m s}^{-1}\) and when it has travelled a distance of \(x\) m beyond \(A\) its speed is \(v \text{ m s}^{-1}\). The engine develops a constant power of \(8\) kW and resistances are modelled by a force of \(0.25v^2\) N opposing the motion.
  1. Show that \(\frac{480v^2}{v^3 - 32000} \frac{dv}{dx} = -1\). [5]
  2. Find the speed of the motor-cycle when it has travelled \(500\) m beyond \(A\). [6]
OCR M3 2009 June Q5
11 marks Challenging +1.2
\includegraphics{figure_5} Each of two identical strings has natural length \(1.5\) m and modulus of elasticity \(18\) N. One end of one of the strings is attached to \(A\) and one end of the other string is attached to \(B\), where \(A\) and \(B\) are fixed points which are \(3\) m apart and at the same horizontal level. \(M\) is the mid-point of \(AB\). A particle \(P\) of mass \(m\) kg is attached to the other end of each of the strings. \(P\) is held at rest at the point \(0.8\) m vertically above \(M\), and then released. The lowest point reached by \(P\) in the subsequent motion is \(2\) m below \(M\) (see diagram).
  1. Find the maximum tension in each of the strings during \(P\)'s motion. [3]
  2. By considering energy,
    1. show that the value of \(m\) is \(0.42\), correct to 2 significant figures, [5]
    2. find the speed of \(P\) at \(M\). [3]
OCR M3 2009 June Q6
13 marks Standard +0.3
\includegraphics{figure_6} A particle \(P\) of mass \(m\) kg is attached to one end of a light inextensible string of length \(L\) m. The other end of the string is attached to a fixed point \(O\). The particle is held at rest with the string taut and then released. \(P\) starts to move and in the subsequent motion the angular displacement of \(OP\), at time \(t\) s, is \(\theta\) radians from the downward vertical (see diagram). The initial value of \(\theta\) is \(0.05\).
  1. Show that \(\frac{d^2\theta}{dt^2} = -\frac{g}{L} \sin \theta\). [2]
  2. Hence show that the motion of \(P\) is approximately simple harmonic. [2]
  3. Given that the period of the approximate simple harmonic motion is \(\frac{4}{3}\pi\) s, find the value of \(L\). [2]
  4. Find the value of \(\theta\) when \(t = 0.7\) s, and the value of \(t\) when \(\theta\) next takes this value. [4]
  5. Find the speed of \(P\) when \(t = 0.7\) s. [3]
OCR M3 2009 June Q7
13 marks Standard +0.3
\includegraphics{figure_7} A hollow cylinder has internal radius \(a\). The cylinder is fixed with its axis horizontal. A particle \(P\) of mass \(m\) is at rest in contact with the smooth inner surface of the cylinder. \(P\) is given a horizontal velocity \(u\), in a vertical plane perpendicular to the axis of the cylinder, and begins to move in a vertical circle. While \(P\) remains in contact with the surface, \(OP\) makes an angle \(\theta\) with the downward vertical, where \(O\) is the centre of the circle. The speed of \(P\) is \(v\) and the magnitude of the force exerted on \(P\) by the surface is \(R\) (see diagram).
  1. Find \(v^2\) in terms of \(u\), \(a\), \(g\) and \(\theta\) and show that \(R = \frac{mu^2}{a} + mg(3\cos\theta - 2)\). [7]
  2. Given that \(P\) just reaches the highest point of the circle, find \(u^2\) in terms of \(a\) and \(g\), and show that in this case the least value of \(v^2\) is \(ag\). [4]
  3. Given instead that \(P\) oscillates between \(\theta = \pm\frac{1}{5}\pi\) radians, find \(u^2\) in terms of \(a\) and \(g\). [2]
OCR M3 2010 June Q1
6 marks Moderate -0.3
A small ball of mass \(0.8\) kg is moving with speed \(10.5\) m s\(^{-1}\) when it receives an impulse of magnitude \(4\) N s. The speed of the ball immediately afterwards is \(8.5\) m s\(^{-1}\). The angle between the directions of motion before and after the impulse acts is \(\alpha\). Using an impulse-momentum triangle, or otherwise, find \(\alpha\). [6]
OCR M3 2010 June Q2
7 marks Standard +0.3
\includegraphics{figure_2} Two uniform rods \(AB\) and \(BC\) are of equal length and each has weight \(100\) N. The rods are freely jointed to each other at \(B\), and \(A\) is freely jointed to a fixed point. The rods are in equilibrium in a vertical plane with \(AB\) horizontal and \(C\) resting on a rough horizontal surface. \(C\) is vertically below the mid-point of \(AB\) (see diagram).
  1. By taking moments about \(A\) for \(AB\), find the vertical component of the force on \(AB\) at \(B\). Hence find the vertical component of the contact force on \(BC\) at \(C\). [3]
  2. Calculate the magnitude of the frictional force on \(BC\) at \(C\) and state its direction. [4]
OCR M3 2010 June Q3
8 marks Standard +0.3
A uniform smooth sphere \(A\) moves on a smooth horizontal surface towards a smooth vertical wall. Immediately before the sphere hits the wall it has components of velocity parallel and perpendicular to the wall each of magnitude \(4\) m s\(^{-1}\). Immediately after hitting the wall the components have magnitudes \(u\) m s\(^{-1}\) and \(v\) m s\(^{-1}\), respectively (see Fig. 1). \includegraphics{figure_1}
  1. Given that the coefficient of restitution between the sphere and the wall is \(\frac{1}{4}\), state the values of \(u\) and \(v\). [2]
Shortly after hitting the wall the sphere \(A\) comes into contact with another uniform smooth sphere \(B\), which has the same mass and radius as \(A\). The sphere \(B\) is stationary and at the instant of contact the line of centres of the spheres is parallel to the wall (see Fig. 2). The contact between the spheres is perfectly elastic. \includegraphics{figure_2}
  1. Find, for each sphere, its speed and its direction of motion immediately after the contact. [6]
OCR M3 2010 June Q4
11 marks Standard +0.3
\(O\) is a fixed point on a horizontal plane. A particle \(P\) of mass \(0.25\) kg is released from rest at \(O\) and moves in a straight line on the plane. At time \(t\) s after release the only horizontal force acting on \(P\) has magnitude $$\frac{1}{2400}(144 - t^2) \text{ N} \quad \text{for } 0 \leqslant t \leqslant 12$$ and $$\frac{1}{2400}(t^2 - 144) \text{ N} \quad \text{for } t \geqslant 12.$$ The force acts in the direction of \(P\)'s motion. \(P\)'s velocity at time \(t\) s is \(v\) m s\(^{-1}\).
  1. Find an expression for \(v\) in terms of \(t\), valid for \(t \geqslant 12\), and hence show that \(v\) is three times greater when \(t = 24\) than it is when \(t = 12\). [8]
  2. Sketch the \((t, v)\) graph for \(0 \leqslant t \leqslant 24\). [3]
OCR M3 2010 June Q5
11 marks Standard +0.8
\includegraphics{figure_5} Particles \(P_1\) and \(P_2\) are each moving with simple harmonic motion along the same straight line. \(P_1\)'s motion has centre \(C_1\), period \(2\pi\) s and amplitude \(3\) m; \(P_2\)'s motion has centre \(C_2\), period \(\frac{4}{3}\pi\) s and amplitude \(4\) m. The points \(C_1\) and \(C_2\) are \(6.5\) m apart. The displacements of \(P_1\) and \(P_2\) from their centres of oscillation at time \(t\) s are denoted by \(x_1\) m and \(x_2\) m respectively. The diagram shows the positions of the particles at time \(t = 0\), when \(x_1 = 3\) and \(x_2 = 4\).
  1. State expressions for \(x_1\) and \(x_2\) in terms of \(t\), which are valid until the particles collide. [3]
The particles collide when \(t = 5.99\), correct to \(3\) significant figures.
  1. Find the distance travelled by \(P_2\) before the collision takes place. [4]
  2. Find the velocities of \(P_1\) and \(P_2\) immediately before the collision, and state whether the particles are travelling in the same direction or in opposite directions. [4]
OCR M3 2010 June Q6
12 marks Standard +0.8
A bungee jumper of weight \(W\) N is joined to a fixed point \(O\) by a light elastic rope of natural length \(20\) m and modulus of elasticity \(32\,000\) N. The jumper starts from rest at \(O\) and falls vertically. The jumper is modelled as a particle and air resistance is ignored.
  1. Given that the jumper just reaches a point \(25\) m below \(O\), find the value of \(W\). [5]
  2. Find the maximum speed reached by the jumper. [4]
  3. Find the maximum value of the deceleration of the jumper during the downward motion. [3]
OCR M3 2010 June Q7
17 marks Challenging +1.2
\includegraphics{figure_7} A particle \(P\) is attached to a fixed point \(O\) by a light inextensible string of length \(0.7\) m. A particle \(Q\) is in equilibrium suspended from \(O\) by an identical string. With the string \(OP\) taut and horizontal, \(P\) is projected vertically downwards with speed \(6\) m s\(^{-1}\) so that it strikes \(Q\) directly (see diagram). \(P\) is brought to rest by the collision and \(Q\) starts to move with speed \(4.9\) m s\(^{-1}\).
  1. Find the speed of \(P\) immediately before the collision. Hence find the coefficient of restitution between \(P\) and \(Q\). [3]
  2. Given that the speed of \(Q\) is \(v\) m s\(^{-1}\) when \(OQ\) makes an angle \(\theta\) with the downward vertical, find an expression for \(v^2\) in terms of \(\theta\), and show that the tension in the string \(OQ\) is \(14.7m(1 + 2\cos\theta)\) N, where \(m\) kg is the mass of \(Q\). [6]
  3. Find the radial and transverse components of the acceleration of \(Q\) at the instant that the string \(OQ\) becomes slack. [4]
  4. Show that \(V^2 = 0.8575\), where \(V\) m s\(^{-1}\) is the speed of \(Q\) when it reaches its greatest height (after the string \(OQ\) becomes slack). Hence find the greatest height reached by \(Q\) above its initial position. [4]